When Google App Engine (GAE) launched in preview for 20,000 developers in early 2008, few would have guessed that it would reach 100,000+ active developers per month and 200,000+ active apps per month in just a few years. Cloud computing was still on the rise in Gartner's Hype Cycle and hadn't quite reached the Peak of Inflated Expectations stage.
App Engine offered a fully-managed platform that allowed you to run your application on top of Google's infrastructure, handling the "deep-in-the-weeds" infrastructure work for you so you could instead focus on developing your application quickly. In exchange, you were limited, at least in the beginning, to developing your app using pre-defined and configured services. But once you accepted this minor trade-off, you benefited with a faster time to market for your critical application and required less time and money to deploy your app in the cloud. Additionally, GAE was the first serverless development model which meant that Google handled the thorny infrastructure operations and servers effortlessly and automatically scaled up and down as the workload required. It is a simply brilliant and efficient development model.
Since then, App Engine has quietly become the platform of choice for thousands of companies, from startups to large enterprises; including companies like Snapchat, Ubisoft, Motorola, Coca-Cola and Best Buy, who recognized the benefits and developed their applications on GAE.
Some industries may require that your application run in a cloud environment that meets certain regulatory or statutory requirements. HIPAA for health data and FedRAMP for federal government engagements are two or many examples. Additionally, data sovereignty laws may be involved depending on where your data resides in the cloud. All clouds are not created equal.
Although Google has datacenters around the globe, some regions, such as mainland China, may not be accessible. These geographic limitations restrict deployment options, impact revenue growth and introduces operational risk.
App Engine has a different billing model than your traditional occupancy based IaaS model, which has benefits but also some challenges depending on your application.
The App Engine pricing model charges you by the API call, which makes it very affordable when you're just starting to develop your application. App Engine also provides a free quota, so if you don't have many users in the beginning, you don't pay anything.
However, when your application starts to gain traction and scale, your Google bill can become very expensive very quickly. When usage increases over a certain point, an IaaS billing model with a fixed price per virtual machine might be the better option.
AppScale is an easy-to-manage serverless platform that enables you to quickly and easily build, deploy, and run your application, just like App Engine. How do we do this? Simply put, we're the open source version of App Engine, where we faithfully implement App Engine's APIs but allow you to deploy your application to any infrastructure. With AppScale, you can avoid all of the challenges mentioned above by running your app on-premises, in a Chinese cloud like Alibaba or Tencent, or on Azure, AWS, or even Google Compute Engine (GCE).
Besides providing you application portability with our multi cloud platform, you'll also see significant cost savings on your Google bill, ranging from 35% to 70%, depending on your situation. When leveraging AppScale, you still benefit from developing your application using the App Engine model but can also deploy your application on a virtual machine (VM) on the infrastructure of your choice. Instead of paying per transaction, you now pay per VM. Even switching from GAE to AppScale on GCE can save you significant expense.
To make it easy for you to compare the cost of running your GAE application in Google vs. deploying your app on AppScale in another public or private cloud, we've created the GAE Cost Calculator. For each cloud provider, we have chosen a commonly used number of VMs if you were to migrate your application using AppScale's serverless platform. The calculator only require 5 variables to calculate the cost of running on AppScale on AWS, GCE, Azure, IBM Cloud, or Alibaba Cloud.
Have your App Engine bill ready, and then visit the GAE Cost Calculator and try it out for free to see your savings. For detailed information on how to use the calculator, check out the step-by-step instructions.
Still have questions? We're always here to help, just contact us and we'll get back to you asap.


Will Romney's Secret Service protection hinder Mormon temple visits?
But expert says LDS rite would not impede protection operation.
Washington • Mitt Romney has gone to church several times since the Secret Service began guarding the Republican nominee, but he's yet to take in the more sacred Mormon rite: visiting an LDS temple.
As the first Mormon heading a major-party presidential ticket, Romney's newfound position with a Secret Service detail raises a question of how his around-the-clock protection would affect his ability to visit The Church of Jesus Christ of Latter-day Saints' most holy buildings.
While the general public can worship at an LDS meetinghouse, only church members in good standing are allowed inside temples, including one just outside Washington, D.C., in a Maryland suburb.
"There's no way the Secret Service is going to put up with that [restriction], with all due respect," says Ronald Kessler, author of "In the President's Secret Service: Behind the Scenes With Agents in the Line of Fire and the Presidents They Protect." "They will get permission and, of course, the church will understand and want him protected. They will go in."
Kessler says protectees, like presidents or presidential candidates, do get a say in the level of security they need, though they also understand the threats against them at all times and are smart enough to know it would be too risky to go it alone.
"It would be ludicrous if his Secret Service couldn't protect him in any area like that where there are so many people. He would be a perfect target," Kessler says. "Even Mormons can be criminals."
Romney's campaign referred questions about the candidate's security detail to the Secret Service, which also declined comment, noting that it doesn't speak publicly about its means and methods of securing its charges.
LDS Church spokesman Scott Trotter, likewise, said the church wouldn't comment on security matters.
Other Mormons have held positions where security is mandatory, most notably Senate Majority Leader Harry Reid, who has at least two Capitol Police officers with him at all times.
A Capitol Police spokeswoman, Sgt. Kimberly Schneider, declined to say how Reid's security is handled with regard to LDS temples.
"We don't discuss our security operations related to members of Congress," she said.
Reid does attend temple services on occasion, according to people close to the Nevada Democratic senator, but they were not clear if his security accompany him into the temple.
It wouldn't be an option for Romney with Secret Service protection, which, in the case of the president, stands close guard even in the secure compound of the White House.
But there might be other ways for Romney to attend services if he desired.
Bill Nixon, a Washington lobbyist and president of the Mount Vernon, Va., LDS stake, says the easy solution likely could be to send Romney in with temple-recommend-carrying agents.
"I imagine that there will be a very pragmatic solution to this," Nixon said, "not the least of which [could be that] some of many, many very sharp LDS members of the Secret Service could accompany on that specific kind of detail."


text:
Jobs 1 To 25 of 196 Page 1 of 8 Page [1]23456-10 Next
Soccer Sports Monitor- Sports Club New York
Job Description RESPONSIBILITIES - Monitor all activities on the court and maintain the sign-up list for 4 on 4 Soccer play. - Act as a referee and enforce both soccer rules and club policy. - Maintain the courts in a clean and orderly fashion by...
From EQUINOX HOLDINGS, INC.
EQUINOX HOLDINGS, INC.
Job Description RESPONSIBILITIES - Monitor all activities on the court and maintain the sign-up list for 4 on 4 Soccer play. - Act as a referee and enforce both soccer rules and club policy. - Maintain the courts in a clean and orderly fashion by picking up towels, cups, bottles and trash. - All other duties as assigned, including assisting with various leagues. - Demonstrate an approachable, service oriented, friendly, outgoing demeanor at all times, establish solid member relationships. Additional Information AS A MEMBER OF THE EQUINOX TEAM YOU WILL RECEIVE: - We offer competitive salary,...
Umpires, Referees, and Other Sports Officials
General Utility Worker - Citi Field - Suites
Overview Aramark (NYSE: ARMK) proudly serves the world's leading educational institutions, Fortune 500 companies, world champion sports teams, prominent healthcare providers, iconic destinations and cultural attractions, and numerous municipalities...
From Aramark Corporation
Aramark Corporation
Overview Aramark (NYSE: ARMK) proudly serves the world's leading educational institutions, Fortune 500 companies, world champion sports teams, prominent healthcare providers, iconic destinations and cultural attractions, and numerous municipalities in 19 countries around the world. Our 280,000 team members deliver innovative experiences and services in food, facilities management and uniforms to millions of people every day. We strive to create a better world by making a positive impact on people and the planet, including commitments to engage our employees; empower healthy consumers; build...
Maintenance and Repair Workers, General
Library Page - Library for the Performing Arts / Research Access Services
Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more that 100 years. As a Library Page at The New York Public...
From The New York Public Library
Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more that 100 years. As a Library Page at The New York Public Library you will do that and much more. A Page position is meant to give you knowledge about what it means to work within a library system. You will have access to a wealth of knowledge and experience which will allow you to decide if the library is where you want to begin your career. We are looking for candidates that demonstrate a strong work ethic,...
Library Assistants, Clerical
Library Page - Macomb's Bridge Library
Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more than 100 years. As a Library Page at The New York Public...
Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more than 100 years. As a Library Page at The New York Public Library you will do that and much more. A Page position is meant to give you knowledge about what it means to work within a library system. You will have access to a wealth of knowledge and experience which will allow you to decide if the library is where you want to begin your career. We are looking for candidates that demonstrate a strong work ethic,...
Library Page - Stephen A Schwarzman Building / Access Services
Curatorial Assistant (Part-Time/Temporary)
Overview The Music Division is one of the world's preeminent music collections--documenting the art of music in all its diversity, providing access to music materials for students, scholars, and artists from around the world . Reporting to the...
Overview The Music Division is one of the world's preeminent music collections--documenting the art of music in all its diversity, providing access to music materials for students, scholars, and artists from around the world . Reporting to the Curator of Music and Recorded Sound, the Curatorial Assistant will serve to assist the Curatorial team of the 2021 exhibition on Lou Reed. The position would ideally start ASAP and run through the opening of the exhibition in March 2021. Principal Responsibilities Maintaining an object list of all items to be included in the exhibition (as originals and...
Library Page - George Bruce Library
Library Page - Bloomingdale Library
Library Page - 67th Street Library
Job Description Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more than 100 years. As a Library Page at The New...
Job Description Overview Do you love books and love helping people? The New York Public Library has been an essential provider of free books, information, ideas and education for all New Yorkers for more than 100 years. As a Library Page at The New York Public Library you will do that and much more. A Page position is meant to give you knowledge about what it means to work within a library system. You will have access to a wealth of knowledge and experience which will allow you to decide if the library is where you want to begin your career. We are looking for candidates that demonstrate a...
Outreach Coordinator (Part-Time)
About the Opportunity The mission of our schools is to educate responsible citizen scholars for success in a college of their choice and a life of active citizenship. Core to achieving this mission are the people who work in our schools every day....
From Democracy Prep Public Schools
Democracy Prep Public Schools
About the Opportunity The mission of our schools is to educate responsible citizen scholars for success in a college of their choice and a life of active citizenship. Core to achieving this mission are the people who work in our schools every day. Our Outreach Team works to build candidate pipelines and ensures that we have the strongest talent pool possible for our school and network based roles. Who You Are: - An enthusiastic teammate with an unwavering commitment to the Democracy Prep mission of college success and authentic civic engagement for our scholars. - An organized, data-driven...
Social and Human Service Assistants
Trades Generalist
Under the supervision of the Regional Safety & Health Manager or other higher-level personnel, the Trades Generalist performs skilled journey level maintenance, repair, and inspection to gasoline, electric, and diesel-powered equipment and performs...
From New York State
Under the supervision of the Regional Safety & Health Manager or other higher-level personnel, the Trades Generalist performs skilled journey level maintenance, repair, and inspection to gasoline, electric, and diesel-powered equipment and performs necessary welding in the course of fabricating parts or repairing frames and other components. The duties and responsibilities include, but are not limited to, the following: 1. Diagnoses and determines causes of faulty operation of various motorized equipment and machinery 2. overhauls ignition and fuel systems including carburetors, starters,...
General Utility Worker - Citi Field - Concessions
Library Information Assistant (Part Time) - Stapleton...
Overview The New York Public Library (NYPL) has been an essential provider of free books, information, ideas, and education for all New Yorkers for more than 100 years. NYPL's neighborhood libraries in the Bronx, Manhattan, and Staten Island-many...
Overview The New York Public Library (NYPL) has been an essential provider of free books, information, ideas, and education for all New Yorkers for more than 100 years. NYPL's neighborhood libraries in the Bronx, Manhattan, and Staten Island-many of which date to Andrew Carnegie's visionary philanthropy at the turn of the 20th century-are being transformed into true centers of educational innovation and service, and vital community hubs that provide far more than just free books and materials. The Library is seeking a Library Information Assistant to provide service to patrons of all ages....
Infantryman - Management Training
Company: Army National Guard Contact: Post Date: 01/08/20 Phone: Job Type: Part Time Fax: Reference: NYoBcx2iC2 Mailing Address: AGE REQUIREMENTS: Must be between the ages of 17 and 35The Infantry is the backbone of the Army. These Soldiers fill...
From National Guard
Company: Army National Guard Contact: Post Date: 01/08/20 Phone: Job Type: Part Time Fax: Reference: NYoBcx2iC2 Mailing Address: AGE REQUIREMENTS: Must be between the ages of 17 and 35The Infantry is the backbone of the Army. These Soldiers fill the literal boots on the ground who are responsible for taking or holding ground during any combat operation. You ll attack, repel, and capture enemy ground forces using advanced weapons and tactics. Job Duties - Assist in reconnaissance operations - Employ, fire, and recover anti-personnel and anti-tank mines - Operate weapon systems under various...
Post Production Center Manager
Posting Number 2020-7082 Post Production Center Manager Location : Location US-NY-New York Department Film Center Production (Administrators) School/Division Tisch School of the Arts (TS1035) Compensation Grade Band 52 Union N/A FT/PT Full-Time...
From New York University (inc)
New York University (inc)
Posting Number 2020-7082 Post Production Center Manager Location : Location US-NY-New York Department Film Center Production (Administrators) School/Division Tisch School of the Arts (TS1035) Compensation Grade Band 52 Union N/A FT/PT Full-Time Category Marketing/Advertising/Public Relations/Media/Video/Audio Position Summary Manage day-to-day operations of Post Production area for for Tisch School of the Arts: develop and implement operational and creative policies and practices to ensure student productions run smoothly and successfully and that the safety and security of faculty, students,...
Executive Administrative Assistant - Global Commercial Services
description The GCS US Large Market Client Group manages strategic corporate payment relationships with clients, including many multi-national organizations and acquires new corporate payments customers with annual sales of $300M+. This position is...
From American Express Company
description The GCS US Large Market Client Group manages strategic corporate payment relationships with clients, including many multi-national organizations and acquires new corporate payments customers with annual sales of $300M+. This position is responsible for providing administrative support to 2 virtual VPs of Account Development, Large Market East in GCS. Job responsibilities include: Organizing and maintaining complex calendar using a high level of tact and integrity. Managing the logistics of key meetings, including room booking, catering, production of meeting materials, etc....
Conservation Recreation Facility Supervisor 1 (Seasonal) Multiple Positions
CRFS 1s are responsible for the entire operation of a small facility, including subordinate staff. Will function as the Caretakers and must conduct themselves in an ethical and professional manner as the face of the agency for visitors to our...
CRFS 1s are responsible for the entire operation of a small facility, including subordinate staff. Will function as the Caretakers and must conduct themselves in an ethical and professional manner as the face of the agency for visitors to our recreation facilities providing customer service that will enhance the public's stay. - Must be willing to work undefined hours, ensuring continued operation of your park, with a limited, if any, assigned amount of overtime. Must work all weekends and holidays through the summer, unless otherwise excused. Required to work through Labor Day weekend, as...
First-Line Supervisors of Mechanics, Installers, and Repairers
CRFS 3s are responsible for the entire operation of a large facility, including subordinate staff. They function as caretakers and are to conduct themselves in an ethical and professional manner as the face of the agency for visitors to our...
CRFS 3s are responsible for the entire operation of a large facility, including subordinate staff. They function as caretakers and are to conduct themselves in an ethical and professional manner as the face of the agency for visitors to our recreation facilities, providing customer service that will enhance the public's stay. - Must be willing to work undefined hours, ensuring continued operation of your park, with a limited, if any, assigned amount of overtime. Must work all weekends and holidays through the summer, unless otherwise excused. Required to work through Labor Day weekend, as...
Security Shift Supervisor New York City, New York $35.00 Security Shift Supervisor Location: New York City, New York | Salary: $35.00 | Posted: 13 Jan 2020 | Closes: 13 Feb 2020 | Job Type: Permanent & Full-time | Business Unit: US Secure Solutions...
From G4S
Security Shift Supervisor New York City, New York $35.00 Security Shift Supervisor Location: New York City, New York | Salary: $35.00 | Posted: 13 Jan 2020 | Closes: 13 Feb 2020 | Job Type: Permanent & Full-time | Business Unit: US Secure Solutions | Region / Division: North America | Reference: G4S/TP/2985559/181781 Apply now Job Introduction: PLEASE CLICK APPLY OR TEXT "JOBS" TO 561-660-9797 FROM YOUR SMARTPHONE! The world's leading private security organization, G4S, has an immediate job opportunity for an Shift Supervisor. As a Shift Supervisor, you will act as the G4S liaison with the...
First-Line Supervisors of Protective Service Workers, All Other
AQUATICS SPECIALIST
If you're both an animal lover and a people person, a position in one of our stores might be a great fit. We're looking for people with a strong commitment to helping pets and their parents experience their very best lives together. When you join...
From Petco
If you're both an animal lover and a people person, a position in one of our stores might be a great fit. We're looking for people with a strong commitment to helping pets and their parents experience their very best lives together. When you join us, you'll be part of a great team, working together to achieve sales goals while doing work you truly love. Position Purpose: To ensure that proper aquatic care and husbandry is being provided to our aquatic life in the store. Act as the primary resource in their assigned store regarding all aquatic life education and care. The position will also be...
Lifeguards, Ski Patrol, and Other Recreational Protective Service Workers
Asset Protection CoordLP
Job Description Summary: The Asset Protection Coordinator ("APC") is primarily responsible for preventing and deterring theft and limiting the loss of Company assets/products in the store(s) to which he or she is assigned, through provision of...
From CVS Health Corporation
CVS Health Corporation
Job Description Summary: The Asset Protection Coordinator ("APC") is primarily responsible for preventing and deterring theft and limiting the loss of Company assets/products in the store(s) to which he or she is assigned, through provision of attentive customer service and the effective implementation of other loss prevention programs and strategies. While the APC reports directly to a Store Manager ("SM"), he or she will also work closely with the Asset Protection and Analytics Department to understand and implement loss prevention programs and strategies with the assigned store(s). The APC...
DIRECT SUPPORT PROFESSIONAL - ALL SHIFTS AVAILABLE (OPWDD) (Full Time)
Provides close supervision for individuals in accordance with the individual treatment plan. If individual is on a 1:1 supervision status, provides continual supervision as noted in plan. Completes transfer form when transferring individual's care...
From Sco Family of Services
Sco Family of Services
Provides close supervision for individuals in accordance with the individual treatment plan. If individual is on a 1:1 supervision status, provides continual supervision as noted in plan. Completes transfer form when transferring individual's care to another staff. Participates in the individuals' program planning meetings; reviews and signs off on Annual and Semi-Annual Treatment Plans. Maintains individuals' possessions in a neat and orderly manner; communicates needs for the individuals to the management; completes documentation in accordance with program procedures. __ Transports...
Commercial Bank- Corporate Client Banking & Specialized Industries - Senior Administrative Assistant
Responsibilities: - Expense management - keep track of invoices and processing of expense claims and related record-keeping - Provide back up support for a senior executive, working closely with executive's assistant - Help manage the calendars of...
From JPMorgan Chase & Co.
Responsibilities: - Expense management - keep track of invoices and processing of expense claims and related record-keeping - Provide back up support for a senior executive, working closely with executive's assistant - Help manage the calendars of the team's Relationship Executives and work effectively with other administrators and meeting participants to efficiently and effectively schedule meetings. Adept at working around conflicts in a proactive and skilled manner and keeping manager aware of updates. - Coordinate staff onboarding and off boarding which includes requesting equipment setup...
Front Desk Monitor-RPT (ROS777)
Volunteers of America, Inc. (VOA, Inc.) is a national, faith-based human services organization that was founded in New York City in 1896 and has remained at the forefront of social service ever since, helping individuals and families in need...
From Volunteers of America-Greater New York
Volunteers of America-Greater New York
Volunteers of America, Inc. (VOA, Inc.) is a national, faith-based human services organization that was founded in New York City in 1896 and has remained at the forefront of social service ever since, helping individuals and families in need overcome obstacles to living safe, healthy and productive lives in the community. The largest of the affiliates, Volunteers of America-Greater New York (VOA-GNY) serves tens of thousands of men, women, and children in need every year, through 80 programs staffed by 1,300 skilled and dedicated staff, throughout New York City, Westchester County and...
Registered Poetry Therapist
Senior Executive Assistant
The Senior Executive Assistant provides support to another individual or group of individuals by handling correspondence, managing calendars and appointments, arranging conferences and conference calls, making travel arrangements and providing...
From Citigroup Inc.
Citigroup Inc.
The Senior Executive Assistant provides support to another individual or group of individuals by handling correspondence, managing calendars and appointments, arranging conferences and conference calls, making travel arrangements and providing other administrative tasks. Responsibilities: - Directly supports a C16 level manager and generally supports an entire department - Manages day to day calendar(s) of senior management - Coordinates all aspects of executive level manager appointments, meetings, receptions and conference calls - May prepare and review materials and correspondence for...
Miami University offers courses to help you get started or improve your Marketing/Marketing Management, General skills.
Entry Level Jobs (less than $30K) (28)
High Income Jobs ($80K-$99K) (17)
Administration & Support Services (24)
Transportation & Storage (12)
Real Estate & Rentals (9)
Social and Human Service Assistants (46)
Managers, All Other (25)
First-Line Supervisors of Protective Service Workers, All Other (24)
Executive Secretaries and Executive Administrative Assistants (20)
Library Assistants, Clerical (17)
Retail Loss Prevention Specialists (9)
Business Operations Specialists, All Other (8)
Maintenance and Repair Workers, General (8)
Compliance Managers (5)
Umpires, Referees, and Other Sports Officials (4)
High school diploma or equivalent (196)

redpajama_set_name: RedPajamaCommonCrawl


Choosing which Cloud will best serve your organisation is an important discussion. Recently, Claremont and Timico joined to offer a webinar on two often overlooked factors: licensing and support. Joining with Claremont, we had the opportunity to talk about this most important consideration of Cloud migration. Alongside Jonathan Stewart, Delivery Director at Claremont, we gave an honest reflection of the pros and cons of each Cloud offering.
You can find a full recording of the Webinar here.
There's been significant growth – in 2018 Cloud made up nearly 50% of IT spend, this is predicted to grow to 60-70% by 2020. Microsoft are reducing development of new features in the on-premise versions of their traditional server products, focusing instead on maximising the functionality of the Cloud based versions. With this level of investment and commitment, the Cloud is fast becoming a staple of IT proficiency. It is a vital requirement to keeping pace with modern business demands.
What is often called 'elastic computing' is an essential benefit of the Public Cloud for your business. This gives the ability to scale-up or scale-out workloads and servers very easily and quickly – this is beneficial as it allows you to respond to the demands of your users/customer requirements fast whilst minimising costs.
An example could be a retail application. As we approach Christmas, most retailers experience a significant spike in business – it's imperative there is the ability to scale out on demand to meet performance requirements, controlling your costs by making sure they scale back down once the demand peak falls off.
Efficiency is a benefit to every business. With the Public Cloud, the speed you can deploy services is greatly increased. We use scripted, prebuilt templates to deploy complex solutions in a rapid, repeatable way. We typically refer to this as "infrastructure as code".
An example is Azure Building Blocks, which simplifies scripted deployment of services by using sets of simple parameters to build out complex infrastructure without having to create large numbers of complex template scripts.
Serverless apps such as Azure Functions make for even faster deployment, as they don't need a server to run on. With serverless apps, a developer simply uploads their code as a function which can then be tested and deployed within minutes. A further benefit is that you only pay per execution, so you're not paying for an ongoing service 24/7. Also, zero administration is required, because there is no need to maintain an underlying server OS.
For any SME, providing a highly available infrastructure becomes costly quickly. Looking at Exchange as an example, to provide a highly available exchange environment on premise with the minimum configuration needed for high availability effectively doubles the cost of the exchange environment. Whereas if you're running in Exchange Online, high availability is built into service. Equally, with infrastructure services running in Azure, the ability to provide high availability is easy to provide across not just data centres but entire global regions if required.
Importantly, you pay for what you use. Making sure you only consume the resource you need is a key consideration of Cloud. By being alert to opportunities to reduce cost, Public Cloud allows you to align your infrastructure, so when you're not using a resource, it deprovisions and scales back to a minimum specification.


text:
\section{Introduction and History} \label{sec:intro}

Consider a stochastic process $X=(X_t)_{t\geq 0}$ starting in $x\in\mathbb{R}^d$. The \emph{probabilistic symbol} is the right-hand side derivative of the characteristic function of $(X_t-x)_{t\geq 0}$, that is,
\begin{align} \label{probsymbol1}
     q(x,\xi):=- \lim_{t\downarrow 0} \frac{\mathbb{E}^xe^{i(X_t-x)'\xi}-1}{t}.
\end{align}
The characteristic function carries all the information of the marginal distribution \emph{at time} $t$. Hence, it is not surprising, that the right-hand side derivative \eqref{probsymbol1} might be used to analyze the development of the process \emph{over time.}  From the point of view of Markov processes, Formula \eqref{probsymbol1} can be interpreted as follows: one plugs a (complex) exponential function into the generator of the process.
In this spirit the formula was used for the first time in 1998 by Jacob \cite{nielsursprung} in order to
calculate the functional analytic symbol of a Feller process satisfying some additional properties (see below).
Since then, this formula, related maximal inequalities and results on path properties have been generalized to further classes of processes.
Within the Markovian framework the most general class for which this symbol is known to exist is the class of L\'evy-type processes \cite{levymatters3} which are sometimes called It\^o processes in the sense of Çinlar, Jacod, Protter and Sharpe \cite{vierleute}.
A canonical class for a further generalization would be Hunt semimartingales. In \cite{withMartynas} we have shown that within this class, there exist processes which do not admit a symbol.
It has been an open question for over 10 years, whether or not L\'evy-type processes are the most general class for which the symbol exists and can be used. This is also point (d) in the open problems section of the monograph \cite{levymatters3}.  We answer this question in the present paper: we prove the existence of a Markov semimartingale, which is not a L\'evy-type process but which admits a symbol. This process gives rise to a class of processes of this kind.
Furthermore, we show that there is no Hunt process admitting a symbol which is not a L\'evy-type process (Theorem \ref{thm:main}). In this sense the above conjecture is true.

Summing up: Under very weak conditions every L\'evy-type process admits a symbol. There are no other Hunt semimartingales admitting a symbol. But if we move on another step and consider Markov semimartingales we again find processes for which the symbol does exist. However, in this case the symbol does not carry the valuable information on the process anymore.

Let us first fix some notation: a \emph{Markov process} $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t\geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0},\mathbb{P}^x)_{x\in\mathbb{R}^d}$ is defined in the sense of Blumenthal-Getoor (cf. \cite{blumenthalget}). This means in particular that the following formula holds
\begin{align} \label{universalmp}
  P_{s,t}^w(x,B) = P_{0,t-s}^{x}(x,B)=:P_{t-s}(x,B)
\end{align}
where $P_{s,t}^w(x,B)$ is the regular version of $\mathbb{P}^w(X_t\in B \, | \, X_s=x)$ with $w,x\in\mathbb{R}^d$, $s\leq t$ and $B$ is a Borel set in $\mathbb{R}^d$. Processes of this kind are sometimes called Markov families or universal Markov processes (cf. \cite{bauerwt}, \cite{niels3}) in the literature. We assume that every universal Markov process is normal, that is,  $\mathbb{P}^x(X_0=x)=1$.
We associate a semigroup $(T_t)_{t\geq 0}$ of operators on $B_b(\mathbb{R}^d)$ with every (universal) Markov process by setting
\[
    T_t u(x):= \mathbb{E}^x u(X_t), \quad t\geq 0,\; x\in \mathbb{R}^d.
\]
Let us denote by $C_\infty(\mathbb{R}^d)$ the space of all functions $u:\mathbb{R}^d\to\mathbb{R}$ which are continuous and vanish at
infinity. We call $(T_t)_{t\geq 0}$ a Feller
semigroup and $(X_t)_{t\geq 0}$ a \emph{Feller process}, if the semigroup is strongly continuous, that is, $\norm{T_tu-u}_\infty\to 0$ for $t\downarrow 0$, and if the following condition is satisfied:
\begin{align}
  T_t:C_\infty(\mathbb{R}^d) \to C_\infty(\mathbb{R}^d) \text{ for every }t\geq 0.
\end{align}

Without loss of generality, we assume all Feller processes we encounter to be c\`adl\`ag (cf. \cite{revuzyor} Theorem III.2.7). The \emph{generator} of the Feller process $(A,D(A))$ is the closed operator given by
\begin{gather}\label{generator}
    Au:=\lim_{t \downarrow 0} \frac{T_t u -u}{t} \qquad\text{for\ \ } u\in D(A)
\end{gather}
where the \emph{domain} $D(A)$ consists of all $u\in C_\infty(\mathbb{R}^d)$ for which the limit \eqref{generator} exists uniformly. A Feller process is called \emph{rich} if $C_c^\infty(\mathbb{R}^d)\subseteq D(A)$, that is, the test functions are contained in the domain of the generator.
By a classical result due to P.\ Courr\`ege \cite{courrege} it is known that the generator of a rich Feller process is (restricted to the test functions $C_c^\infty(\mathbb{R}^d)$) a \emph{pseudo differential operator}, i.e.,\ $A$ can be written as
\begin{gather} \label{pseudo}
    Au(x)= - \int_{\mathbb{R}^d} e^{ix'\xi} q(x,\xi) \widehat{u}(\xi) \,d\xi, \qquad u\in C_c^\infty(\mathbb{R}^d)
\end{gather}
where $\widehat{u}(\xi)=(2\pi)^{-d}\int e^{-iy'\xi}u(y) dy$ denotes the Fourier transform. The function $q:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{C}$ is locally bounded and, for fixed $x$, a continuous negative definite function in the sense of Schoenberg in the co-variable $\xi$ (cf. \cite{bergforst} Chapter II). This is equivalent to the fact that it admits a L\'evy-Khintchine representation
\begin{align} \label{lkfx}
  q(x,\xi)=
  -i \ell(x)'  \xi + \frac{1}{2} \xi'Q(x) \xi -\int_{w\neq 0} \left( e^{i \xi' w}
  -1 - i \xi' w \cdot \chi(w)\right)N(x,dw)
\end{align}
where $\ell(x)=(\ell^{(j)}(x))_{1\leq j \leq d} \in \mathbb{R}^d$, $Q(x)=(Q^{jk}(x))_{1\leq j,k \leq d}$ is a symmetric positive semidefinite matrix and $N(x,dw)$ is a measure on $\mathbb{R}^d\setminus\{0\}$ such that $\int_{w\neq 0} (1 \wedge \norm{w}^2) \,N(x,dw) < \infty$ and $\chi:\mathbb{R}^d\to\mathbb{R}$ is a cut-off function. The function $q:\mathbb{R}^d\times \mathbb{R}^d\to \mathbb{C}$ which is often written as $q(x,\xi)$ is called the \emph{symbol} of the operator. For details on the rich theory of the interplay between processes and their symbols we refer the reader to \cite{levymatters3}  and \cite{niels3}.

In \cite{nielsursprung} and for a more general class of Feller processes in \cite{schilling98pos} it was shown that the formula \eqref{probsymbol1} can be used to calculate the symbol $q(x,\xi)$ of the process.  For L\'evy processes this is the characteristic exponent.

A (universal) Markov process $X$ is called \emph{Markov semimartingale}, if $X$ is for every $\mathbb{P}^x$ a semimartingale. If such a process is in addition quasi-left-continuous (cf. \cite{jacodshir} Definition I.2.25), it is called \emph{Hunt semimartingale}. In \cite{mydiss} we have shown that every rich Feller process is a Hunt semimartingale and even a \emph{L\'evy-type process}, that is, a Markov semimartingale with characteristics of the following form
\begin{align} \begin{split} \label{chars}
  B^{(j)}_t(\omega)&=\int_0^t \ell^{(j)}(X_s(\omega)) \ ds \\
  C^{jk}_t(\omega)&=\int_0^t Q^{jk} (X_s(\omega)) \ ds \\
  \nu(\omega;ds,dy)&=N(X_s(\omega),dy)\ ds
\end{split}\end{align}
for every starting point $x\in\mathbb{R}^d$ with respect to a fixed cut-off function $\chi$. Here $\ell$, $Q$ and $N$ are as above. The triplet $(\ell,Q,n:=\int_{y\neq 0} (1\wedge \norm{y}^2) \ N(\cdot,dy)$) is called the \emph{differential characteristics} of the process. The class of semimartingales which are not necessarily Markovian, but having characteristics as described in \eqref{chars} are called \emph{homogeneous diffusions with jumps}  (cf. \cite{jacodshir} Definition III.2.18).

Since the symbol and some related indizes have proven to be useful in analyzing global and path properties, we have generalized the notion of the symbol using a generalization of formula \eqref{probsymbol1}, which can be used even for processes which do not admit a generator, namely homogeneous diffusions with jumps.

\begin{definition} \label{def:symbol}
Let $X$ be a stochastic process. Fix a starting point $x$ and define $\sigma=\sigma^x_k$ to be the first exit time from a compact ball with radius $k$ and $x$ as center
  \[
    \sigma:= \inf \{ t \geq 0 : \| X_t-x \| > k \}.
  \]
For $\xi \in \mathbb{R}^d$ we call $p:\mathbb{R}^d\times \mathbb{R}^d\to \mathbb{C}$ given by
\begin{align} \label{probsymbol2}
     p(x,\xi):=- \lim_{t\downarrow 0}\mathbb{E}^x \frac{e^{i(X^\sigma_t-x)'\xi}-1}{t}
\end{align}
the \emph{(probabilistic) symbol} of the process, if the limit exists and coincides for every choice of $k>0$.
\end{definition}

In \cite{generalizedindizes} we have shown that the symbol exists for homogeneous diffusions with jumps having differential characteristics which are locally bounded and finely continuous (cf. \cite{blumenthalget},  Theorem II.4.8). Like the functional analytic symbol in the Feller case it is a state-space dependent continuous negative definite function in the sense of Schoenberg. Instead of the compact balls, one can use general compact neighborhoods. In the definition, it is demanded that the limit exists for all $k>0$. If this is the case, it is easily seen that the limit is indeed the same for every $k>0$ and even for every compact neighborhood of $x$. The stopping time is needed in order to have the process bounded at least on $\llbracket 0, \sigma \llbracket$.
In order to calculate the symbol, \emph{fine continuity} (and local boundedness) are the most general requirements. However, since classical continuity is much more natural, we demand this property from now on. Continuity implies both: fine continuity and local boundedness.  Hence, Theorem 3.6 of \cite{generalizedindizes} yields that the symbol exists for every L\'evy-type process with continuous differential characteristics.

\section{Preliminaries from Advanced Calculus} \label{sec:calculus}

For the readers convenience we recall some concepts and results on advanced calculus on the real line (cf.  \cite{kankrue} and \cite{natanson}).

\begin{definition}\label{def:dinider}
Let $f:[a,b] \to \mathbb{R}$ be a function, and let $x_0 \in [a,b]$ for $a,b \in \mathbb{R}$ and $a<b$. Then
\begin{itemize}
\item[(a.)] the \emph{upper right Dini derivative} of $f$ at $x_0$ is defined by
$$
D^+f(t):= \limsup_{x \downarrow x_0} \frac{f(x)-f(x_0)}{x-x_0},
$$
\item[(b.)] the \emph{lower right Dini derivative} of $f$ at $x_0$ is defined by
$$
D_+f(t):= \liminf_{x \downarrow x_0} \frac{f(x)-f(x_0)}{x-x_0},
$$
\item[(c.)] the \emph{upper left Dini derivative} of $f$ at $x_0$ is defined by
$$
D^-f(t):= \limsup_{x \uparrow x_0} \frac{f(x)-f(x_0)}{x-x_0},
$$
\item[(d.)] the \emph{lower left Dini derivative} of $f$ at $x_0$ is defined by
$$
D_-f(t):= \liminf_{x \uparrow x_0} \frac{f(x)-f(x_0)}{x-x_0}.
$$
\end{itemize}
\end{definition}

\begin{definition}\label{def:derivednr}
The number $c$ (finite or infinite) is called a \emph{derived number} of the function $f$ at the point $x_0$ if there exists a sequence $h_n\to 0$ and
\[
\lim_{n\to \infty} \frac{f(x_0+h_n)-f(x_0)}{h_n}=c.
\]
\end{definition}

Subsequently, we write $\lambda$ for Lebesgue measure and $\lambda^*$ for the corresponding outer measure.  \\
In fact, all sets on which we will use the following results will be measurable.   The first result is  \cite{kankrue}  Lemma 1.2.3 while the second one is a combination of Lemma 1.2.3 and Lemma 1.2.5 of \cite{kankrue}.

\begin{lemma} \label{lem:lemma2}
Let $f$ be a strictly increasing function on $[a,b]$ and let $p\geq 0$. If at every point $x$ of a set $E\subseteq [a,b]$, there exists at least one derived number $Df(x)$ such that $Df(x)\leq p$ then
\[
\lambda^*(f(E))\leq p \cdot \lambda^*(E).
\]
\end{lemma}

\begin{lemma} \label{lem:lemma3}
Let $f$ be a strictly increasing function. If at every point $x$ of the set $E\subseteq [a,b]$, there exists $f'(x)=p$ then
\[
\lambda^*(f(E))=p\cdot \lambda^*(E).
\]
\end{lemma}

The concept of singular functions (Definition 7.1.44 \cite{kankrue}) is fundamental for the reasoning in the subsequent sections. More specifically, in order to prove the main theorem of this paper,  we use the fact that a singular function possesses at least one point where a Dini derivative is infinite.  Although, this seems to be clear since the most prominent examples like the Cantor function or Minkowski's question mark function on $[0,1]$ possess infinite derivatives in \textit{all} points where the derivative is not zero, it was shown in \cite{spanier} that there exists a singular function with a derivative that takes non-zero finite values on an uncountable dense set whose intersection with any interval $(a,b)$ possesses Hausdorff dimension one.

\begin{proposition} \label{prop:derived}
Let $g:\mathbb{R}_+ \to \mathbb{R}$ be an increasing, singular function which is not constantly zero. Then there exist at least one point where $g$ possesses an infinite upper right (resp. upper left, lower right, lower left) Dini derivative.
\end{proposition}

\begin{proof}
We prove the result only for the upper right Dini derivative $D^+$ since all other cases, i.e. , $D_+,D^-,D_-$, work analogously.\\
Let $t \in \mathbb{R}_+$ with $g(t) \neq 0$.
We consider $g$ on the interval $I:=[t-0.5, t+0.5 ]$. Let us assume that $g$ possesses a finite upper right Dini derivative in every point of $I$. For $x \in I$, we consider $f(x):=g(x)+x$ which is a strictly increasing, continuous function with $f'(x)=1$ almost everywhere, i.e., in a set of measure $1$. We define the disjoint sets
\begin{align*}
B&:= \{ x \in I : f'(x)=1\},\\
E_j &:= \{ x \in I : j \leq D^+f(x) < j+1\} \text{ for } j \in \mathbb{N},
\end{align*}
and obtain that $I = B \cup \left( \bigcup_{j=1}^\infty E_j \right)$.  Moreover, the sets $B$ and $(E_j)_{j \in \mathbb{N}}$ are measurable by Theorem 3.6.5 of \cite{kankrue}, and since $f$ is continuous and strictly increasing the sets $f(B),f(E_1),...$  form a disjoint decomposition of $f(I)$ into measurable sets. The sets $(E_j)_{j \in \mathbb{N}}$ do all have Lebesgue measure zero, hence, outer measure zero.  We obtain
\begin{align*}
\lambda(f(I)) &= \lambda(f(B  \cup E_1 \cup...)) \\
&=\lambda(f(B)  \cup f(E_1) \cup...) \\
&= \lambda(f(B))\\
&= \lambda (B)\\
&=1
\end{align*}
where we used Lemma \ref{lem:lemma2} in the third and Lemma \ref{lem:lemma3} in the fourth equation.  Thus, since $f$ maps intervals to intervals, we obtain that  $f(t-0.5)=c$ and $f(t+0.5)=c+1$ for a $c \in \mathbb{R}$,  and conclude that $g \equiv 0$ on $I$ by monotonicity. This is a contradiction.
\end{proof}

\section{The Deterministic Case} \label{sec:deterministic}

As we have pointed out above it was believed that for the symbol to exist in the Markovian framework, it was necessary to consider a L\'evy-type process. Now we construct a counter-example, that is, a Markov semimartingale, which is not L\'evy-type but having a symbol.
Most people would probably start with Brownian Motion, transforming and/or time-shifting it. Our process is different, since it does not even depend on random, i.e.,  it is a deterministic function. Rather one should say `a family of functions' satisfying the deterministic version of \eqref{universalmp}. We have dealt with processes of this kind in \cite{detmp2} and \cite{detmp1}. In the following we will work one-dimensional.

\begin{lemma}
Given a family of functions $f_x:[0,\infty[\to\mathbb{R}$, $x\in\mathbb{R}$, the corresponding stochastic process $X_t^x:=f_x(t)$ is a time-homogeneous universal Markov process, iff for every $s,t\geq 0$ and $x,y\in\mathbb{R}$ such that $f_x(s)=f_y(t)$ we obtain
\begin{align} \label{timehom}
  f_x(s+h)=f_y(t+h)
\end{align}
for $h\geq 0$.
\end{lemma}

For a deterministic process - which is allowed to start in every point $x\in\mathbb{R}$ - the symbol boils down to
\[
-\lim_{t\downarrow 0} \left(\frac{\cos((X^x_t-x)\xi)-1}{t}+ i\frac{\sin((X^x_t-x)\xi)}{t}\right).
\]
Whether we use the version \eqref{probsymbol1} or \eqref{probsymbol2} does not make any difference here. Furthermore, since (for $\xi\neq 0$) the function $y\mapsto \sin((y-x)\xi)$ is locally diffeomorph around zero, the symbol exists if and only if the process is right-differentiable at zero in every starting point $x$. In this case the cosine-part becomes zero.

\begin{example} \label{example:det1}
Writing as usual every $y\in\mathbb{R}$ as $y=[y]+\{y\}$, where $[\cdot]$ denotes the floor function and $\{y\}\in[0,1[$ we define
\begin{align}\label{loop}
X_t^x:= [x]+\{x+t\}.
\end{align}
Starting from $x$ the process drifts with constant speed upwards, but jumps to $[x]$ directly before reaching $[x]+1$. The symbol of this process exists and is equal to $-i\xi$. This means that from the point of view of the symbol, this process is a L\'evy process (!) or with other words: the symbol does not see the fixed time of discontinuity. \\

\begin{center}
\begin{tikzpicture}[x=.5cm, y=.5cm,domain=-9:9,smooth]

   \draw [color=gray!50]  [step=5mm] (0,0) grid (16,5);

   \draw[->,thick] (0,0) -- (16,0) node[below]{t};
   \draw[->,thick] (0,0) -- (0,5) node[above]{} ;

     \draw (2,-.1) -- (2,.1) node[below=4pt] {$\frac{1}{2}$};
     \draw (6,-.1) -- (6,.1) node[below=4pt] {$\frac{3}{2}$};
      \draw (10,-.1) -- (10,.1) node[below=4pt] {$\frac{5}{2}$};
      \draw (14,-.1) -- (14,.1) node[below=4pt] {$\frac{7}{2}$};

     \draw (-.1,2) -- (.1,2) node[left=4pt] {$\scriptstyle x=\frac{1}{2}$};
      \draw (-.1,4) -- (.1,4) node[left=4pt] {$\scriptstyle 1$};

   \node[below left]{$\scriptstyle0$};

\draw (0,2) -- (2,4);
\draw (2,0) -- (6,4);
\draw (6,0) -- (10,4);
\draw (10,0) -- (14,4);
\draw (14,0) -- (16,2);
\draw[fill=black] (2,0) circle(0.5mm) ;
\draw[fill=black] (6,0) circle(0.5mm) ;
\draw[fill=black] (10,0) circle(0.5mm) ;
\draw[fill=black] (14,0) circle(0.5mm) ;
\draw (2,4) circle(0.5mm) ;
\draw(6,4) circle(0.5mm) ;
\draw (10,4) circle(0.5mm) ;
\draw (14,4) circle(0.5mm) ;

\end{tikzpicture}
\end{center}
Admittedly, this process does (at first) not look Markovian, since in the definition \eqref{loop} the starting point $x$ appears. In fact this is just to simplify notation. Being at time $s$ at the point $y$, we know that at time $t>s$ we are at $[y]+\{y+(t-s)\}$, without any knowledge on the fact where the process has started at time zero.
\end{example}

Now we prove our main result in the deterministic setting.  To this end we will make use of a few concepts and results of \cite{detmp1} which we recall here for the readers convenience:
The main idea of that article was to encode the information on a deterministic Hunt process in so called generating paths: Let $\Phi:\mathbb{R}\to\mathbb{R}$ be bijective, strictly increasing, continuous and such that $\Phi(0)=0$. In this case a Hunt semimartingale is given by
\begin{equation} \label{genpath}
X_t^x(\omega):=\Phi(t+\Phi^{-1}(x)),\hspace{10mm} \text{ for every } \omega \in \Omega.
\end{equation}
The idea is to shift the function $\Phi$ to the left and to the right.
For processes of this kind the question whether the right-hand side derivative at zero exists for every starting point is equivalent to the existence of the right-hand side derivative of $\Phi$ in every point.

Obviously, not every deterministic Hunt process can be written in this way. However, in \cite{detmp1} we have shown that for every deterministic one-dimensional Hunt process we can find a decomposition of $\mathbb{R}$ into a countable number of disjoint intervals [one point intervals are allowed] such that on every interval the process is either constant or it can be characterized by a strictly monotone (increasing or decreasing) function $\Phi$ by \eqref{genpath}. In particular every deterministic one-dimensional Hunt process is a semimartingale. For a deterministic Hunt process the characteristics $C$ and $\nu$ are zero and $B_t^x=X_t^x-x$. Therefore, such a process is L\'evy-type iff $B$ (and hence $X$) is absolutely continuous w.r.t. Lebesgue measure.

\begin{theorem}
Let $X$ be a deterministic one-dimensional Hunt process. If the symbol of $X$ exists, that is, if the limit \eqref{probsymbol1} [or equivalently \eqref{probsymbol2}] is finite for every $\xi,x$, the process is a L\'evy-type process.
\end{theorem}

\begin{remark}
The theorem tells us that there is no (one-dimensional, deterministic) Hunt semimartingale which is not L\'evy-type, but which admits a symbol. Therefore, this is the desired border. However, if one leaves quasi-continuity behind, that is, one considers an even bigger class of processes, one finds again processes admitting a symbol (cf. the example above). In these cases the symbol looses its applicability (See Section \ref{sec:final}).
\end{remark}

\begin{proof}
At first we simplify the problem. We have recalled above that every one-dimensional deterministic Hunt process can be written using generating path on a countable number of disjoint intervals. For the details we refer to \cite{detmp1}. It is enough to consider one of these intervals on which the process is not constant. W.l.o.g. we can even consider a single strictly increasing monotone function $\Phi:[0,1]\to[0,1]$. First we separate this function into two monotone increasing functions $g$ and $h$ which are singular respectively absolutely continuous. A deterministic Hunt process is a L\'evy-type process, iff $\Phi$ is absolutely continuous. Therefore, it remains to show $g=0$. We do not know whether $g=\Phi-h$ has a finite right-derivative at every point [it could happen that there is a point where $g$ and $h$ are not right-differentiable, but the sum is.]
However,  since the derivative of $\Phi$ exists by assumption,  $D^+\Phi$ exists in every point. Since $g$ and $h$ are both strictly increasing we derive that $D^+g>0$ and $D^+h>0$.  Thus, we know that the upper right Dini derivative of $g$ is finite in every point.  By Proposition \ref{prop:derived} this is a contradiction and the statement follows.
\end{proof}

\section{The General Case} \label{sec:general}

A Hunt semimartingale is a Markov process $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t \geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0}, \mathbb{P}^x)_{x \in \mathbb{R}^d}$ which is also a $\mathbb{P}^x$-semimartingale and quasi-left-continuous.  Moreover, for the Markov filtration $(\mathcal{M}_t)_{t \geq 0}$ it holds true that $\mathcal{M}_t = \mathcal{M}_{t+}$ (see Definition 3.2 of \cite{vierleute}).\\
Within this framework, the following definition is useful:
\begin{definition}\label{def:addi}
Let $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t \geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0}, \mathbb{P}^x)_{x \in \mathbb{R}^d}$ be a Markov process. We call a process $Y=(Y_t)_{t\geq 0}$ additive if
\begin{itemize}
\item[(i.)] $Y_0=0$ a.s.
\item[(ii.)] for every $s,t \geq 0$ we have
$$
Y_{s+t} = Y_s+Y_t \circ \theta_s \text{  a.s.}
$$
\end{itemize}
In particular, we call $Y$ strongly additive if instead of $(ii.)$
\begin{itemize}
\item[(ii.')] for every $t \geq 0$ and all $(\mathcal{M}_t)$-stopping times $\rho$ we have
$$
Y_{\rho+t}=Y_{\rho}+ Y_t \circ \theta_\rho \text{  a.s.}
$$
\end{itemize}
holds true.
\end{definition}
\noindent
We denote by $\mathcal{V}$ (resp. $\mathcal{V}^+$) the set of all real-valued processes that are càdlàg, adapted and possess paths of finite variation (resp. increasing paths), and we denote by $\mathcal{V}_{ad}$ (resp. $\mathcal{V}^+_{ad}$) the set of all additive processes belonging to $\mathcal{V}$ (resp.  $\mathcal{V}^+$).

By Theorem 6.27 of \cite{vierleute} there exists for every Hunt semimartingale a continuous, strictly increasing, strongly additive $F \in \mathcal{V}$ (Although,  in 6.27 $F$ is stated to be additive only, by the remark below 6.14 , one can assume $F$ to be strongly additive), measurable functions $\ell: \mathbb{R}^d \to \mathbb{R}^d$, $Q:\mathbb{R}^d \to \mathbb{R}^{d \times d}$ and a positive kernel $N$ from $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$ into $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$ such that the characteristics of $X$ are of the form
\begin{align}
B^j_t&= \int_0^t \ell^{(j)}(X_s)  \ dF_s \label{b} \\
C^{jk}_t&= \int_0^t Q^{jk}(X_s)  \ dF_s  \label{c} \\
\nu(\omega; dt,dx) &= dF_t(\omega) N(X_t(\omega),dx).\label{K}
\end{align}

The proof of the following lemma is very similar to the one of Theorem 4.4 of \cite{mydiss}.  Of course it is possible enlarge the statement to the multidimensional case but we omit that since it is notationally more involved.
\begin{lemma}\label{Lemma 1}
Let $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t \geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0}, \mathbb{P}^x)_{x \in \mathbb{R}}$ be a Hunt semimartingale with characteristics stated in (\ref{b})-(\ref{K}), and let
$$
\sigma:= \inf \{ t \geq 0 : \| X_t-x \| > k \}
$$
for some starting point $x$ and $k \geq 0$.
Then the following equation holds true for $\xi \in \mathbb{R}$:
\begin{align*}
 \mathbb{E}^x \left( e^{i(X^\sigma_t-x)\xi } -1 \right)
 &=\mathbb{E}^x \left[ \int_{0}^t 1_{\llbracket 0,\sigma \llbracket}\left(i \xi e^{i(X^\sigma_{s-}-x)\xi}\ell(X_s)-  \frac{1}{2}\xi^2 e^{i(X^\sigma_{s-}-x)\xi} Q(X_s) \right.\right.\\
&+ \left. \left.\int_{ \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big) \ N(X_s(\omega),dy) \right)\ dF_s(\omega)\right].
\end{align*}
\end{lemma}
\begin{proof}
The left-continuous process $X^\sigma_{t-}$ is bounded, the stopped jumps $(\Delta X)^\sigma$ are the jumps of the stopped process $(\Delta X^\sigma)$ and $X^\sigma$ admits the stopped characteristics:
\begin{align*}
B^\sigma_t(\omega)& = \int_0^t \ell (X_s(\omega)) 1_{\llbracket 0, \sigma \rrbracket}(\omega, s) \ dF_s(\omega) \\
C_t^\sigma(\omega)&=\int_0^t Q(X_s(\omega)) 1_{\llbracket 0, \sigma \rrbracket}(\omega, s) \ dF_s(\omega) \\
\nu^\sigma(\omega;dF_s,dy)&:=1_{\llbracket 0, \sigma \rrbracket}(\omega, s) \ N(X_s(\omega),dy) \ dF_s(\omega).
\end{align*}
One can now set the integrand at the right endpoint of the stochastic support to zero, as we are integrating with respect to a continuous measure:
\begin{align*}
B^T_t(\omega)&=\int_0^t \ell (X_s(\omega)) 1_{\llbracket 0, \sigma \llbracket}(\omega, s) \ dF_s(\omega) \\
C_t^\sigma(\omega)&=\int_0^t Q(X_s(\omega)) 1_{\llbracket 0, \sigma \llbracket}(\omega, s) \ dF_s(\omega) \\
\nu^\sigma(\omega;dF_s,dy)&=1_{\llbracket 0, \sigma \llbracket}(\omega, s) \ N(X_s(\omega),dy) \ dF_s(\omega).
\end{align*}
Using It\^o's formula we obtain
\begin{align*}
 \mathbb{E}^x \left( e^{i(X^\sigma_t-x)'\xi } -1 \right)
&= \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dX^\sigma_s \right)\\
&+  \mathbb{E}^x\left(\frac{1}{2} \int_{0+}^t-\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} \ d[X^\sigma,X^\sigma]_s^c \right)\\
&+  \mathbb{E}^x\left(e^{-ix'\xi} \sum_{0<s\leq t} \Big(e^{i \xi'X_s^\sigma}-e^{i\xi'X^\sigma_{s-}}-i\xi e^{i\xi' X^\sigma_{s-}} \Delta X^\sigma_s \Big)\right)\\
&=  \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dX^\sigma_s \right)\\
&+  \mathbb{E}^x\left(\frac{1}{2} \int_{0+}^t-\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} \ dC^\sigma_s \right)\\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)'\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big)\ \mu^{X^\sigma}(\cdot;ds,dy)\right) \\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}} \Big(  e^{i (X_{s-}-x)'\xi}( -i\xi'y \cdot(1-\chi(y)))\Big) \ \mu^{X^\sigma}(\cdot;ds,dy) \right).
\end{align*}
Now, we want to consider the first summand from above. To this end,  we use the canonical decomposition of the semimartingale (see \cite{jacodshir} Theorem II.2.34):
$$
X_t=X_0 + \left(X^\sigma_t\right)^c + \int_0^t \chi(y)y \ \left(\mu^{X^\sigma}(\cdot \ ;ds,dy)-\nu^\sigma(\cdot \ ;ds,dy)\right) +\int_0^t y(1-\chi(y)) \ \mu^{X^\sigma}(\cdot \ ;ds,dy) +B^\sigma_t.
$$
Thus, the linearity of the integral provides
\begin{align*}
\mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dX^\sigma_s \right)&= \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dX^{\sigma,c}_s \right)\\
&+ \mathbb{E}^x\left( \int_0^t\int_{y\neq 0} i\xi \Big(e^{i(X_{s-}-x)'\xi}  \chi(y)y \Big) \mu^{X^\sigma}(\cdot;ds,dy)-\nu^\sigma(\cdot;ds,dy))\right) \\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  i \xi e^{i (X_{s-}-x)'\xi}y(1-\chi(y)) \ \mu^{X^\sigma}(\cdot;ds,dy)\right) \\
&+ \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dB^\sigma_s \right)\\
\end{align*}
First, we show that
\[
\mathbb{E}^x \left[\int_{0+}^t i \xi e^{i(X_{s-}-x)'\xi} \ d\left(X_s^c \right)^\sigma \right]= 0.
\]
The integral $e^{i(X_{t-}-x)'\xi}  \bullet X_t^c$ is a local martingale, since $X_t^c$ is a local martingale. To see that it is indeed a martingale, we calculate the following:
In the first two lines the integrand is now bounded, because $\ell$ and $Q$ are locally bounded and $\norm{X^{\sigma}_s(\omega)}<k$ on $\roi{0}{\sigma(\omega)}$ for every $\omega \in \Omega$.
For the martingale preservation in the first term we obtain
\begin{align*}
\left[e^{i(X^\sigma-x)'\xi} \bullet X^{\sigma,c}, e^{i(X^\sigma-x)'\xi} \bullet X^{\sigma,c} \right]_t
&= \left[e^{i(X^\sigma-x)'\xi} \bullet X^{c,\sigma}, e^{i(X^\sigma-x)'\xi} \bullet X^{c,\sigma} \right]_t  \\
&= \left[(e^{i(X^\sigma-x)'\xi} \bullet X^{c})^\sigma, (e^{i(X^\sigma-x)'\xi} \bullet X^{c})^\sigma \right]_t \\
&= \left[e^{i(X^\sigma-x)'\xi} \bullet X^{c}, e^{i(X^\sigma-x)'\xi} \bullet X^{c} \right]^\sigma_t \\
&= \left(\int_0^t (e^{i(X^\sigma_s-x)'\xi})^2  \  d[X^{c},X^{c}]_s\right)^\sigma \\
&= \int_0^t \left(e^{i(X^\sigma_s-x)'\xi}\right)^2 1_{\llbracket 0 , \sigma \llbracket}(s) \ d[X^{c},X^{c}]_s \\
&=\int_0^t \left(e^{i(X^\sigma_s-x)'\xi}\right)^2 1_{\llbracket 0 , \sigma \llbracket}(s) \ d\left(\int_0^s Q(X_{r}) \ dF_r \right) \\
&= \int_0^t \left(e^{i(X^\sigma_s-x)'\xi}\right)^2 1_{\llbracket 0 , \sigma \llbracket}(s)Q(X_s)  \ dF_s
\end{align*}
where we used several well known facts about the square bracket.  Hence,
$$
\mathbb{E}^x \left(\left[e^{i(X^\sigma-x)'\xi} \bullet X^{\sigma,c}, e^{i(X^\sigma_t-x)'\xi} \bullet X_t^{\sigma,c} \right]_t\right) < \infty, \quad t\geq 0,
$$
and Theorem 2.41 of \cite{mydiss} provides that $e^{i(X^\sigma_t-x)'\xi} \bullet X_t^{\sigma,c}$ is an $L^2$-martingale which is zero at zero and therefore, its expected value is constantly zero.  \\
Moreover,  one easily sees that $e^{i(X_{s-}-x)'\xi}y\chi(y)$ is in the class $F_p^2$ of Ikeda and Watanabe, and we conclude by Theorem 2.21 b) of \cite{mydiss} that
$$
\int_0^t\int_{y\neq 0} \Big(e^{i(X_{s-}-x)'\xi}  \chi(y)y \Big) \mu^{X^\sigma}(\cdot;ds,dy)-\nu^\sigma(\cdot;ds,dy))
$$
is also a martingale. Thus, in summation we obtain
\begin{align*}
 \mathbb{E}^x \left( e^{i(X^\sigma_t-x)'\xi } -1 \right)
&=  \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dX^\sigma_s \right)+  \mathbb{E}^x\left(\frac{1}{2} \int_{0+}^t-\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} \ dC^\sigma_s \right)\\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)'\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big)\ \mu^{X^\sigma}(\cdot;ds,dy)\right) \\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}} \Big(  e^{i (X_{s-}-x)'\xi}( -i\xi'y \cdot(1-\chi(y)))\Big) \ \mu^{X^\sigma}(\cdot;ds,dy) \right)\\
&=  \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  i \xi e^{i (X_{s-}-x)'\xi}y(1-\chi(y)) \ \mu^{X^\sigma}(\cdot;ds,dy)\right) \\
&+ \mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dB^\sigma_s \right)+  \mathbb{E}^x\left(\frac{1}{2} \int_{0+}^t-\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} \ dC^\sigma_s \right)\\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)'\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big)\ \mu^{X^\sigma}(\cdot;ds,dy)\right) \\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}} \Big(  e^{i (X_{s-}-x)'\xi}( -i\xi'y \cdot(1-\chi(y)))\Big) \ \mu^{X^\sigma}(\cdot;ds,dy) \right)\\
&=\mathbb{E}^x \left(\int_{0+}^t i \xi e^{i(X^\sigma_{s-}-x)'\xi} \ dB^\sigma_s \right) +  \mathbb{E}^x\left(\frac{1}{2} \int_{0+}^t-\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} \ dC^\sigma_s \right)\\
&+ \mathbb{E}^x\left( \int_{]0,t]\times \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)'\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big)\ \mu^{X^\sigma}(\cdot;ds,dy)\right)
\end{align*}
By the associativity of the stochastic integral and the properties of the compensator of a random measure the following equation holds true
\begin{align*}
 \mathbb{E}^x \left( e^{i(X^\sigma_t-x)'\xi } -1 \right)
 &=\mathbb{E}^x \left[ \int_{0}^t \left(i \xi e^{i(X^\sigma_{s-}-x)'\xi}\ell(X_s)1_{\llbracket 0,\sigma \llbracket}-  \frac{1}{2}\xi^2 e^{i(X^\sigma_{s-}-x)'\xi} Q(X_s)1_{\llbracket 0,\sigma \llbracket} \right.\right.\\
&+ \left. \left.\int_{ \{y\neq 0\}}  \Big(e^{i (X_{s-}-x)'\xi}(e^{i\xi'y}-1 -i \xi'y\chi(y))\Big)1_{\llbracket 0,\sigma \llbracket} \ N(X_s(\omega),dy) \right)\ dF_s(\omega)\right].
\end{align*}
\end{proof}

\begin{lemma}\label{Lemma 2}
Let $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t \geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0}, \mathbb{P}^x)_{x \in \mathbb{R}}$ be a Hunt-process. Let $(F_t)_{t \geq 0}$ be an increasing, continuous and strongly additive stochastic process adapted to $(\mathcal{F}_t)_{t\geq 0}$ where $\mathcal{F}_t:=\sigma(X_s: s\leq t))$. \\
Then
$$F_t= \int_0^t h(X_u) du +S_t$$
where $h$ is a positive $\mathcal{M}$-measurable function and $(S_t)_{t \geq 0}$ is an increasing, continuous and strongly additive, singular process adapted to $(\mathcal{F}_t)_{t \geq 0}$.
\end{lemma}
\begin{proof}
We consider the function $F_\cdot(\omega): \mathbb{R}_+ \to \mathbb{R}; t \mapsto F_t(\omega)$ which is continuous and increasing for almost all $\omega \in \Omega$. By Theorem 7.1.45 of \cite{kankrue} there exists an increasing, absolutely continuous function $A_\cdot(\omega)$ and a continuous, increasing, singular function $S_\cdot(\omega)$ such that
$$
F_t(\omega) =A_t(\omega) + S_t(\omega)
$$
for $t \geq 0$ and almost all $\omega \in \Omega$.\\
As defined in the proof of 7.1.45 of \cite{kankrue} it is $A_t(\omega)= \int_0^t F_u'(\omega) du$.  It holds true for $s,t \geq 0$ that
\begin{align*}
F_{s+t}'(\omega) &= \lim_{h \to 0} \frac{F_{s+t+h}(\omega)-F_{s+t}(\omega)}{h}\\
&= \lim_{h \to 0} \frac{F_{t+h}(\theta_s(\omega))-F_{t}(\theta_s(\omega))}{h}\\
&= F_t'(\theta_s(\omega)).
\end{align*}
In the second equality we have used the additivity of $F$.  With that, we are able to derive the additivity of $(A_t)_{t\geq 0}$ since
\begin{align*}
A_{s+t} &= \int_0^{s+t} F'_u \ du \\
&= \int_0^t F'_u \ du + \int_t^{s+t} F_u' \  du \\
&= A_t + \int_0^s F_u' \circ \theta_t \ du \\
&= A_t+A_s \circ \theta_t.
\end{align*}
Hence, $(A_t)_{t \geq 0} \in \mathcal{V}_{ad.}^+(\mathcal{F}_t)$,
and Proposition 3.56 of \cite{vierleute} provides the existence of a positive, $\mathcal{M}$-measurable function $h$ such that
$$
A_t= \int_0^t h(X_u) du.
$$
With this representation of $A$ one easily computes as above that $A$ is even strongly additive because $X$ is strongly Markovian. Hence, $(S_t)_{t \geq 0}$ with $S_t=F_t-A_t$ is strongly additive.
\end{proof}

\begin{theorem} \label{thm:main}
Let $(\Omega, \mathcal{M}, (\mathcal{M}_t)_{t \geq 0}, (X_t)_{t\geq 0}, (\theta_t)_{t \geq 0}, \mathbb{P}^x)_{x \in \mathbb{R}^d}$ be a Hunt semimartingale, i.e., a strongly Markovian, quasi-left-continuous $\mathbb{P}^x$-semimartingale.  Moreover, let the functions $\ell$ and $Q$ as mentioned in (\ref{b}) and (\ref{c})  be continuous, and let the function
$$
x \mapsto \int_{\{y \neq 0\}} (1\wedge y^2) \ N(x,dy)
$$ be continuous.\\
If the symbol of $X$, namely
\begin{equation}\label{1}
p(x,\xi) = - \lim_{t \downarrow 0} \mathbb{E}^x \left[ \frac{e^{i(X^\sigma_t-x)'\xi}-1}{t}\right]
\end{equation}
exists for all $x, \xi \in \mathbb{R}^d$, then $X$ is a Lévy-type process.
\end{theorem}

\begin{proof}
In this proof we only consider the one-dimensional case. The multivariate one works alike, but it is notationally more involved. \\
The stopping time $\sigma$ mentioned in (\ref{1}) is defined by
$$
\sigma:= \inf \{ t \geq 0 : \| X_t-x \| > k \}
$$
for some starting point $x$ and $k \in \mathbb{R}_+$.  As we have already pointed out in the introduction, the limit does not depend on the particular choice of $k$. Hence, we will be able to choose a particular $k$ later in this proof. \\
At first, let us consider the expression
$$
\mathbb{E}^x \left[ \frac{e^{i(X^\sigma_t-x)\xi}-1}{t}\right].
$$
Lemma $\ref{Lemma 1}$ provides the following equation:
\begin{align*}
\mathbb{E}^x \left[ e^{i(X^\sigma_t-x)\xi}-1\right]
=&\mathbb{E}^x \left[  \int_0^t i\xi e^{i(X_{s-}-x)\xi}\ell(X_s)1_{\llbracket 0, \sigma \llbracket}- \frac{1}{2}\xi^2 e^{i(X_{s-}-x)\xi} Q(X_s)1_{\llbracket 0, \sigma \llbracket}\right. \\
&\hspace{0.5cm} +\left. \int_{\{ y \neq 0 \}} e^{i(X_{s-}-x)\xi} 1_{\llbracket 0, \sigma \llbracket}\left(e^{i\xi y} -1 -i\xi y \chi(y)\right) \ N(X_s,dy)\ dF_s \right].
\end{align*}
So, when considering the symbol we obtain
\begin{align*}
p(x,\xi) &= - \lim_{t \downarrow 0} \frac{1}{t}  \mathbb{E}^x \left[ e^{i(X^\sigma_t-x)\xi}-1\right]\\
&= \lim_{t \downarrow 0} \mathbb{E}^x \left[  \int_0^t -i\xi e^{i(X_{s-}-x)\xi}\ell(X_s)1_{\llbracket 0, \sigma \llbracket}+ \frac{1}{2}\xi^2 e^{i(X_{s-}-x)\xi} Q(X_s)1_{\llbracket 0, \sigma \llbracket}\right. \\
&\hspace{0.5cm} -\left. \int_{\{ y \neq 0 \}} e^{i(X_{s-}-x)\xi} 1_{\llbracket 0, \sigma \llbracket}\left(e^{i\xi y} -1 -i\xi y \chi(y)\right) \ N(X_s,dy)\ dF_s \right].\\
\end{align*}
We define
$$
Y_s^{\xi,x}:= -i\xi e^{i(X_{s}-x)\xi}\ell(X_{s})+ \frac{1}{2}\xi^2 e^{i(X_{s}-x)\xi} Q(X_{s})- \int_{\{ y \neq 0 \}} e^{i(X_{s}-x)\xi} \left(e^{i\xi y} -1 -i\xi y \chi(y)\right) \ N(X_s,dy).
$$
Hence,  by the continuity of $F$ the existing limit in (\ref{1}) is given by
\begin{equation}\label{2}
p(x,\xi)= \lim_{t \downarrow 0} \mathbb{E}^x \left[ \int_0^t Y_s^{\xi,x} 1_{\llbracket 0, \sigma \llbracket} \ dF_s \right].
\end{equation}
Let us consider the process $(F_t)_{t \geq 0}$: by Lemma \ref{Lemma 2} there exists a decomposition of the form
$$
F_t=\int_0^t g(X_u) du +S_t, \quad t \geq 0
$$
where $g$ is a positive, measurable function and $(S_t)_{t \geq 0}$ is an increasing, continuous, strongly additive, singular process.
Thus, we decompose (\ref{2}) as follows:
\begin{align}
p(x,\xi) = \lim_{ t\downarrow 0}\left( \frac{1}{t}\mathbb{E}^x \left[ \int_0^t Y_u^{\xi,x}g(X_u)1_{\llbracket 0, \sigma \llbracket} \ du \right]+ \frac{1}{t}\mathbb{E}^x \left[ \int_0^t Y_u^{\xi,x}1_{\llbracket 0, \sigma \llbracket} \ dS_u \right]\right) \label{3}
\end{align}
For the first summand of (\ref{3}), Theorem 4.4 of \cite{mydiss} provides the existence of the limit, and since (\ref{3}) exists, the limit
\begin{equation*}
\lim_{ t\downarrow 0} \frac{1}{t}\mathbb{E}^x \left[ \int_0^t Y_u^{\xi,x}1_{\llbracket 0, \sigma \llbracket} \ dS_u \right]
\end{equation*}
exists for all $\xi, x \in \mathbb{R}$, and, therefore, the same is true for
\begin{equation}\label{4}
\lim_{ t\downarrow 0} \frac{1}{t}\mathbb{E}^x \left[ \int_0^t \Re \left(Y_u^{\xi,x}\right)1_{\llbracket 0, \sigma \llbracket} \ dS_u \right],
\end{equation}
and
\begin{equation}\label{4.2}
\lim_{ t\downarrow 0} \frac{1}{t}\mathbb{E}^x \left[ \int_0^t \Im \left(Y_u^{\xi,x}\right)1_{\llbracket 0, \sigma \llbracket} \ dS_u \right].
\end{equation}
First, we consider the case where there exists a $x_0 \in \mathbb{R}$ such that  $Q(x_0) \neq 0$ or $N(x_0, dy) \neq 0$:\\
the function $q^{\xi,x}$ defined by
\begin{align*}
z \mapsto  \Re \left( -i\xi e^{i(z-x)\xi}b(z)+\frac{1}{2}\xi^2 e^{i(z-x)\xi} c(z)-\int_\mathbb{R} e^{i(z-x)\xi} \left(e^{i\xi y} -1 -i\xi y \chi(y)\right) \ N(z,dy) \right)
\end{align*}
is continuous, and satisfies
$$
q^{\xi,x_0}(x_0)=\frac{1}{2}\xi^2 c(x_0)+\int_\mathbb{R}  1-\cos(y\xi)  \ N(x_0,dy) >0.
$$
The continuity provides the existence of  a $\delta>0$ such that $q^{\xi,x_0}(z) >\varepsilon$ for all $\| z- x_0 \| \leq \delta$ and an $\varepsilon >0$. As mentioned at the beginning of this proof we again consider the stopping time
$$
\sigma:= \inf \{ t \geq 0 : \| X_t-x_0 \| > k \},
$$
and we set $k = \delta$.
Thus,
\begin{align*}
q^{\xi,x_0}(X_t)1_{\llbracket 0, \sigma \llbracket}> 0 \quad \text{ a.s.}
\end{align*}
Now, we again consider the limit in (\ref{4}):
$$
\lim_{ t\downarrow 0} \frac{1}{t}\mathbb{E}^{x_0} \left[ \int_0^t  \Re \left(Y_u^{\xi,x_0}\right)1_{\llbracket 0, \sigma \llbracket} \ dS_u \right]=\lim_{ t\downarrow 0} \frac{1}{t}\mathbb{E}^{x_0} \left[ \int_0^t  \Re \left(q^{\xi,x_0}(X_u) \right) 1_{\llbracket 0, \sigma \llbracket}\ dS_u \right]
$$
which exists for all $\xi,x\in \mathbb{R}$. Moreover $(S_t)_{t \geq 0}$ is a path-wise singular function.
In order to establish the statement, we want to show that $(S_t)_{t \geq 0} \equiv 0$ (in this case $X$ is a Lévy-type process).  Thus, we assume that there exists a set $M \in \mathcal{M}$ with $\mathbb{P}(M) > 0$ such that $S_\cdot (\omega) \neq 0$ for all $\omega \in M$. \\
By Proposition \ref{prop:derived} for all $\omega \in M$ there exists a $t(\omega) \geq 0$ with
$$
D_+ S_{t(\omega)} = \infty.
$$
We define
$$
\tau := \inf\{ u \geq 0: D_+ S_u = \infty \} \quad \forall\omega \in \Omega
$$
which is an $( \mathcal{M}_t)_{t \geq 0}$-stopping time since $D_+S_u$ is $ \mathcal{M}_{u+}$-measurable and the filtration is right-continuous.
Thus,
\begin{equation}\label{5}
\lim_{t \downarrow 0} \frac{1}{t} \mathbb{E}^{X_{\tau(\omega)}(\omega)} \left[ \int_0^t \Re \left(Y_u^{\xi,x_0}\right)1_{\llbracket 0, \sigma \llbracket}  \ dS_u\right]
\end{equation}
exists for all $x,\xi \in \mathbb{R}$ and $\omega \in M$. \\
By applying Fatou's lemma for the conditional expectation the following inequality holds true
\begin{align*}
&\lim_{t \downarrow 0} \frac{1}{t} \mathbb{E}^{X_{\tau(\omega)}(\omega)} \left[ \int_0^t \Re \left(Y_u^{\xi,x_0}\right)1_{\llbracket 0, \sigma \llbracket}  \ dS_u\right] \\
=&\lim_{t \downarrow 0} \frac{1}{t} \mathbb{E}^{x_0} \left[ \int_0^t \Re \left(Y_u^{\xi,x_0}\circ \theta_\tau \right)1_{\llbracket 0, \sigma \llbracket}\circ \theta_\tau  \ dS_u\circ \theta_\tau \; \bigg| \; \mathcal{M}_\tau\right] \\
\geq &\mathbb{E}^{ x_0} \left[ \liminf_{t \downarrow 0} \frac{1}{t}\int_0^t \Re \left(Y_u^{\xi,x_0}\circ \theta_\tau \right)1_{\llbracket 0, \sigma \llbracket}\circ \theta_\tau  \ dS_u\circ \theta_\tau \; \bigg| \; \mathcal{M}_\tau\right] \\
\geq &\mathbb{E}^{ x_0} \left[ \liminf_{t \downarrow 0} \left( \inf_{u \in [0,t]} \Re \left(Y_u^{\xi,x_0}\circ \theta_\tau \right)1_{\llbracket 0, \sigma \llbracket}\circ \theta_\tau  \right) \ \frac{S_t\circ \theta_\tau}{t} \; \bigg| \; \mathcal{M}_\tau \right] \\
= &\mathbb{E}^{ x_0} \left[ \liminf_{t \downarrow 0} \left( \inf_{u \in [0,t]} \Re \left(Y_u^{\xi,x_0}\circ \theta_\tau \right)1_{\llbracket 0, \sigma \llbracket}\circ \theta_\tau  \right) \ \frac{S_{\tau + t} - S_\tau }{t} \; \bigg| \; \mathcal{M}_\tau \right]
\end{align*}
where we have used the strong additivity of $S$ in the last equation.\\
Since $\Re \left(Y_u^{\xi,x_0}\circ \theta_\tau \right)1_{\llbracket 0, \sigma \llbracket}\circ \theta_\tau  >\varepsilon$ and $(S_{\tau + t} - S_\tau)/t \to \infty $ for $t\downarrow 0$ by the definition of $\tau$ this is a contradiction.\\
In case that $c, N \equiv 0$ there exists  $x_0 \in \mathbb{R}$ with $b(x_0) \neq 0$. Otherwise the processes would be constantly zero. We set
\begin{align*}
q^{\xi,x}(z)&:= \Im \left( i\xi e^{i(z-x)\xi}b(z)- \frac{1}{2}\xi^2 e^{i(z-x)\xi} c(z)-\int_\mathbb{R} e^{i(z-x)\xi} \left(e^{i\xi y} -1 -i\xi y \chi(y)\right) \ N(z,dy) \right)\\
&= \xi \cos \left((z-x)\xi\right)b(z)
\end{align*}
which is continuous, and satisfies
$$
q^{\xi_0,x_0}(x_0)=\xi_0 b(x_0) >0 \quad a.s.
$$
for $\xi_0 = \sign \left(  b(x_0) \right)$.
The rest of the proof is analogous to the case above but using of (\ref{4.2}) instead of (\ref{4}).
\end{proof}

\section{Generalizations Beyond Hunt Processes} \label{sec:final}

We have seen above that under some mild technical assumptions within the Hunt framework the following holds: the process admits a probabilistic symbol iff it is a L\'evy-type process.  However, in Section \ref{sec:deterministic} we have constructed a Markov semimartingale, which is not L\'evy-type but which admits a symbol.  In this short section we show, that the symbol does not contain the same information on the process as before. We focus on one particular property of the paths, but the same holds true for various other properties of the process.

\begin{example} \label{example:det2}

Let us for simplicity consider again a deterministic Markov process as in Example \ref{example:det1}.  Starting at time zero in zero, the paths looks at follows:
\begin{center}
\begin{tikzpicture}[x=.5cm, y=.5cm,domain=-9:9,smooth]

   \draw [color=gray!50]  [step=5mm] (0,0) grid (8,10);

   \draw[->,thick] (0,0) -- (8,0) node[below]{t};
   \draw[->,thick] (0,0) -- (0,10) node[above]{} ;

     \draw (2,-.1) -- (2,.1) node[below=4pt] {$\scriptstyle 1$};
      \draw (4,-.1) -- (4,.1) node[below=4pt] {$\scriptstyle 2$};
     \draw (6,-.1) -- (6,.1) node[below=4pt] {$\scriptstyle 3$};

   \foreach \c in {1,4,9}{
     \draw (-.1,\c) -- (.1,\c) node[left=4pt] {$\scriptstyle\c$};
     }

   \node[below left]{$\scriptstyle0$};

\draw (0,0) -- (2,0.5);
\draw (2,1) -- (4,1.5);
\draw (4,4) -- (6,4.5);
\draw (6,9) -- (8,9.5);
\draw[fill=black] (2,1) circle(0.5mm) ;
\draw[fill=black] (4,4) circle(0.5mm) ;
\draw[fill=black] (6,9) circle(0.5mm) ;
\draw (2,0.5) circle(0.5mm) ;
\draw(4,1.5) circle(0.5mm) ;
\draw (6,4.5) circle(0.5mm) ;
\draw (8,9.5) circle(0.5mm) ;

\end{tikzpicture}
\end{center}
A drift with incline 1/2 is disrupted by deterministic jumps. The paths from other starting points are added in a time-homogeneous manner.
Starting in zero, for $t\to \infty$ our process behaves like $t^2$.
The symbol of the process is: $p(x,\xi)=\psi(\xi)=-\ell\xi$, where in this special case $\ell=(1/2)$.
In the class of L\'evy-type processes one considers
\[
H(R):=\sup_{y\in\mathbb{R}} \sup_{\norm{\varepsilon}\leq 1} \abs{p\left( y, \frac{\varepsilon}{R} \right)} = \frac{\ell}{R},
\]
and derives the so called generalized Blumenthal-Getoor index as follows:
\[
\beta_0:=\sup\left\{\lambda \geq 0: \limsup_{R\to\infty} R^\lambda H(R)=0\right\} = 1.
\]
A typical application for indices of this kind deals with the behavior of the paths when $t$ tends to infinity.  For L\'evy-type processes it holds that (cf. \cite{generalizedindizes} Theorem 3.11)
\[
\lim_{t\to\infty} t^{-1/\lambda} (X-x)_t^*=0 \text{ for all } \lambda<\beta_0.
\]
This is obviously wrong for the process considered here, although the symbol and the corresponding index do exist.  In fact, one can mimic totally different kinds of behavior for $t\to\infty$, by using different jump structures.  Various other results, like the maximal inequality \cite{generalizedindizes} Proposition 3.10 fail to hold also.
\end{example}

If one needs a probabilistic example (for some reason), one can make e.g. the jump sizes random.  In any case, the example shows that if we leave quasi-continuity behind, the symbol might exist, but is of no use anymore.  Hence, within the framework where the symbol is useful, we have found a characterization for its existence.

redpajama_set_name: RedPajamaArXiv


text:
News · Schools
Pompano Beach High's Make Our Schools Safe Club Selected as Finalists for Invent2Prevent Competition in Washington D.C.
By Sharon Aron Baron on December 20, 2022 No Comments
MOSS Club Members from Pompano Beach High School.
By Sharon Aron Baron
The Pompano Beach High Make Our Schools Safe Club (MOSS) has been selected as a finalist for the Fall 2022 Invent2Prevent competition in Washington, D.C.
The I2P program, an EdVenture Partners initiative, challenges high school and college students to create and implement their own solutions to prevent acts of targeted violence, hate, or terrorism in their communities.
The MOSS club, made up of Ally Vagelos, Zoey Katz, Sofia Ramirez, Ayla Friedman, and Krystal Felix, will present their project, "Bonding Buddies," to a panel of senior executives.
"Bonding Buddies" is an initiative aimed at eliminating social isolation and fostering relationships among Pompano Beach Elementary School youth. The program, which pairs older students with younger ones as mentors, uses activities like gift-giving, bracelet-making, and icebreakers to build relationships and provide stability to students in need.
MOSS Club Members of Pompano Beach High School during the project they created.
Make Our Schools Safe is a national non-profit organization dedicated to protecting students and teachers at school. It was created in 2018 after the Marjory Stoneman Douglas school shooting took the lives of 17 victims, including founder Lori Alhadeff's daughter Alyssa.
According to Invent2Prevent, this year's competition was the most difficult yet, with teams bringing forward progressive, creative, and innovative projects and initiatives.
Send your news to Parkland's #1 Award-Winning News Source, Parkland Talk. Don't miss reading Coral Springs Talk, Coconut Creek Talk, Margate Talk, and Tamarac Talk.
Sharon Aron Baron
Sharon Aron Baron is a Parkland resident and editor of Talk Media. She has been covering Parkland news since 2012. Parkland Talk was created to provide News, Views, and Entertainment for the residents of Parkland.
News2023.01.28Mah Jongg Tournament Raises Awareness For Women's Heart Health
News2023.01.26Parkland Holds Commemoration Ceremony for Marjory Stoneman Douglas Community on Feb. 14
News2023.01.25Parkland Man Pleads Guilty to $4.1 Million Fraud Scheme
News2023.01.22UPDATE: Multiple Injuries Reported in Crash on Northbound Sawgrass Expressway
Beyond Nail Lounge
Casa Decor and More
Marcos Fuentes RE/MAX Select


text:
470 Killed, 820 Disappeared And 3,250 Abducted In Eastern Nigeria By Security Forces In 220 Days, January10th – August 2021
FeaturedPress Releases August 12, 2021 admin
… CP Abutu Yaro and DCP Abba Kyari are "false accusers and executioners-in-chief" and responsible for death of 240 in 220 days ….soldiers cleared...
… CP Abutu Yaro and DCP Abba Kyari are "false accusers and executioners-in-chief" and responsible for death of 240 in 220 days
….soldiers cleared grounds for jihadist Fulani Herdsmen who massacred 250 native Igbo Christians in Igbo Land since Jan 2021
Intersociety, Nigeria: Tuesday, 10th August, 2021: The Nigerian security forces especially personnel of the Nigeria Police Force and soldiers of the Nigerian Army and their Muslim dominated superior and field commanders posted or deployed to Eastern Nigerian States of Abia, Anambra, Imo, Ebonyi, Rivers, Enugu, Cross River, Akwa Ibom and Delta have killed or supervised the mass killing of not less than 470 untried and uninvestigated citizens of Eastern Nigeria. The killings had taken place in the past 220 days or 1st Jan to 10th August 2021, a period of seven months and ten days. Most of the victims are Igbo Judeo-Christian citizens. The patterns and trends of the killings are also unknown to municipal and international human rights and humanitarian laws including the int'l rules of engagement guiding peacetime or conflict situation policing operations. The killings had increased to 350 as at June 2021 and recorded further increase of more 50 deaths following intensification of same especially in Imo, Abia and Ebonyi States. Fresh 80 captivity deaths were shockingly discovered few days ago and was said to have occurred secretly at the Nigeria Police Force former FSARS Headquarters (Abattoir), Abuja. The slain citizens, who were uninvestigated and untried were controversially accused of "being members of IPOB/ESN" leading to their secret executions around June and early July 2021. Fifty-seven of the 80 captivity deaths were reported to have been perpertated in a single night by the IRT squad headed by suspended DCP Alhaji Abba Kyari.
It Is 580 Deaths If 110 Slain Victims Of Obigbo Army Massacre Are Added: The 470 fatalities or victims of the security forces massacre in Eastern Nigeria since January 2021 did not include 110 deaths arising from the Obigbo massacre of Oct/Nov 2020 and if included, it would be correct to say that not less than 580 civilians were killed by security forces especially by soldiers of the Nigerian Army in Eastern Nigeria between 21st October 2020 and 10th August 2021, a period of nine months and twenty days. Among the latest causalities of the Obigbo Army massacre are five of the 110 residents of the area secretly abducted by soldiers and held in perpetual captivity at the Army's Alpha Military Commando Base in Suleja, Niger State and their names are: Citizens Thomas Ekwueme, Pius Onochie, Obinna Akapuru, Monday Ifeanyi and Tobechukwu Udoka. Citizen Tobechukwu Udoka died in the morning of on 30th July 2021 at the Mogadishu (Abacha) Army Barracks in Abuja where he was rushed to after vomiting blood. The breakdown of the total of 580 deaths in Eastern Nigeria since Oct/Nov 2020 showed that Imo took the lead with at least 160 deaths, Rivers at least 150 deaths including 110 dead victims of the Obigbo Army massacre, Abia at least 90 deaths, DCP Alhaji Abba Kyari led IRT at least 80 Abattoir captivity deaths; totaling 480 deaths. The remainder, numbering over 100 deaths is spread across Ebonyi, Akwa Ibom, Anambra, Cross River and Enugu States. It must also be remembered that over 90% of the killings by the security forces are defenseless and unarmed citizens, randomly accused, falsely labeled and stigmatized on account of their Igbo tribe and Judeo-Christian faith or religion. Over 60% of the ESN operatives killed which constitutes less than 10% of the total killings, are also found to have been killed outside the law, having been arrested unarmed and killed instantly or killed while in detention or captivity.
CP Abutu Yaro And DCP Abba Kyari Named The False Accusers & Executioners-in-Chief: The present Commissioner of Police Imo State, CP Abutu Yaro has been closely monitored by powerful rights groups for months and found to be "ESN/IPOB Phobia CP" as over 90% of information accompanying his indiscriminate arrest and parading of dozens of unarmed and defenseless Igbo citizens including Christian and Traditional religious leaders and their centers and symbols of worship contained lies of unpardonable, shocking and provoking proportions. The CP is so hateful of the Igbo tribe and her citizens that every arrest made in Imo must be linked to 'IPOB/ESN'. Even in simple offenses of affray or traffic or intra or inter personal quarrelling, land grabbing or criminal trespass, they are magnified and over-bloated to suit being labeled "IPOB/ESN terrorism" case. The CP is also widely seen as the most lying CP in present Eastern Nigeria and has earned himself the infamous title of "False Accuser-in-Chief" .His reckless and indiscriminate false labeling and stigmatization of defenseless citizens, without proper investigations have also gravely led to killing of not less than 160 uninvestigated and untried citizens in Imo State in the past 220 days and also aided the killing by Abba Kyari's IRT at Abuja's FSARS's Abattoir of at least 80 Igbo citizens mainly picked up and falsely labeled in Imo State. The Imo CP, therefore, shares with DCP Alhaji Abba Kyari and Brig Gen Raymond Utsasa, Commander of 34 Brigade, the title of "Executioners-in-Chief" for killing 240 Igbo citizens outside the law and modern policing and soldiering norms.
Profiles Of Senior Police Officers Involved In Mass Killings In Eastern Nigeria:(a) CP Abutu Yaro, CP, Imo State: Born on 10th Oct 1964 in Dekina, Kogi State and enlisted into the Nigeria Police Force on 15th March 1988 with Enlistment No AP30455, promoted to CP on 21st Dec 2019 and retires on 15th March 2023. He holds Bachelor's Degree in Pub Admin. (b)CP Janet Agbede, CP Abia State: Born on 2nd Jan 1963 in Akoko North, Ondo State and enlisted into the NPF on 15th March 1988 with Enlistment No AP30557 and promoted to CP on 21st Dec 2019, retires on 2nd Jan 2023. She holds B.Ed. in English Language. (c) CP Aliyu Garba, CP, Ebonyi State: Born on 21st June 1963 in Kontangora, Niger State, enlisted into the NPF on 3rd March 1990 with Enlistment No AP36647, promoted to CP in 2021, retires on 20th June 2023. He holds Bachelor's Degree in History. (d)CP Ndatsu Aliyu Mohammed, immediate past CP, Enugu State: Born on 18th Sept 1962 in Kwali, Federal Capital Territory, Abuja and enlisted into the NPF on 3rd March 1990 with Enlistment No AP36750, promoted to CP on 16th Jan 2019, retires on 18th Sept 2022. He holds Bachelor's Degree in Political Science. (e) DCP Kolo Yusuf, DC STS/TIU: Born on 7th Feb 1968 in Edati, Niger State, joined NPF on 1st May 1988 with Enlistment No.AP42108, promoted to DC on 1st May 2018, retires on 1st May 2023. He holds LLB, BL in Law. (e) DCP Alhaji Abba Kyari, DC IRT: Born on 17th March 1975 in Gujba, Yobe State and enlisted into NPF on 1st May 2000 with Enlistment No AP57296, promoted DCP on 4th April 2018, retires on 17th March 2035. He holds Bachelor's Degree in Geography. (f) Col Abubakar Abdullahi, Deputy Army Public Relations Officer, 82 Division of the Nigerian Army, Enugu
820 Abducted Igbo Citizens Disappeared Since Jan 2021:
It is also our finding that not less than 820 Igbo citizens have been disappeared by security forces since January 2021 in Eastern Nigeria. Of the 820 disappeared citizens, Army and other branches of the Armed Forces accounted for 500, Police 270 including 80 victims of the Abuja former FSARS Abattoir captivity killings and DSS 50. Victims of enforced or involuntary disappearances are citizens abducted by security forces without the knowledge of their loved ones and recourse to the established processes and procedures such as indenting their names and other particulars in the official records, allowing their families, physicians and lawyers access to them in custody or granting them administrative bail or diligently charging them to Court. They represent 'victims of enforced disappearances' because in most cases they are hardly returned alive to their families and tortured to death or shot and killed in custody with their bodies disappeared permanently to erase traces.
How Abba Kyari And His IRT Massacred 80 Igbo Detainees At FSARS Abattoir, Abuja
The Int'l Society for Civil Liberties and Rule of Law is shocked to disclose the recent secret killing of at least 80 uninvestigated and untried Igbo citizens abducted by the trio of the CP Abutu Yaro, CP, Imo State, the Imo Governor, Hope Uzodimma and the suspended DCP Abba Alhaji Kyari, DCP in charge of IGP Usman Alkali Baba's "Intelligence Response Team". The secret killings had taken place at odd hours at the former FSARS Headquarters, Abattoir, Abuja, which now houses DCP Abba Kyari's "Intelligence Response Team" and DCP Kolo Yusuf's "Special Tactical Squad" and "Tactical Intelligence Unit". The three NPF murderous squads are operationally under the NPF's Directorate of Force Intelligence Bureau headed by DIG Tijani Baba. The recent custodial killing of detained Igbo citizens was disclosed to Intersociety by an activist legal practitioner who is very conversant with the matter and has been following same. The not less than 80 slain Igbo Judeo-Christian detainees were part of hundreds abducted mainly in Imo State under the direct supervision and false labeling and stigmatization of the Imo CP, Abutu Yaro and handed over to DCP Abba Kyari Team for "further interrogation and investigation in Abuja".
At abduction, their mobile phones were seized leading to electronic disappearance of the victims' bank deposits and tracking down of their friends, business associates, in-laws and relatives, numbering dozens who were tricked, trapped, tracked and abducted. As if that was not enough, the '57 of them were forced out of their cells at a single night and massacred around June and early July 2021'. Most of the slain 57 citizens were abducted between March, April and May 2021 and among them was Citizen Cletus Ebole, abducted along with his wife, Ifeyinwa Ebole on 28th March 2021 in Imo State and labeled by CP Abutu Yaro as "a native doctor preparing charms for IPOB/ESN". The wife of the slain is still held incommunicado by another section of the IRT and presently nursing bullet leg injuries. Dozens of others were also shot and killed in the Abattoir's captivity, bringing the total killed to not less than 80. The slain victims were abducted and held incommunicado for not less than four months without trial or access to their families, physicians and lawyers leading to their custodial killings and permanent disappearance of their bodies till date. Information about their secret killings leaked few days ago after 15 of the remaining survivors in that section were arraigned at an "Abuja Area Court" on 2nd August 2021 and "remanded indefinitely" at Keffi Prisons in Nasarawa State.
Intersociety was also reliably informed that seven of the 24 survivors left in that section of the IRT had earlier left after each of them was forced to part with N1m or N1.1m, remaining seventeen of them who could not afford, out of who fifteen were "arraigned" before an Abuja Area Court with six freed and nine remaining. The nine remanded survivors are: Citizens Anthony Mba, Vincent Kalu, Iloejebechukwu Ikeji, Ezike Chukwuebuka Victor, Bede Obizue, Uchenna Ugwu, Okeke Boniface, Enemuo Vitus and Okechukwu Oguguo. The two others remaining at that section of IRT are Pastor Ebele Lemoha and Uche Ekwuoha. Among dozens of others still held incommunicado at different sections of Abuja's former FSARS, now IRT are four married women including Mrs. Maria Ezedialu, abducted in Owerri since Feb 2021 from where she was transferred to IRT, Abuja, Mrs. Ngozi Umeadi, a nursing mother arrested at a hospital in Onitsha since Feb 2021 where she was receiving a post child delivery treatment, Mrs. Ngozi Dave, abducted in Lagos whose legs and hands have been irreparably battered by leg chains and handcuffs; and Mrs. Ifeyinwa Ebole, whose husband, Cletus Ebole, was among the 57 secretly executed in the IRT captivity.
3,250 Judeo-Christian Igbo Citizens Unlawfully Arrested Or Abducted Since Jan 2021
No fewer than 3,250 mostly Igbo citizens have been unlawfully arrested or abducted by Nigerian security forces since January 2021 and if counted from Oct/Nov 2020 with addition of 550 abducted victims of the Obigbo Army massacre, then it will be correct to say that a total of 3,800 uninvestigated and untried Igbo citizens were unlawfully arrested or abducted between 21st Oct 2020 and 10th August 2021, a period of nine months and twenty days. The not less than 3,800 unlawfully arrestees or abductees included many of the 580 killed and all the 820 disappeared citizens; out of which 2,400 were taken into custody alive and released or still detained alive. Also out of the 1,400 others, the whereabouts of 820 are unknown or presumed dead in custody while 580 others are traceably found to have to been killed. The breakdown of the 3,250-3800 abductees further showed that Imo and Rivers States recorded the highest number with 900 abductions each, followed by Abia State 470, Ebonyi State 400, Anambra 300 State, IRT-Abuja 150, Delta, Enugu and Cross River 150 each and Akwa Ibom 100; totaling 3,800 abductions including 550 abductions arising from the Obigbo Army massacre and abductions. The abductions in Eastern Nigeria also recorded 690 additions in 58 days from 2,560 recorded as at 17th June 2021, excluding 550 recorded during the Obigbo Army massacre and abductions.
Nigerian Soldiers Cleared Grounds For Jihadist Herdsmen Killing Of 250 Igbo Christians In 220 Days
Despite our strong warning on 25th March 2021 that "Nigerian Army and other security forces are turning into ethnic militias for radical Muslims" especially the Jihadist Fulani Herdsmen, soldiers of the Nigerian Army are now responsible for clearing grounds for Jihadist Herdsmen attacks and killings in Igbo Land. For the first time in the history of Igbo Nation and her territory, not less than 250 native Igbo Christians have been hacked to death by Jihadist Fulani Herdsmen since January 2021. Between Jan and April, not less than 120 Igbo Christians were hacked to death and from 1st June to 18th July, no fewer than 80 more Igbo Christians were killed and in the past 23 days or between 19th July and 10th August 2021, 50 additional Igbo Christians have been killed. Soldiers of the Nigerian Army have severally and openly been accused of aiding anti Christian butcheries by Jihadist Fulani Herdsmen in places such as Miango (Irigwe) and Kauru/Zango-Kataf areas of Plateau and Southern Kaduna States. In Eastern Nigeria, same strong accusations have rented the air. This is to the extent that where communities mobilize their youths to disengage the killer-Herdsmen, soldiers will swiftly storm such communities and lay them under siege, providing grounds for the Jihadists to come back and launch attacks; and claiming that they are "searching for or combating "ESN/IPOB terrorists or members". These have resulted in abduction of several defenseless citizens and killing of scores of others. A typical case in point is Egudene (Agu-Udene)-Ekpofu in Nkanu East, Enugu State where soldiers stormed and attacked on Friday 23rd July 2021 along with Akpugo and Amagunze communities and killed scores in revenge for sacking of Jihadist Fulani Herdsmen in March 2021. As if that was not enough, the rampaging soldiers, drafted from 82 Division, Enugu also forcefully convened a village meeting at Obodo Nworie Market Square in the same Community and ordered the Community's Traditional Ruler and the interim PG to 'compile the list of all the youths in the Community and hand them over or risk having their villages and homes bombed and burned. The soldiers led by a Northern Muslim officer also told the Community's leaders that all the youths of the Community are "ESN/IPOB members and must be fished out and handed over to them urgently".
Signed: For: Int'l Society for Civil Liberties and Rule of Law:
Emeka Umeagbalasi, Chinwe Umeche, Esq., Obianuju Igboeli, Esq., Chidimma Udegbunam, Esq.


text:
'use strict';

module.exports = {
  up: (queryInterface, Sequelize) => {
    return queryInterface.sequelize.query('UPDATE users AS u INNER JOIN groups ON groups.id=u.group_id SET u.group_id=(SELECT id FROM groups WHERE company_id=-1 AND code="superadmin") WHERE u.username="superadmin" AND u.company_id=-1 AND groups.company_id!=-1;')
      .catch(function(err) {

      })
  },

  down: (queryInterface, Sequelize) => {
    return Promise.resolve();
  }
};

redpajama_set_name: RedPajamaGithub


As a Chartered Financial Planning firm as you would expect we have developed strong technical knowledge and expertise in pensions, investments, ISA's, life assurance/protection and tax & estate planning. So what makes us different?
it's about what's important to you.
We want to get to know you, your interests, your values, your goals and your most important relationships. That way we can help you achieve your financial goals.
Looking after someone's accumulated wealth is one of the most personal and privileged things you can do.
That's why we take time to get to know you, and give you time to get to know us. To build a relationship that will work over your lifetime and your children's lifetimes.