Bethesda Softworks LLC v. Interplay Entertainment Corporation - The Vault Fallout Wiki - Everything you need to know about Fallout 76, Fallout 4, New Vegas and more!
Bethesda Softworks LLC v. Interplay Entertainment Corporation was a legal dispute between Bethesda Softworks, the current owner of the Fallout franchise, and Interplay, the publisher of the original Fallout games. It concerned alleged trademark infringement in connection with past and future Fallout products. The dispute ended in January 2012 with a settlement in which Bethesda acquired full rights to Fallout MMO.
In November 2006, Interplay, headed by Herve Caen, filed a Form 8-K filing to the United States Securities and Exchange Commission (SEC) regarding a potential Fallout massively multiplayer online game. In April 2007, Bethesda Softworks, the developer of Fallout 3, purchased full rights to the Fallout IP for $5.75 million USD. While Bethesda now owned the rights to the Fallout MMO IP as well, clauses in the purchase agreement state allowed Interplay to license the rights to the development of the MMO. Specific requirements were stated in the agreement that if not met, Interplay would immediately lose and forfeit its license rights for Fallout. Development must have begun within 24 months of the date of the agreement (April 4, 2007), and Interplay must have secured $30 million within that time frame or forfeit its rights to license. Interplay would furthermore need to launch the MMOG within 4 years of the beginning of development, and pay Bethesda 12 percent of sales and subscription fees for the use of the IP.
Bethesda Softworks claimed that Interplay is in breach of the Fallout MMO agreement for failure to commence full scale development by April 4, 2009 and to secure certain funding for the game. Interplay disputed these claims and maintained that it had the right to develop the Fallout MMO Project V13, together with Bulgaria-based Masthead Studios.
While Interplay was permitted to sell Fallout, Fallout 2 and Fallout Tactics after the sale of the IP to Bethesda, the contract required all advertising, packaging and other promotional material to be sent to Bethesda for approval first, which, according to the lawsuit, was never done regarding either the Fallout Trilogy pack, or any of the releases through Good Old Games, Steam and other digital distribution platforms. Bethesda also claimed that the name Trilogy constitutes unfair competition, since it suggests that the pack includes Fallout 3.
Interplay claimed that Bethesda's termination of its contract with Glutton Creeper Games regarding the license to develop and publish Fallout Pen and Paper d20 was illegal. Bethesda held that Interplay had no rights to license it in the first place, since it was forbidden to do so by the 2004 Exclusive Licensing Agreement.
According to Interplay, Bethesda interfering in their digital download deals with GameTap, GOG, etc. by sending these companies letters claiming that Interplay has no rights to the Fallout titles constituted unreasonable interference/withholding of permission. Interplay argued that it is Bethesda that is in fact in breach of the contract, which makes the sale of the rights to the franchise null and void, bringing back into effect the former contract, that would only give Bethesda the rights to publish three Fallout games, including Fallout 3 and Fallout New Vegas.
On April 2, 2009 Interplay announced a binding letter of intent with Masthead Studios, a Bulgarian-based developer, to fund the development of a post-apocalyptic MMO codenamed Project V13, which had been all but confirmed to be the aforementioned Fallout MMO. Masthead and Interplay teams will work together under the direction and control of Interplay to complete development of the project.
On April 15, 2009, it was announced that Bethesda Softworks moved to rescind the Fallout MMORPG license.
On September 8, 2009 Bethesda filed its original lawsuit against Interplay in the Maryland District Court in Baltimore.
In October 2009, Interplay filed its own counter-lawsuit, arguing Bethesda is in breach of contract and the contract to sell the rights to the series to Bethesda is null and void and Interplay owns the franchise again.
In December 2010, Bethesda added the claim that Interplay only licensed the rights to use the Fallout name and not any other elements of the Fallout games. This claim is disputed by Interplay.
On December 10, 2009, Bethesda's motion for preliminary injunction was denied.
On June 9, 2011, Bethesda filed for another injunction on grounds that there was infringement of the license that hurt's Bethesda's gameplay. This is on the assumption that the story in Fallout Online will alter the story to a damaging level to Bethesda Softwork's future developments such as the speculated Fallout 4. In August 2011, the injunction was denied, and Bethesda appealed the ruling to the United States Court of Appeals.
On January 9, 2012 Bethesda Softworks announced that the long-lasting litigation between Bethesda and Interplay, over the rights to Fallout Online ended with a settlement in which Bethesda acquired full rights to the MMO in exchange for paying Interplay $2 million. Interplay's rights to sell the original Fallout games would also expire on December 31, 2013.
This page was last edited on 11 June 2018, at 03:58.


"The very tone of your response" blah blah blah. The point is you're having your cake and eating it... No I wouldn't serve ice by hand nor serve spirits in a lager glass. So what? That's not the point. Glass Houses and Stones come to mind - exactly how? With regard to Inez or you? You don't know the half of what she's been through.
As for me - I haven't made ANY personal insults. I commented on an anonymouns poster pontificating behind 'name and address withheld' who relies on an eleven year old's backing for support rather than to have the courage to be up front about who he (or she) actually is when putting down someone else. YOU have entirely missed the point here. IF. IF. IF you had had the courage to make your earlier statment in person - I for one wouldn't have had a problem with it.
And besides, when it comes to criticising Inez... what have you done to broadcast the reality of tied pub life and what have you done to highlight the need for change and the need hundreds of other lessees have for a voice?


You are here : Home Could Diwali now be the festival of 'solar lights'?
Hindus in India and around the globe adorned their houses with lamps, shared feasts and exchanged gifts to celebrate Diwali, the festival of lights. For Hindus, light symbolizes the triumph of good over darkness or of knowledge over ignorance.
In many Hindu traditions, the sun -- our planet's source of light -- is a major source of energy and power; now, as the world seeks to address climate change, Diwali's celebration of light takes on a very contemporary significance.
Sensitivity to the environment is also an inherent part of the Hindu spiritual worldview. The Mahabharata, Ramayana, Vedas, Upanishads, Bhagavad Gita, Puranas and Smriti contain the earliest messages for the preservation of environment and ecological balance. Hindu traditions and teachers have always advocated for care and respect of the natural world. As we are growing more materialistic, we are in danger of losing this connection.
Up until now, India's reliance on coal for almost three-fourths of its electricity supply has hindered its transition to clean energy. Part of the reason for India's reliance on coal for so much of its power is to be found underground, in the country's enormous reserves of the fossil fuel -- the fourth-largest in the world. But, like many other countries, India is also blessed with another abundant resource: sunshine.
India is a land endowed with abundant free solar energy. Using the country's deserts and farmlands and taking advantage of its 300-plus sunny days a year, India can easily generate around 1,000 gigawatts (GW) of solar power -- by using only 0.5% of its land.
Progress such as this is vital if we are to achieve the UN's Sustainable Development Goals and the goals of the Paris Agreement on climate change.
India's dense population and high solar insolation provide ideal conditions for the exponential growth of solar power as a future energy source. With GDP growing at about 8%, solar photovoltaic systems are the only renewable energy resource that can bridge the 'gap' between supply of and demand for energy, especially in rural India, where close to 80,000 villages are still not electrified.
Solar energy can transform India and help to bring about decentralized distribution of energy, thereby empowering people at the grassroots level and eliminating the need for costly expansion of transmission and distribution systems.
For its part, the Indian government has taken several measurable steps toward solar transition. The Hindu community is becoming more involved. In March earlier this year, spiritual leader and Art of Living founder Sri Sri Ravi Shankar joined forces with other faith leaders and institutions to launch an Interfaith Solar Alliance in order to mobilize Hindu religious leaders to solarize ashrams and advocate the benefits of solar to their followers around the world.
In the lead-up to last year's Paris negotiations, with the Bhumi Project at Oxford, I helped to organize a People's Pilgrimage focused on climate change to a number of Hindu sacred sites. Prominent Hindu groups and leaders also endorsed the Hindu Declaration on Climate Change.
India's future, and the world's future, depends in large part on how well we use that ancient gift of the sun.


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You are here: Home / Skin Care / Fight Acne with an Aloe and Honey Face Mask!
With only two ingredients, this aloe and honey face mask recipe in quick, easy, cheap, and effective–all four things you are looking for in an acne treatment.
Yield: 1 awesome face mask! If you want to adjust the amount, just make sure you keep the ratio the same. You want equal amount of aloe gel to honey. No biases here!
Combine the 2 ingredients in a bowl.
Let it work its magic for a good 20-25 minutes.
There ya have it! An easy way to get smooth skin in no time!
Taste Test That Honey and Aloe!
Sneak a little honey into your mouth before using it all in your aloe face mask mixture –and every day, for that matter. Research reveals that daily consumption of honey can do a whole bunch of things for your body: (1) prevent cancer; (2) boost your immune system; and (3) relieve your allergy systems. Because honey is bursting with antioxidants that strengthen the immune system, it can do a whole lot more than just sit on your face, fighting dry skin and acne. Whether mixed into meals, drunk in tea or taken straight from a spoon, honey should go in your mouth too!
And what do you know, you can also reap aloe's other health benefits by taking it orally. As with honey, aloe's skin-healing ability is just the tip of the medicinal iceberg: when consumed, aloe can help a lot of gastrointestinal issues–from IBS to acid reflux disease. Just be careful with how much aloe you eat (or drink, for that matter); too much and your digestive track will not be able to handle it.
Who would have thought? An edible face mask as well (and I bet you it's sweet, given that honey!). Go forth to slather it on!
When getting ready to extract gel from my aloe plant do I cut off an entire leaf from my plant?
Great! Sooo loving this,I hope it works for me.
This skin care regimen looks quick and simple; but how often should it be carried out in a day/week?
I hv black marks on face bez of pimples bt nw I don't hv any pimples on my face only those black marks r there does it really work's…!!?? I want to remove it within 1 week how can I do it with aloe?? Can u please suggest me….!!
how long is the shelf life of this in the fridge? 1 week? 3 days? if i want to use it 2-3x a week.
Wow! Thanks for that. Can't wait to try it.
I tried this face mask several times already and it did really work great on my skin. This face mask is effective on my skin that it provides a healthy glow. Thank for sharing, greatly appreciated. I will stop by again next time and will check some of your beauty related blogs.


text:
Alfred Morris Interview
Why One Of The NFL's Best Running Backs Hates Running
Chris Aung-Thwin
AskMen: What's your favorite cheat food?
AM: This summer I went 30 days or a month and a half without eating sweets — you'd think coming off of that you'd be like, "Man, I want sweets! I haven't had anything in so long!" but now I don't even desire it that much. I do love Reese's though. I love peanut butter and I love the combination of peanut butter and chocolate — so if I had to choose something, I'd say Reese's sticks or Reese's cups... or if not Reese's, then ice cream.
AskMen: If you had to get the average guy in shape, what's the first thing you'd have him do?
AM: Change his diet. Like I said, everyone knows they should be exercising. A lot of people will go exercise — say a constant jog for an hour where they burn about 300 calories — and then that same person will go to a fast food place or Starbucks or something and get a meal that's a 1,000 calories or a drink that's 500 or 700 calories. The activity was pointless! I would definitely get you to change your diet first. Start making healthier choices a habit. You can still treat yourself, but everything in moderation. You need to find that balance.
AskMen: What advice would you give to parents?
AM: They have to know that they're in control. I see a kid who's not eating right and the parents say, "Well, I can't make them eat right." You are the parent and you can point them in the right direction. Eat more of this, eat less of that — find a nutritional balance. Kids want to eat junk — snacks, cakes, fried foods — but kids can't pay for meals. It's up to parents to point them in the right direction.
AskMen: What's the exercise you hate most (but know is good for you)?
AM: Any cardio. I hate running. Well, [laughing] I'm a running back so I like running short distances — the most I'll run is 100 yards. Cardio is grueling, it's excruciating — it's a grind. But I know it's going to make me better. I want to be in better shape than everyone else. There are people who are bigger, faster, stronger than me but I'm going to outlast you. If it's the fourth quarter, you might be bigger and stronger than me, but it doesn't matter. You're tired. You're not going to be anywhere near 100%. I'm winning because I'm outlasting you. I take pride in my cardio… even though I don't like it at all. I wish you could take a pill or something and be in the best shape ever, but it's not that easy.
Why Pounding That Post-Workout Protein Shake Might Not Be The Best Idea
Don't Waste Your Gains. Here's How To Do Your Pre-Workout Nutrition Right.
Andrew Luck On Learning To Take Hits
A Crash Course In Fitness From NFL Superstars
How Bobby Flay Learned To Be Healthy

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}\label{sec1}

There are various notions of average in financial mathematics and ergodic theory. For instance, in the first one, the discounted averages determine the price of the contingent claims using a risk-neutral pricing model \cite{13}, while the latter demonstrates the tight connection between a time average and an ensemble average using Birkhoff's ergodic theorem. While the log normal distributions play an important part in many branches of scientific knowledge, it is sensible to expect that using the new notion of log ergodic process along with Birkhoff ergodic theorem will open a new window to the research at least in the field of mathematical finance models. Therefore in this paper we address the questions: What positive processes can be brought into models of mathematical finance using the notion of log ergodic process? Secondly, what is the connection between ergodic averages and financial averages?

By definition, a stochastic process is ergodic in the mean, or simply mean ergodic, if it's ensemble average and time average are equal in the long run \cite{1,59}.

Since the development of expected value, economics theory and mathematical finance have existed. However, there hasn't been much focus on the connection between expected value and time average. Ergodic economics is a method of approaching economic theory from the viewpoint of ergodic theory. Therefore, the fundamental query is whether or not the time average of the stochastic processes employed in economic models is equal to the expectation of those processes? An affirmative response suggests a different approach to solving economic and mathematical finance problems. The study of ergodic economics focuses on the economic implications of these phenomena. A thorough analysis by Ole Peters was presented in this case in December 2019 \cite{89}. Additionally, since 2011, the London Mathematical Laboratory has conducted specialized research on ergodic economics \cite{90}. A research was done on modeling blockchain-enabled economics using the stochastic dynamical systems \cite{100}. The problem of ergodicity necessitates a reassessment of how probabilities are used in economics, according to Nassim Nicholas Taleb \cite{92}.

The following questions arise when looking at financial stochastic processes from an ergodic theory point of view: Which financial models are ergodic? Which non-ergodic financial models may be used to create an ergodic model? Some processes with specific properties have been shown to be ergodic, or at least ergodic in the mean. Markov processes with stationary distribution, are ergodic, according to Shalizi \cite{59}. Oesook Lee demonstrated a specific example of the mixing and ergodic properties for generalized Ornstein-Uhlenbeck processes in her paper \cite{93}. Trabelsi explored the ergodic properties of $CIR$ model and demonstrated that it has the ergodic recurrence property\footnote{An irreducible, non-periodic Markov chain with a stationary distribution is said to be recurrent if it converges to its stationary distribution for almost all initial points.}, which is known as mean reversion property in mathematical finance \cite{62}. The $CIR$ process is ergodic and has a stationary distribution \cite{62}. For a model (process) to be ergodic it is necessary to possess Markov property with a stationary distribution. Additionally, the model must possess the mean recurrence property in order to be ergodic \cite{39,40,35}.

The main purpose of this paper is to address these questions and to provide a family of processes for studying financial markets. These processes, together with their ergodic properties, are helpful for managing investment risk. Thus, we select some stochastic processes from the viewpoint of ergodic theory that can be applied to mathematical finance.

The rest of the paper is organized as follows:\\
Some necessary concepts from ergodic theory are introduced in section \ref{sec2}. In section \ref{sec3}, we present the concept of ergodicity in economics. In section \ref{4}, we define log-ergodicity and the ergodic maker operator and discuss their properties. The main theorem of the paper is presented in section \ref{5}, where we also examine financial models from the standpoint of ergodic theory. The main theorem is proved in section \ref{se6}. In section \ref{se7}, we shall discuss the applications of log-ergodic processes in financial theory along with an example. The results and recommendations for further studies are presented in section \ref{se8}.

\section{Ergodicity and Ergodic theorem of Birkhoff}\label{sec2}
Throughout the remainder of this text, we use the filtered probability space
$(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_t)_{t\geq0})$,  in which   $\Omega$ is the space of events, $\mathcal{F}$ is a $\sigma$-algebra,  $\mathbb{P}$ is an invariant probability measure (see \cite{1} for definition), and $\mathcal{F}_t$ is a filteration which represents the information of the financial market up to time $t$, and is generated by Wiener process $W_t$
(i.e. $\mathcal{F}_t=\sigma\{W_s \lvert s\leq t\}$).

Suppose $\{X_t\}_{t\geq0}$ is a Markov homogeneous stochastic process. There are two requirements for the stochastic process $X_t$ to be ergodic. First, the time average and its ensemble average should be equal. Second, the time average and ensemble average of the autocorrelation function of the process should also be equal \cite{106}. The process is referred to as mean ergodic if just the first criterion met.

\begin{theorem}(Birkhoff)\label{Bir}
	Let $X_t$ be a stochastic process and $\phi$ be a bounded, ergodic, measure-preserving transformation with respect to invariant probability measure $\mathbb{P}$. Then,

	the limit
	\begin{equation}\label{1}
		\tilde{\phi}(\omega):=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T X_t(\phi(\omega))dt,
	\end{equation}
	exists almost surely and,  $\tilde{\phi}=\int_\Omega X_t d\mathbb{P}$.
\end{theorem}
\begin{proof}
	Since $\phi$ is bounded, without loss of generality by discretization, we consider the limit for the integer values of $T$. Discrete-time stochastic process
	$(\int_n^{n+1}X_t(\phi(\omega))dt)_{n\geq0}$
	is stationary
	\cite{59,35}. Therefore, from the classical statement of Birkhoff theorem \cite{1}, It follows that the limit (\ref{1}) exists. For the proof of the second part we refer the reader to \cite{1,59}.
\end{proof}

Considering $\phi$ as the identity function $\phi=I$ in theorem \ref{Bir}, we get
\begin{align}\label{tav}
	\tilde{I}(\omega)=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T X_t(\omega) dt.
\end{align}
We call $\tilde{I}$ the time average of the process $X_t$, and denote it by $<X>$.

Suppose that $X_t$ is a Hölder continuous positive stochastic process of order $0<\beta<\infty$. i.e.
\begin{equation*}
	\exists b>0;\quad \lvert X_t-X_s\rvert\leq b\lvert t-s\rvert^{\beta},\quad \forall t,s>0.
\end{equation*}
According to exponential decay of correlation theorem \cite{1}, there exist positive number $\Lambda<1$, and $\alpha(X_s,X_t)$ such that, the correlation coefficient $\mathbf{cor}(X_s,X_t)$ satisfies the following relationship.
\begin{align*}
	\mathbf{cor}(X_s,X_t)&\leq \alpha(X_s,X_t)\Lambda^n,\quad \forall n\geq 1.
\end{align*}
Therefore, from the definition of correlation we have:
\begin{align}
	\frac{\mathbb{C}(X_s,X_t)}{\Lambda^n\sqrt{\mathbb{V}ar[X_t]\mathbb{V}ar[X_s]}}&\leq \alpha(X_s,X_t),\label{alpha}
\end{align}
where $\mathbb{C}(X_s,X_t)$ is the covariance of $X_t$ and $X_s$.

\section{Ergodicity in economics}\label{sec3}
Let the process $X_t$ represent an investor's wealth process. The primary problem of ergodic economics is to analyze the evolution of this process. Ergodic economics makes the assumption that investor choices will optimize the time-average growth rate of $X_t$. From Daniel Bernoulli's conjecture (1738) \cite{105} it follows that the utility of each additional dollar has an almost inverse relationship with the number (units) of dollars that an investor currently has \cite{89}. Therefore, we conclude that the growth rate is governed by the differential equation $dU(X)=\frac{1}{X}dX$ with initial condition $U(0)=\ln(X_0)$, and the solution $U(X)=\ln(X)$, in which $X_0\neq 0$ \cite{89}. Under these circumstances, let g be the growth rate of $X_t$ and write $g_{t,X}=\frac{\Delta U(X)}{\Delta t}$.
Take two savings accounts as an illustration; one pays 4\% annually, the other 1\% monthly. It is obvious that the first account offers $\frac{4}{12}\%$ interest per month.
The growth rate of the initial investment over time will determine which of these two savings accounts you should choose. To choose between these two accounts, one does not take the interest rate into account. He or she picks an account with a greater rate of capital growth over time. Ergodicity is generally violated by processes of type $X_t$. However, their growth rates are ergodic \cite{89}. We observe that the time average of the growth rate is introduced as the changing rate of mathematical expectation of $U(X)$. This leads us to the following definition.
\begin{definition}
	The time average of the growth rate of a stochastic wealth process $X_t$ is
	$$<g_{t,X}>=\frac{\mathbb{E}[\Delta U(X)]}{\Delta t},$$
	where $U(X)=\ln(X)$.
\end{definition}
A simple example of the wealth process of the investor, $X_t$, is the geometric Brownian motion \cite{13}, which is widely used in mathematical finance and other fields.
\begin{example}
	Consider the process
	\begin{equation*}
		X_t=X_0\exp\{(\mu-\frac{1}{2}\sigma^2)t+\sigma W_t\},\quad X_0=x\neq 0.
	\end{equation*}
	Where the constants $\mu$ and $\sigma$ are the drift and volatility coefficients of the process, and $W_t$ is a standard Wiener process. We observe that:
	\begin{align}
		U(X)&=\ln(X_t),\label{3}\\
		\ln(X_t)&=\ln(X_0)+(\mu-\frac{\sigma^2}{2})t+\sigma W_t.\label{2}
	\end{align}
	From \ref{2}, It follows that $U(X)$ has a linear growth with respect to time. Thus, it's time average of growth rate is:
	\begin{equation*}
		<g_{t,X}>=\frac{\mathbb{E}[\Delta U(X)]}{\Delta t}=\mu-\frac{1}{2}\sigma^2.
	\end{equation*}
	Therefore, maximizing the rate of change of the logarithmic utility function \ref{3} is equivalent to maximizing the time average of the wealth growth rate \cite{89}.

	Due to the necessity of using $U(\cdot)$, from now on, we will study the logarithm of some processes used in financial theory.
\end{example}

\section{Logarithmic ergodicity}\label{4}
In this section, we introduce the concept of log-ergodicity of a positive stochastic process and investigate some of its properties.

\begin{definition}(Log-ergodicity)\label{logergodic}
	We say that the positive stochastic process $X_t$ is logarithmically ergodic (log-ergodic), if its logarithm is mean ergodic. More precisely, the positive stochastic process $X_t$ is log-ergodic if the process $Y_t=\ln(X_t)$ satisfies
	\begin{equation}\label{limer}
		\overline{<Y>}:=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T (1-\frac{\tau}{T})\mathbb{C}_{yy}(\tau)d\tau=0, \quad \forall\tau\in[0,T].
	\end{equation}
	Where $\mathbb{C}_{yy}(\tau)$ is the covariance of  $Y_\tau$.
\end{definition}
The periods of financial market cycles \footnote{Market cycle refers to the bear and bull phases of the market.} have an increasing trend, according to experience \cite{83}. This phenomenon is influenced by variables such as market liquidity, interest rates, and $\cdots$. For instance, the rising trend of the stock index  S$\&$P500 bull market cycles from 1932 to 2022 may be observed in figure \ref{gr1}. The graphic was created using data from the Trading View \footnote{\url{https://www.tradingview.com}} and Yahoo Finance \footnote{\url{https://finance.yahoo.com}} websites.

Even though Brownian motion is not of bounded variation process, these variations are controlled by market participants and policymakers in the financial markets \cite{103}. Issuing currencies, supplying and removing the market with liquidity, adopting stringent legislation, and $\cdots$ are a few of these controls \cite{103,104}. In this paper, we present a coefficient that serves as a measurement of the extent to which of these factors govern the market. Ergodic circumstances will predominate in the market in the long term since this coefficient almost eliminates the randomness of the price process of risky asset. We define the Ergodic Maker Operator in a way that includes a parameter, which indicates how much the growth rate of the upward market cycle trend is under control.
\begin{figure}[H]
	\begin{center}
		\captionsetup{justification=centering}
		\includegraphics[scale=0.5]{figure1.eps}
		\caption{\small{
				S\&P500 index bull market cycle period trend.}\label{gr1}
		}
	\end{center}
\end{figure}
For the stochastic process $X_t$, we denote the mean square convergence with $ms\lim_{t\rightarrow\infty}X_t=X$ and the convergence in probability by\\ $st\lim_{t\rightarrow \infty}X_t=X$.
\begin{proposition}\label{ID}
	Suppose that for the random process $X_t$, we have\\ $\lim_{t\rightarrow \infty}\mathbb{E}[X_t]=k$ and $\lim_{t\rightarrow \infty}\mathbb{V}ar[X_t]=0$. Then we have:
	\begin{equation*}
		ms\lim_{t\rightarrow\infty}X_t=k.
	\end{equation*}
\end{proposition}
\begin{proof}
	See \cite{76}.
\end{proof}

The Wiener process (Brownian motion) can be inhibited (controlled) under specific circumstances, as demonstrated by the following theorem (\cite{76}).
\begin{theorem}\label{IDV}
	For $\beta>\frac{1}{2}$ it is observed that:
	\begin{equation}
		ms\lim_{T\rightarrow\infty}\frac{W_t}{t^\beta}=0.
	\end{equation}
\end{theorem}
\begin{proof}
	Assuming $X_t=\frac{W_t}{t^{\beta}}$, for every $t>0$ we have:
	$\mathbb{E}[X_t]=\frac{\mathbb{E}[W_t]}{t^{\beta}}=0$, and
	$$\mathbb{V}ar[X_t]=\frac{1}{t^{2\beta}}\mathbb{V}ar[W_t]=\frac{t}{t^{2\beta}}=\frac{1}{t^{2\beta-1}}.$$
	Since $\lim_{t\rightarrow \infty}\frac{1}{t^{2\beta-1}}=0$, it follows from proposition \ref{ID} that,\\ $ms\lim_{t\rightarrow\infty}\frac{W_t}{t^{\beta}}=0$.
\end{proof}
\begin{corollary}\label{IDV2}
	For $Z_t=\int_0^tW_sds$, and $\beta>\frac{3}{2}$, we have:
	\begin{equation*}
		ms\lim_{t\rightarrow\infty}\frac{Z_t}{t^{\beta}}=0.
	\end{equation*}
\end{corollary}
\begin{proof}
	For all $t>0$ we have $\frac{\mathbb{E}[Z_t]}{t^{\beta}}=0$, and 	$$\frac{1}{t^{2\beta}}\mathbb{V}ar[Z_t]=\frac{t^3}{3t^{2\beta}}=\frac{1}{3t^{2\beta-3}}.$$
	Now we get the result from proposition \ref{ID}.
\end{proof}
\begin{corollary}
	For all $\beta>\frac{1}{2}$ we have:
	\begin{equation*}
		st\lim_{t\rightarrow \infty}\frac{W_t}{t^{\beta}}=0.
	\end{equation*}
\end{corollary}
\begin{proof}
	See \cite{76}.
\end{proof}

According to these results, the coefficient $\frac{1}{t^{\beta}}$, with $\beta>\frac{3}{2}$, inhibits the variations of Wiener process.

\subsection{Properties of positive log-ergodic process}

Each positive stochastic process $X_t$ can be represented exponentially \cite{43,21}. i.e.
\begin{equation}\label{X}
	X_t=xe^{Y_t},\quad X_0=x.
\end{equation}
or,
\begin{equation*}
	\ln(X_t)=\ln(x)+Y_t,\quad Y_0=0.
\end{equation*}
In this paper, we assume that $Y_t$ is an Itô process. From \cite{13}, we define:
\begin{align}\label{itop}
	Y_t^\prime:=\ln(X_t)&=Y_0^\prime+\int_0^t\sigma_s dW_s+\int_0^t \mu_s ds;\quad Y_0^\prime=\ln(x)+Y_0,
\end{align}
where $W_t$ is a standard Wiener process, and $\mu_t$ and $\sigma_t$, which are integrable functions of time $t$ and $y$, respectively, are drift and volatility coefficients.

The process $Y_t$ is not necessarily ergodic since it is not always stationary \cite{13}. This process includes a deterministic part, a random part, and a constant part. We denote the deterministic part by $D_t$ and the random part by $R_t$. Therefore, it can be written:
\begin{align*}
	Y_t^\prime&=\underbrace{Y_0^\prime}_{\text{constant}}+\underbrace{\int_0^t\mu_s ds}_{D_t}+\underbrace{\int_0^t\sigma_s dW_s}_{R_t},\\
	Y_t^\prime&=Y_0^\prime+D_t+R_t.
\end{align*}
Now we introduce the Ergodic Maker Operator  (EMO).
\begin{definition}\label{deferc}
	Let $W_t$ be a standard Wiener process and $\beta>\frac{3}{2}$. For $t\in[0,T]$, we define the ergodic maker operator of the process $Y_t^\prime$ as
	\begin{equation}\label{xi}
		\xi_{t,W_t}^{\beta}[Y_t^\prime]:=0\cdot Y_0^\prime+\frac{W_t}{t^{\beta}}\cdot D_t+\frac{1}{t^{\beta}}\cdot R_t.
	\end{equation}

\end{definition}
\begin{definition}
	We define the inhibition degree $\beta$ as
	\begin{equation}\label{beta}
		\beta:=\begin{cases}
			\alpha, &\text{if} \quad \alpha>\frac{3}{2},\\
			\frac{3}{2}+\lvert\alpha\rvert, &\text{if} \quad \alpha<\frac{3}{2},
		\end{cases}
	\end{equation}
	where $\alpha$ satisfies \ref{alpha}.
\end{definition}
\begin{remark}
	We define the concept of Log-ergodicity together with EMO in order to facilitate the analysis of mathematical models of financial markets, similar to the idea of log-normality simplifies the related computations.
\end{remark}
Note that the estimation of the smallest possible value for $\beta$ could also be an interesting topic to cover in future studies.
\begin{lemma}\label{mm}

	$\xi_{t,W_t}^\beta[Y_t^\prime]$ is a stationary stochastic process.
\end{lemma}
\begin{proof}
	Using the definition \ref{deferc} yields
	\begin{equation}\label{zt}
		Z_t:=\xi_{t,W_t}^\beta[Y_t^\prime]=Z_0+\frac{1}{t^{\beta}}\int_0^t\sigma_s dW_s+\frac{W_t}{t^{\beta}}\int_0^t\mu_sds,\quad Z_0=0.
	\end{equation}
	Calculating the expectation of the process $Z_t$ for times $t$ and $t+\tau$, we have:
	\begin{align*}
		\mathbb{E}[Z_t]=\mathbb{E}[Z_{t+\tau}]=0,\quad \forall t,\tau>0.
	\end{align*}
	Therefore, the stochastic process $Z_t$ is stationary.
\end{proof}
\begin{proposition}
	Consider the positive stochastic processes $H_t$ and $G_t$, and
	let $\xi_{t,W_t}^\beta[\cdot]$ be the ergodic maker operator defined by \ref{xi}. Suppose that we have\\ $Y_t=\ln(H_t)$ and $Z_t=\ln(G_t)$. The following statements hold.
	\begin{itemize}
		\item [1.] For all $a\in \mathbb{R}$ we have:
		\begin{equation}\label{xi1}
			\xi_{t,W_t}^\beta[aY_t]=a\xi_{t,W_t}^\beta[Y_t].
		\end{equation}
		\item [2.]
		\begin{equation}\label{xi2}
			\xi_{t,W_t}^\beta[Y_t+Z_t]=\xi_{t,W_t}^\beta[Y_t]+\xi_{t,W_t}^\beta[Z_t].
		\end{equation}
		\item [3.]
		$\xi_{t,W_t}^\beta[Y_t\cdot Z_t]=D_t^z \xi_{t,W_t}^\beta[Y_t]+R_t^z H_t$. Where $H_t$ is a stochastic process to be found in the course of the proof.
	\end{itemize}
\end{proposition}
\begin{proof}
	\begin{itemize}
		\item [1.] Using the definition \ref{xi} yields
		\begin{align*}
			\xi_{t,W_t}^\beta[aY_t]&= \frac{W_t}{t^{\beta}}\big(a\int_0^t \mu_s ds\big)+\frac{1}{t^\beta}\big(a\int_0^t \sigma_sdW_s\big)\\
			&=a\Big(\frac{W_t}{t^{\beta}}\big(\int_0^t \mu_s ds\big)+\frac{1}{t^\beta}\big(\int_0^t \sigma_sdW_s\big)\Big)\\
			&=a\big(\frac{W_t}{t^\beta}D_t+\frac{1}{t^\beta}R_t\big)\\
			&=a\xi_{t,W_t}^\beta[Y_t].
		\end{align*}
		\item[2.] Let $D_t^y$ and $D_t^z$ be the deterministic parts of the processes $Y_t$ and $Z_t$, and $R_t^y$ and $R_t^z$ be the random parts of the processes $Y_t$ and $Z_t$, respectively. We have:
		\begin{align*}
			\xi_{t,W_t}^\beta[Y_t+Z_t]&= \frac{W_t}{t^\beta}(D_t^y+D_t^z)+\frac{1}{t^\beta}(R_t^y+R_t^z)\\
			&=\frac{W_t}{t^\beta}D_t^y+\frac{1}{t^\beta}R_t^y+\frac{W_t}{t^\beta}D_t^z+\frac{1}{t^\beta}R_t^z\\
			&=\xi_{t,W_t}^\beta[Y_t]+\xi_{t,W_t}^\beta[Z_t].
		\end{align*}
		\item [3.] Using the proof of \ref{xi2} we have:
		\begin{align*}
			\xi_{t,W_t}^\beta[Y_tZ_t]&=\xi_{t,W_t}^\beta[D_t^yD_t^z+D_t^yR_t^z+R_t^yD_t^z+R_t^yR_t^z].
		\end{align*}
		Using \ref{xi1} it can be written:
		\begin{align*}
			\xi_{t,W_t}^\beta[Y_tZ_t]&=\xi_{t,W_t}^\beta[D_t^yD_t^z]+\xi_{t,W_t}^\beta[D_t^yR_t^z]+\xi_{t,W_t}^\beta[R_t^yD_t^z]+\xi_{t,W_t}^\beta[R_t^yR_t^z]\\
			&=\frac{W_t}{t^\beta}(D_t^yD_t^z+R_t^yR_t^z)+\frac{1}{t^\beta}(D_t^yR_t^z+R_t^yD_t^z)\\
			&=D_t^z\big(\frac{W_t}{t^\beta}D_t^y+\frac{1}{t^\beta}R_t^y\big)+R_t^z\big(\frac{W_t}{t^\beta}R_t^y+\frac{1}{t^\beta}D_t^y\big)\\
			&=D_t^z \xi_{t,W_t}^\beta[Y_t]+R_t^z H_t.
		\end{align*}
		Where $H_t=\frac{W_t}{t^\beta}R_t^y+\frac{1}{t^\beta}D_t^y$.
	\end{itemize}
\end{proof}
\begin{theorem}
	Consider the positive stochastic processes $H_t$ and $G_t$, and suppose that, 	$Y_t=\xi_{t,W_t}^\beta[\ln(H_t)]$, and $Z_t=\xi_{t,W_t}^\beta[\ln(G_t)]$. The following statements hold.
	\begin{itemize}
		\item [1.]
		For all $\gamma,\nu\in \mathbb{R}$ we have:
		\begin{equation}
			\overline{<\gamma Y+\nu Z>}=\gamma^2\overline{<Y>}+\nu^2\overline{<Z>}.\label{I}
		\end{equation}
		\item[2.]
		\begin{equation}
			\overline{<Y\cdot Z>}=0.\label{II}
		\end{equation}
	\end{itemize}
\end{theorem}
\begin{proof}
	\begin{itemize}
		\item[1.]
		Using the definition \ref{limer} and lemma \ref{mm} we have:
		\begin{align*}
			\overline{<\gamma Y+\nu Z>}=&\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\big(\mathbb{E}[\gamma^2Y_\tau^2+\nu^2Z_\tau^2+2\gamma\nu Y_\tau Z_\tau]\\
			&-\gamma^2\mathbb{E}[Y_\tau]^2-\nu^2\mathbb{E}[Z_\tau]^2\big)d\tau\\
			=&\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})(\mathbb{E}[\gamma^2Y_\tau^2]+\mathbb{E}[\nu^2Z_\tau^2])d\tau\\
			=&\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\gamma^2\mathbb{E}[Y_\tau^2]d\tau\\
			&+\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\nu^2\mathbb{E}[Z_\tau^2]d\tau\\
			=&\lim_{T\rightarrow\infty}\frac{\gamma^2}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{yy}(\tau)d\tau\\
			&+\lim_{T\rightarrow\infty}\frac{\nu^2}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau\\
			=&\gamma^2\overline{<Y>}+\nu^2\overline{<Z>}.
		\end{align*}
		\item[2.]
		We write:
		\begin{align*}
			\overline{<Y\cdot Z>}=&\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})(\mathbb{E}[(Y_\tau Z_\tau)^2]-(\mathbb{E}[Y_\tau Z_\tau])^2)d\tau\\
			=&\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{E}[Y_\tau^2]\mathbb{E}[Z_\tau^2]d\tau.
		\end{align*}
		Integrating by parts yields
		\begin{align*}
			\overline{<Y\cdot Z>}=&\lim_{T\rightarrow \infty}\frac{1}{T}\Big[\mathbb{C}_{zz}(\tau)\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{yy}(\tau)d\tau\\
			&-\int_0^T\frac{d\mathbb{C}_{zz}(\tau)}{d\tau}(\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{yy}(\tau)d\tau)d\tau\Big]=0.
		\end{align*}
	\end{itemize}\leavevmode
\end{proof}
\begin{proposition}
	Let $H_t$ and $G_t$ be two log-ergodic processes with\\
	$\mathbb{E}[H_t]=m<\infty$ and $\mathbb{E}[G_t]= n<\infty$. Then, the following statements hold.
	\begin{itemize}
		\item[1.] $G_t+H_t$ is mean ergodic.
		\item[2.] For all $\nu\in \mathbb{R}$, $\nu H_t$ is log-ergodic.
		\item[3.] 	$G_t\cdot H_t$ is log-ergodic.
	\end{itemize}
\end{proposition}
\begin{proof}
	\begin{itemize}
		\item[1.] Define $K_t=G_t+H_t$. Calculating the covariance of the $K_t$ process we have:
		\begin{align*}
			\mathbb{C}_{kk}&=\mathbb{E}[K_t^2]-(\mathbb{E}[K_t])^2\\
			&=\mathbb{E}[G_t^2+H_t^2+2G_tH_t]-(\mathbb{E}[G_t+H_t])^2\\
			&=\mathbb{E}[G_t^2]+\mathbb{E}[H_t^2]+2\mathbb{E}[G_t]\mathbb{E}[H_t]-(\mathbb{E}[G_t]+\mathbb{E}[H_t])^2\\
			&=\mathbb{E}[G_t^2]-(\mathbb{E}[G_t])^2+\mathbb{E}[H_t^2]-(\mathbb{E}[H_t])^2\\
			&=\mathbb{E}[G_t-\mathbb{E}[G_t]]^2+\mathbb{E}[H_t-\mathbb{E}[H_t]]^2\\
			&=\mathbb{E}[G_t-n]^2+\mathbb{E}[H_t-m]^2\\
			&=n^2+n^2-2n^2+m^2+m^2-2m^2=0.
		\end{align*}
		Placing the calculated covariance in \ref{limer}, we obtain: $\overline{<K>}=0$.
		\item[2.] Using \ref{I} and \ref{limer} we have:
		\begin{align*}
			\overline{<\ln(\nu H)>}=&\overline{<\ln(\nu)+Y>}\\
			=&\overline{<\ln(\nu)>}+\overline{<Y>}=0+\overline{<Y>}=0.
		\end{align*}
		\item[3.] Using \ref{I} and \ref{limer} we have:
		\begin{align*}
			\overline{<\ln(G\cdot H)>}=&\overline{<\ln(G)+\ln(H)>}\\
			=&\overline{<\ln(G)>}+\overline{<\ln(H)>}\\
			=&\overline{<Y>}+\overline{<Z>}=0.
		\end{align*}
	\end{itemize}\leavevmode
\end{proof}
\begin{theorem}\label{thr}
	If the positive stochastic process $X_t$ is log-ergodic, then the process $Z_t=\xi_{t,W_t}^{\beta}[\ln(X_t)]$ has the ergodic recurrence (mean reversion) property.
\end{theorem}
\begin{proof}
	Using the definition of log-ergodicity, it follows that the relation \ref{limer} holds for the process $Z_t$. Therefore, from the classical definition of mean ergodicity, we have \cite{1,59}:
	\begin{equation*}
		\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T Z_t dt=\mathbb{E}[Z_t].
	\end{equation*}
	For $i\in \mathbb{N}\cup\{0\}$, let $t_i$ represent the times that the process $Z_t$ meets it's mean along the path generated by $\omega_0\in \mathcal{F}$ (as shown in figure \ref{fig3}). It can be written:
	\begin{equation*}
		\mathbb{P}(\forall i \in \mathbb{N}\cup\{0\}, \quad Z_{t_i}(\omega_0)=\mathbb{E}[Z_{t_i}(\omega_0)])=1.
	\end{equation*}
	Consequently:
	\begin{equation*}
		\mathbb{P}(\exists t_i>T,\quad Z_{t_i}(\omega_0)=\mathbb{E}[Z_{t_i}(\omega_0)])=1.
	\end{equation*}
	Therefore, it follows from Poincaré's recurrence theorem \cite{1} that the process $Z_t$ has the ergodic recurrence property.
\end{proof}
\begin{figure}[H]
	\begin{center}

		\includegraphics[scale=0.5]{figure2.eps}
		\caption{\small{A zoomed sample path of the process $Z_t$, for small time scales, generated by $\omega_0\in \Omega$ with arbitrary time $t_0$, in which the process $Z_t(\omega_0)$ returns to $\mathbb{E}[Z_{t}(\omega_0)]$.}}
		\label{fig3}
	\end{center}
\end{figure}
\section{Log-ergodic processes: a method for constructing mean ergodic processes}\label{5}
\subsection{Mean reversion models}
\begin{proposition}
	Any stochastic process with mean reverting property is mean ergodic.
\end{proposition}
\begin{proof}
	From \cite{95,96} we know that every mean reverting stochastic process is stationary. Let $r_t(\cdot)$ be a stochastic process with mean reversion property and $\mathbb{E}[r_t]<\infty$. From stationarity it follows that,
	\begin{equation*}
		\mathbb{E}[r_t]=\mathbb{E}[r_{t+\delta}], \quad \forall t,\delta>0.
	\end{equation*}
	Let $\tau_0>0$ represent the time that the process $r_t$ meets it's mean along the path generated by $\omega_0$. According to Poincaré recurrence theorem \cite{1} and theorem \ref{thr}, we can write:
	\begin{equation*}
		r_{\tau_0}(\omega_0)=\mathbb{E}[r_{\tau_0}(\omega_0)].
	\end{equation*}
	Computing the covariance of the process in the time $\tau_0$ we obtain:
	\begin{align}
		\mathbb{C}_{rr}(\tau_0)&=\mathbb{E}[r_{\tau_0}^2]-(\mathbb{E}[r_{\tau_0}])^2<\infty\notag\\

		&=\mathbb{E}[\mathbb{E}[r_{\tau_0}]\mathbb{E}[r_{\tau_0}]]-(\mathbb{E}[r_{\tau_0}])^2\notag\\
		&=(\mathbb{E}[r_{\tau_0}])^2-(\mathbb{E}[r_{\tau_0}])^2=0\notag.
	\end{align}
	Therefore, according to \ref{limer}, the stochastic process $r_t$ has to be mean ergodic.
\end{proof}
\subsection{Main theorem}
In this subsection, we present the key theorem of the paper. In section 6, we will use the results of this section to prove the theorem.
\begin{theorem}(Main theorem)\label{maintheo}
	Suppose that the price process, $S_t$, of an asset has the form:
	\begin{align*}
		S_t&=S_0e^{Y_t},\quad S_0=s,
	\end{align*}
	with
	\begin{align*}
		Y_t^\prime&=Y_0^\prime+\int_0^t \mu_{u,s}du+\int_0^t \sigma dW_u,\quad Y_0^\prime=\ln(s)+Y_0,\quad Y_0=0,\\
		\sigma&=f(V_t).
	\end{align*}
	Where $\mu_{t,s}$ is a function of $t$ and $s$, $V_t$ is an arbitrary random process, and $\sigma$ is a function of the random process $V_t$. Then, the process $S_t$ is log-ergodic with respect to the ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$ defined in \ref{xi}.
\end{theorem}
In the remaining part of this section, we prove some results that will be used in the proof of the main theorem.
\begin{theorem}\label{mainth}
	The stochastic process $Z_t$ defined by \ref{zt} is mean ergodic. In other words, the positive stochastic process $X_t$ is log-ergodic with respect to ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$.
\end{theorem}
\begin{proof}
	It follows from \cite{57,21,13} that $Z_t$ is a Markov process. $Z_t$ is stationary by \cite{58} and \ref{mm}. Therefore, $Z_t$ meets the requirements to be ergodic in the mean, as stated in \cite{59,35,68}. Hence, we first calculate the covariance of the $Z_t$ process.
	\begin{align}
		\mathbb{C}_{zz}(\tau)&=\mathbb{V}ar[Z_\tau]=\mathbb{E}[Z_\tau^2],\notag\\
		&=\frac{1}{\tau^{2\beta}}\mathbb{E}\big[(\int_0^\tau\sigma_sdW_s)^2\big]+\frac{1}{\tau^{2\beta}}\mathbb{E}\big[W_\tau^2(\int_0^\tau\mu_sds)^2\big]\notag\\
		+&\frac{2\mathbb{E}[W_\tau]}{\tau^{2\beta}}\mathbb{E}\big[\int_0^\tau\sigma_sdW_s\int_0^\tau\mu_sds\big]\notag\\

		\text{Itô isometry}\Rightarrow=&\frac{1}{\tau^{2\beta}}\mathbb{E}[\int_0^\tau\sigma_s^2ds]+\frac{1}{\tau^{2\beta-1}}\mathbb{E}[(\int_0^\tau\mu_sds)^2]\notag\\

		=&\frac{1}{\tau^{2\beta}}\int_0^\tau\sigma_s^2ds+\frac{1}{\tau^{2\beta-1}}(\int_0^\tau\mu_sds)^2\notag.

	\end{align}
	Now it suffices to prove that the limit in \ref{limer} is equal to zero.
	\begin{align}
		\overline{<Z>}=&\lim_{T\rightarrow \infty}\frac{1}{T}\big[\int_0^T(1-\frac{\tau}{T})\frac{1}{\tau^{2\beta}}(\int_0^\tau\sigma_s^2ds)d\tau\notag\\
		&+\int_0^T(1-\frac{\tau}{T})\frac{1}{\tau^{2\beta-1}}(\int_0^\tau\mu_sds)^2d\tau.\label{intt}
	\end{align}
	First integral:
	\begin{align}
		&\int_0^T(1-\frac{\tau}{T})\frac{1}{\tau^{2\beta}}(\int_0^\tau\sigma_s^2ds)d\tau\notag\\
		=&\frac{1}{(-2\beta+1)T^{2\beta-1}}(\int_0^T\sigma_s^2ds)-\int_0^T\frac{1}{(-2\beta+1)\tau^{2\beta-1}}\frac{d(\int_0^\tau\sigma_s^2ds)}{d\tau}d\tau\notag\\
		&-\frac{1}{T}\big[\int_0^T\frac{1}{\tau^{2\beta-1}}(\int_0^\tau\sigma_s^2ds)d\tau\big]\notag\\
		=&\frac{1}{(-2\beta+1)T^{2\beta-1}}(\int_0^T\sigma_s^2ds)-\frac{1}{(-2\beta+1)T^{2\beta-1}}(\int_0^T\sigma_s^2ds)\notag\\

		&-\frac{1}{T}\big[\frac{(\int_0^T\sigma_s^2ds)}{(-2\beta+2)T^{2\beta-2}}-\frac{(\int_0^T\sigma_s^2ds)}{(-2\beta+2)T^{2\beta-2}}\big]=0\notag.
	\end{align}
	Second integral:
	\begin{align}
		&\int_0^T(1-\frac{\tau}{T})\frac{1}{\tau^{2\beta-1}}(\int_0^\tau\mu_sds)^2d\tau\notag\\
		=&\frac{1}{(-2\beta+2)T^{2\beta-2}}(\int_0^T\mu_sds)^2-\int_0^T\frac{2(\int_0^{\tau}\mu_sds)}{(-2\beta+2)\tau^{2\beta-2}}\frac{d(\int_0^\tau\mu_s ds)}{d\tau}d\tau\notag\\
		&-\frac{1}{T}\big[\int_0^T\frac{1}{\tau^{2\beta-2}}(\int_0^\tau\mu_sds)^2d\tau\big]\notag\\

		=&\frac{1}{(-2\beta+2)T^{2\beta-2}}(\int_0^T\mu_sds)^2-\frac{1}{(-2\beta+2)T^{2\beta-2}}(\int_0^T\mu_sds)^2\notag\\
		&-\frac{1}{T}\big[\frac{1}{(-2\beta+3)T^{2\beta-3}}(\int_0^T\mu_sds)^2-\frac{1}{(-2\beta+3)T^{2\beta-3}}(\int_0^T\mu_sds)^2\big]\notag\\
		=&0.\notag
	\end{align}
	Consequently, the relation \ref{limer} holds and we have: $\overline{<Z>}=0$. Which means the process $Z_t$ is mean ergodic.
\end{proof}

We can use the Ornstein–Uhlenbeck process \cite{84}, the geometric Brownian motion process \cite{13}, and the Jump-Diffusion process \cite{68} as examples of positive log-ergodic processes in the form of relation \ref{X}, which are all log-ergodic with respect to ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$, according to theorem \ref{mainth}.
\subsection{Log-ergodic Lévy processes}
Let $Y_t$ be a Lévy process. Lévy processes are considered to be well-behaved processes from the perspective of ergodic theory since their increments are stationary and independent. The independence of the increments is what gives the Lévy process its Markov property \cite{86}. Therefore, the process $Y_t$ satisfies the requirements for ergodicity.

Matthias Winkel \cite{86} has stated that every Lévy process is infinitely divisible and for every infinitely divisible distribution there corresponds a Lévy process almost surely. Therefore, suppose that the distribution of $Y_1$ is parameterized by $(\eta,\sigma^2,\nu)$. Using Lévy-Itô theorem \cite{85,86}, we decompose process $Y_t$ as
\begin{equation}
	Y_t=\eta t+\sigma W_t+J_t+M_t.\label{levy}
\end{equation}
Where $W_t$ is a Wiener process, for $t\geq0$, $\Delta Y_t=Y_t-Y_{t^-}$ is a Poisson process with intensity $\nu$,
$$J_t=\sum_{s\leq t}\Delta Y_s\boldsymbol{1}_{\{\lvert \Delta Y_s \rvert >1\}},$$
and $M_t$ is a martingale. Using the ergodic maker operator \ref{xi} for the process $Y_t$, we obtain:
\begin{align*}
	Z_t=\frac{\eta W_t}{t^{\beta-1}}+\frac{W_t}{t^{\beta}}\big[\sigma+J_t+M_t\big].
\end{align*}
\begin{proposition}
	If $Y_t$ is a Lévy process, then $X_t=X_0e^{Y_t}$, with $X_0=x$, is log-ergodic with respect to the ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$.
\end{proposition}
\begin{proof}
	Calculating the covariance of the $Z_t$ process, we have:
	\begin{align*}
		\mathbb{C}_{zz}(\tau)&=\mathbb{E}[Z_\tau^2]\\

	&=\mathbb{E}[\frac{\eta^2 W_\tau^2}{\tau^{2\beta-2}}+\frac{W_\tau^2}{\tau^{2\beta}}[\sigma+J_\tau+M_\tau]^2+\frac{2\eta W_\tau^2}{\tau^{2\beta-1}}[\sigma+J_\tau+M_\tau]]\\

&=\frac{\eta^2}{\tau^{2\beta-3}}+\frac{1}{\tau^{2\beta-1}}[\sigma+J_\tau+M_\tau]^2+\frac{2\eta}{\tau^{2\beta-2}}[\sigma+J_\tau+M_\tau].
\end{align*}
Now it suffices to prove \ref{limer} holds.
\begin{align}
\overline{<Z>}=&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T (1-\frac{\tau}{T})[\frac{\eta^2 }{\tau^{2\beta-3}}+\frac{1}{\tau^{2\beta-1}}[\sigma+J_\tau+M_\tau]^2\notag\\
&+\frac{2\eta}{\tau^{2\beta-2}}[\sigma+J_\tau+M_\tau]]d\tau\notag\\
=&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T (1-\frac{\tau}{T})\frac{\eta^2 }{\tau^{2\beta-3}}d\tau\notag\\
&+\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T (1-\frac{\tau}{T})\frac{1}{\tau^{2\beta-1}}[\sigma+J_\tau+M_\tau]^2d\tau\notag\\
&+\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T (1-\frac{\tau}{T})\frac{2\eta}{\tau^{2\beta-2}}[\sigma+J_\tau+M_\tau]d\tau.\label{th}
\end{align}
First integral:
\begin{align*}
\int_0^T (1-\frac{\tau}{T})\frac{\eta^2 }{\tau^{2\beta-3}}d\tau
=\frac{\eta^2}{(-2\beta+4)T^{2\beta-4}}-[\frac{\eta^2}{(-2\beta+5)T^{2\beta-4}}].
\end{align*}
Second integral:
\begin{align*}
&\int_0^T(1-\frac{\tau}{T})\frac{1}{\tau^{2\beta-1}}[\sigma+J_\tau+M_\tau]^2d\tau\\
=&\frac{[\sigma+J_T+M_T]^2}{(-2\beta+2)T^{2\beta-2}}-\int_0^T\frac{2[\sigma+J_\tau+M_\tau]}{(-2\beta+2)\tau^{2\beta-2}}\frac{d[\sigma+J_\tau+M_\tau]}{d\tau}d\tau\notag\\
&-\frac{1}{T}[\frac{[\sigma+J_T+M_T]^2}{(-2\beta+3)T^{2\beta-3}}-\frac{[\sigma+J_T+M_T]^2}{(-2\beta+3)T^{2\beta-3}}]=0.
\end{align*}
Third integral:
\begin{align*}
&\int_0^T (1-\frac{\tau}{T})\frac{2\eta}{\tau^{2\beta-2}}[\sigma+J_\tau+M_\tau]d\tau\\
=&2\eta\Big[\frac{\sigma+J_T+M_T}{(-2\beta+3)T^{2\beta-3}}-\int_0^T\frac{1}{(-2\beta+3)\tau^{2\beta-3}}\frac{d[\sigma+J_\tau+M_\tau]}{d\tau}d\tau\notag\\
&-\frac{1}{T}[\frac{\sigma+J_T+M_T}{(-2\beta+4)T^{2\beta-4}}-\frac{\sigma+J_T+M_T}{(-2\beta+4)T^{2\beta-4}}]\Big]=0.
\end{align*}
Now we substitute the calculated integrals in \ref{th}, and obtain $\overline{<Z>}=0$. Therefore, the process $Y_t$ defined in \ref{levy} is mean ergodic with respect to the ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$.
\end{proof}
\subsection{Bounded processes}
\begin{proposition}
The paths of the non-negative bounded stochastic process $Y_t$ are mean ergodic with respect to ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$.
\end{proposition}
\begin{proof}
Let $Y_t(\omega_0)$ be the path of the process $Y_t$ generated by $\omega_0\in \Omega$, and\\ $\xi_{t,W_t}^{\beta}[Y_t]=Z_t$. According to the definition of boundedness with respect to $\omega_0$, we have:
\begin{equation*}
\lvert Y_t(\omega_0)\rvert \leq M.
\end{equation*}
For some positive number $M$. Therefore, it can be written:
\begin{align*}
&Y_t(\omega_0)\leq \sup_{t\in[0,T]} Y_t(\omega_0)\\
\Rightarrow& \int_0^T \xi_{t,W_t}^\beta[Y_t(\omega_0)] dt\leq \int_0^T \sup_{t\in[0,T]} \xi_{t,W_t}^\beta[Y_t(\omega_0)] dt\\
\Rightarrow& \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T \underbrace{\xi_{t,W_t}^\beta[Y_t(\omega_0)]}_{Z_t(\omega_0)} dt\leq \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\sup_{t\in [0,T]}\underbrace{\xi_{t,W_t}^\beta[Y_t(\omega_0)]}_{Z_t(\omega_0)} dt.
\end{align*}
From \cite{87}, for the covariance of the process $Z_t$, we consider the following boundary:
\begin{equation*}
-\sqrt{{\mathbb{V}ar}^2\big[\xi_{\tau,W_\tau}^\beta[Y_\tau(\omega_0)]\big]}\leq \mathbb{C}_{zz}(\tau)\leq\sqrt{{\mathbb{V}ar}^2\big[\xi_{\tau,W_{\tau}}^\beta[Y_{\tau}(\omega_0)]\big]}.
\end{equation*}
Thus, by definition \ref{limer} we get:
\begin{align}\label{vv}
&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau\notag\\
\leq&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\big \lvert \mathbb{V}ar\big[\xi_{\tau,W_{\tau}}^\beta[Y_{\tau}(\omega_0)]\big]\big \rvert d\tau.
\end{align}
Let $\sup_{t\in[0,T]} Y_t(\omega)=M_t(\omega)$, and $\inf_{t\in[0,T]} Y_t(\omega)\geq0$,
for any $\omega\in \Omega$. We have:
\begin{align*}
0&\leq Y_t \leq M_t(\omega)\Rightarrow 0\leq \xi_{t,W_t}^\beta[Y_t]\leq \xi_{t,W_t}^\beta[M_t(\omega)]\\
\Rightarrow 0&\leq Z_t^2\leq (\xi_{t,W_t}^\beta[M_t(\omega)])^2\\
\Rightarrow 0&\leq \mathbb{E}[Z_t^2]\leq \mathbb{E}[(\xi_{t,W_t}^\beta[M_t(\omega)])^2].
\end{align*}
It follows from lemma \ref{mm} that:
\begin{equation}\label{varz}
0\leq\mathbb{V}ar[Z_t]\leq\mathbb{E}[(\xi_{t,W_t}^\beta[M_t(\omega)])^2].
\end{equation}
Now, using \ref{varz} in \ref{vv} yields
\begin{align*}
&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau\\
\leq&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{E}[(\xi_{t,W_t}^\beta[M_t(\omega)])^2]d\tau.
\end{align*}
Let $\mu_\xi(t)=\mathbb{E}[(\xi_{t,W_t}^\beta[M_t(\omega)])^2]$. According to \ref{xi}, it follows that $\mu_\xi(t)<\infty$. Thus, we have:
\begin{align*}
\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau\leq
\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mu_\xi(\tau)d\tau.
\end{align*}
To calculate the integral on the right hand side, we write:
\begin{align}
&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mu_\xi(\tau)d\tau\notag\\
&=\lim_{T\rightarrow\infty}\big[\frac{1}{T}\int_0^T\mu_\xi(\tau)d\tau-\frac{1}{T^2}\int_0^T \tau\mu_\xi(\tau)d\tau\big].\label{re}
\end{align}
Using \ref{IDV2} it follows that the first integral in \ref{re} equals to zero. To calculate the second integral, we proceed as follows.
\begin{align*}
&\lim_{T\rightarrow \infty}\frac{1}{T^2}[\frac{1}{2}T^2\mu_\xi(T)-\frac{1}{2}\int_0^T \tau^2 d\mu_\xi(\tau)]\\
=&\lim_{T\rightarrow \infty}\frac{1}{T^2}[\frac{1}{2}T^2\mu_\xi(T)-\frac{1}{2}T^2\mu_\xi(T)]=0.
\end{align*}
Therefore, it follows that:
\begin{align*}
0\leq\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau\leq 0.
\end{align*}
Consequently we have:
\begin{align*}
\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})\mathbb{C}_{zz}(\tau)d\tau=0.
\end{align*}
\end{proof}
\begin{example}
For $t>0$, the following statements hold.
\begin{itemize}
\item [1.] The process $Y_t=\sin(W_t)$ is mean ergodic.
\item [2.] The process $Y_t=\gamma\sin(\mu t+\sigma W_t)$ where $\gamma$, $\mu$, and $\sigma$ are constants, is mean ergodic.
\end{itemize}
\end{example}
\begin{solution}  \textbf{1.}
Using Itô lemma it can be written:
\begin{align*}
&d\sin(W_t)=\cos(W_t)dW_t-\frac{1}{2}\sin(W_t)dt\\
&\mathbb{E}[\sin(W_t)]=-\frac{1}{2}\int_0^t\mathbb{E}[\sin(W_s)]ds\\
&d\mathbb{E}[\sin(W_t)]=-\frac{1}{2}\mathbb{E}[\sin(W_t)]dt\\
&\frac{d\mathbb{E}[\sin(W_t)]}{\mathbb{E}[\sin(W_t)]}=-\frac{1}{2}dt\\
&d\ln(\mathbb{E}[\sin(W_t)])=-\frac{1}{2}dt\Rightarrow \mathbb{E}[\sin(W_t)]=e^{\frac{-t}{2}}.
\end{align*}
Calculating the covariance of $\sin(W_t)$ we get:
\begin{align*}
\mathbb{C}_{yy}(\tau)=e^{-2\tau}-e^{\frac{-\tau}{2}}.
\end{align*}
Now, we calculate the limit in the definition \ref{limer} as follows.
\begin{align*}
&\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T(1-\frac{\tau}{T})(e^{-2\tau}-e^{\frac{-\tau}{2}})d\tau\\
=&\lim_{T\rightarrow \infty}\frac{1}{T}\Big[-\frac{1}{2}e^{-2T}+2e^{\frac{-T}{2}}+\frac{1}{T}[\frac{(2T+1)e^{-2T}}{4}-2(T+2)e^{\frac{-T}{2}}]\Big]\\
=&\lim_{T\rightarrow \infty}\frac{1}{T}\Big[\frac{1}{4Te^{2T}}-\frac{4}{Te^{\frac{T}{2}}}\Big]=0.
\end{align*}
\end{solution}

\begin{solution}  \textbf{2.}
We calculate the expectation (ensemble average) of the process $Y_t$:
\begin{align*}
\mathbb{E}[Y_t]=&\mathbb{E}[\gamma\sin(\mu t+\sigma W_t)]\\
=&\int_{-\infty}^{\infty}\gamma\sin(\mu t+\sigma W_t)f_{ Y}(y)dy\\
=&\int_0^{2\pi} \gamma\sin(\mu t+\sigma W_t)\frac{1}{2\pi}dW_t\\
=&\frac{\gamma}{2\pi}\int_{0}^{2\pi}\sin(\mu t+\sigma W_t)dW_t=0.
\end{align*}
Computing the time average we obtain the following:
\begin{align*}
<Y>&=\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T\gamma\sin(\mu t+\sigma W_t)dt\\
&=\lim_{T\rightarrow\infty}\frac{\gamma}{T}\int_0^T\sin(\mu t+\sigma W_t)dt=0.
\end{align*}
Therefore, we have $<Y>=\mathbb{E}[Y_t]$. Which implies that the process $Y_t$ is mean ergodic.
\end{solution}
\begin{example}
It is proven in paper \cite{108} that a Markov chain that models a process confined to a bounded interval, exhibits ergodic behavior, with the process being constantly attracted to the center of the interval.
\end{example}

\section{Proof of the main theorem: Stochastic Volatility Case}\label{se6}
In this section we prove the main theorem of the paper, which we stated in section \ref{5}.
\begin{theorem}(Main theorem)
Suppose that the price process, $S_t$, of an asset has the form:
\begin{align*}
S_t&=S_0e^{Y_t},\quad S_0=s,
\end{align*}
with
\begin{align*}
Y_t^\prime&=Y_0^\prime+\int_0^t \mu_{u,s}du+\int_0^t \sigma dW_u,\quad Y_0^\prime=\ln(s)+Y_0,\quad Y_0=0,\\
\sigma&=f(V_t).
\end{align*}
Where $\mu_{t,s}$ is a function of $t$ and $s$, $V_t$ is an arbitrary random process, and $\sigma$ is a function of the random process $V_t$. Then, the process $S_t$ is log-ergodic with respect to the ergodic maker operator $\xi_{t,W_t}^\beta[\cdot]$ defined in \ref{xi}.
\end{theorem}
\begin{proof}
We write:
\begin{align*}
Y_t^\prime&=\ln(s)+\int_0^t \mu_{u,s} du+\int_0^t f(V_u) dW_u,\\
Z_t&=\xi_{t,W_t}^\beta[Y_t^\prime]=\frac{W_t\int_0^t \mu_{u,s}du}{t^{\beta}}+\frac{\int_0^t f(V_u)dW_u}{t^{\beta}},\quad Z_0=0.
\end{align*}
Now we calculate the covariance of $Z_t$:
\begin{align*}
\mathbb{C}_{zz}(\tau)=\frac{\mathbb{E}[\int_0^\tau \mu_{u,s}du]}{\tau^{2\beta-1}}+\frac{\mathbb{E}[(\int_0^\tau f(V_u)dW_u)^2]}{\tau^{2\beta}}.
\end{align*}
Next, we prove that \ref{limer} holds.
\begin{align*}
\overline{<Z>}=&\lim_{T\rightarrow \infty}\frac{1}{T}\big[\int_0^T(1-\frac{\tau}{T})\frac{\mathbb{E}[\int_0^\tau\mu_{u,s}du]}{\tau^{2\beta-1}}d\tau\notag\\
&+\int_0^T(1-\frac{\tau}{T})\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}} d\tau\big].
\end{align*}
It follows from theorem \ref{mainth} that the first integral equals to zero. Therefore, it suffices to prove
\begin{align*}
\lim_{T\rightarrow \infty}\frac{1}{T}\big[\int_0^T(1-\frac{\tau}{T})\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}} d\tau\big]=0.
\end{align*}
The volatility term at any time $t$ is bounded \cite{13}. Therefore, there are positive numbers $M_1$, and $M_2$ such that:
$$0<M_1\leq f(V_t) \leq M_2, \quad a.s.$$
Thus, we have:
\begin{align*}
M_1^2&\leq f^2(V_t)\leq M_2^2,\\
\int_0^\tau M_1^2 du&\leq \int_0^\tau f^2(V_u)du\leq \int_0^\tau M_2^2 du,\\
\mathbb{E}[M_1^2\tau]&\leq \mathbb{E}[\int_0^\tau f^2(V_u)du]\leq \mathbb{E}[M_2^2\tau],\\
\frac{M_1^2}{\tau^{2\beta-1}}&\leq \frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}}\leq \frac{M_2^2}{\tau^{2\beta-1}}.
\end{align*}
As a result we get:
\begin{align}
\int_0^T\frac{M_1^2}{\tau^{2\beta-1}}d\tau&\leq \int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}}d\tau\leq \int_0^T\frac{M_2^2}{\tau^{2\beta-1}}d\tau,\notag\\
\frac{M_1^2}{(-2\beta+2)T^{2\beta-2}}&\leq \int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}}d\tau\leq \frac{M_2^2}{(-2\beta+2)T^{2\beta-2}},\notag\\
\lim_{T\rightarrow \infty}\frac{M_1^2}{(-2\beta+2)T^{2\beta-1}}&\leq \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}}d\tau\notag\\
&\leq \lim_{T\rightarrow \infty}\frac{M_2^2}{(-2\beta+2)T^{2\beta-1}},\notag\\
\Rightarrow\quad 0&\leq \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta}}d\tau\leq 0.\label{I1}
\end{align}
Also we have:
\begin{align}
\int_0^T\frac{1}{T}\frac{M_1^2}{\tau^{2\beta-2}}d\tau&\leq \int_0^T\frac{1}{T}\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta-1}}d\tau\leq \int_0^T\frac{1}{T}\frac{M_2^2}{\tau^{2\beta-2}}d\tau,\notag\\
\lim_{T\rightarrow\infty}\frac{M_1^2}{(-2\beta+3)T^{2\beta-1}}&\leq \lim_{T\rightarrow \infty}\frac{1}{T^2}\int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta-1}}d\tau\notag\\
&\leq \lim_{T\rightarrow \infty}\frac{M_2^2}{(-2\beta+3)T^{2\beta-1}},\notag\\
\Rightarrow\quad 0&\leq \lim_{T\rightarrow \infty}\frac{1}{T^2}\int_0^T\frac{\mathbb{E}[\int_0^\tau f^2(V_u)du]}{\tau^{2\beta-1}}d\tau\leq 0.\label{I2}
\end{align}
Now it follows from \ref{I1} and \ref{I2} that $\overline{<Z>}=0$.
\end{proof}

\section{Application of log-ergodic processes in mathematical finance}\label{se7}
The benefit of using log-ergodic processes is that in the long run, time averaging takes the role of expectation in computations. These procedures will be used to model leveraged futures trading and calculating the mean reversion time in subsequent studies. Reducing the randomness edge of a financial model, reduces the risk of trading and therefore allows us to enter or leave a trading position with a lower risk. We will show later in this section that the Black-Scholes partial differential equation is modified by log-ergodic processes, which also incorporate the inhibition degree parameter into this equation.

In the following, we will study the behavior of log-ergodic processes using empirical data and express a novel version of the Black-Scholes partial differential equation by providing an example concerning the simulation of the mean ergodic process $Z_t$.
\begin{example}
Consider the empirical data of Tesla stock price from December 14, 2011 to December 14, 2021. This data was extracted from the Yahoo Finance \footnote{\url{https://finance.yahoo.com}} website. Considering the stock price process, $S_t$, of this company follows the geometric Brownian motion:
\begin{align*}
S_t&=S_0\exp\{(\mu-\frac{1}{2}\sigma^2)t+\sigma W_t\},\quad S_0=s.\\
Y_t^\prime&=\ln(S_t)=Y_0^\prime+(\mu-\frac{1}{2}\sigma^2)t+\sigma W_t,\quad Y_0^\prime=\ln(s).
\end{align*}
Where $\mu$ and $\sigma$ are constants, and $W_t$ is a standard Wiener process. Using the ergodic maker operator for $t>0$, we have:
$$Z_t=Z_0+\frac{(\mu-\frac{1}{2}\sigma^2)W_t}{t^{\beta-1}}+\frac{\sigma W_t}{t^{\beta}},\quad Z_0=0.$$
A random path of $Z_t$ for the data of the Tesla for $\beta=2$ is shown in figure \ref{fig2}.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.55]{figure3.eps}
\caption{\small{The random path of the process $Z_t$ for the Tesla price data for $\beta=2$ and a 300-day time frame.}}
\label{fig2}
\end{center}
\end{figure}
\end{example}
The reference level that we will use in future studies is the zero mean level (the red line in figure \ref{fig2}). Similar to the recent work of Ananova, Cont, and Xu \cite{110}, we divide the path of the process $Z_t$ into two parts: one above and one below the reference level. According to mean reverting property of $Z_t$, the process eventually meets the reference level. We take a sell position if the process path is above the reference level, and a buy position if it is below the reference level.\\
Let $q=\mu-\frac{1}{2}\sigma^2$. Using Itô lemma, we have:
\begin{align}
dZ_t&=\frac{1}{2}\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]dt+\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]dW_t
-\big[\frac{(\beta-1)W_t}{t^{\beta}}+\frac{\beta\sigma W_t}{t^{\beta+1}}\big]dt,\notag\\
&=\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]\big[\frac{1}{2}dt+dW_t\big]-t\big[\frac{(\beta-1)qW_t}{t^{\beta-1}}+\frac{\beta \sigma W_t}{t^{\beta}}\big]dt,\notag\\
&=\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]\big[\frac{1}{2}dt+dW_t\big]-\big[\beta Z_t-\frac{qW_t}{t^{\beta-1}}\big]tdt,\notag\\
\Rightarrow dZ_t&=\underbrace{\Big[\frac{1}{2}\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]-\beta t Z_t+\frac{qW_t}{t^{\beta-2}}\Big]}_{A(t,W_t)}dt+\underbrace{\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]}_{B_t}dW_t,\notag\\
dZ_t&=A(t,W_t)dt+B_tdW_t.
\end{align}
According to the above assumptions, we have the following result:
\begin{proposition}(Ergodic Partial differential equation of Black–Scholes)
Under the assumptions of the Black-Scholes model, the European call option price, $C(Z_t,t)$, relative to the stock price $Z_t=Z$, with respect to short rate $r$, inhibition degree parameter $\beta$, and the strike price $K$ satisfies in the following partial differential equation.
\begin{align*}
&\frac{\partial C}{\partial t}+(A(t,W_t)+r)\frac{\partial C}{\partial Z}+\big(\frac{1}{2}B_t^2-A(t,W_t)\big)\frac{\partial^2 C}{\partial Z^2}-rC=0,\notag\\
&\text{for} \quad 0<\lvert Z\rvert<\infty,\quad 0<t<T,
\end{align*}
together with initial conditions $C(0,t)=0$ and $C(Z,T)=(\lvert Z\rvert-\ln(K))^+$.
\end{proposition}
\begin{proof}
We think of $Z_t$ as the price process of a traded stock and form a risk hedging basket, including $x$ shares with the price $Z$ and one unit of call option with a sell position \cite{13}. For the price $V_t:=V(Z,t)$ of this basket, we have:
\begin{align}
V_t&=-C(Z,T)+xZ,\notag \\
dV_t&=-dC(Z,T)+xdZ.\label{V}
\end{align}
We set $C:=C(Z,T)$. Substituting the dynamics of $Z$ in \ref{V} and using Itô lemma, we have:
\begin{align}
dV_t=&-\frac{\partial C}{\partial t}dt-\frac{\partial C}{\partial Z}A(t,W_t)dt-\frac{\partial C}{\partial Z}B_tdW_t-\frac{1}{2}\frac{\partial^2 C}{\partial Z^2}B_t^2dt\notag\\
&+xA(t,W_t)dt+xB_tdW_t,\notag\\
dV_t=&-\Big[\frac{\partial C}{\partial t}+\frac{\partial C}{\partial Z}xA(t,W_t)-\frac{\partial C}{\partial Z}A(t,W_t)+\frac{1}{2}\frac{\partial^2 C}{\partial Z^2}B_t^2\Big]dt\notag\\
&+\Big[xB_t-\frac{\partial C}{\partial Z}B_t\Big]dW_t.\label{V1}
\end{align}
In order that the portfolio to be risk-free, the coefficient of $dW_t$ must be zero. We therefore have: $x=\frac{\partial C}{\partial Z}$.
Now, substituting the value of $x$ in \ref{V1}, we reach the relation:
\begin{align*}
dV_t=&-\Big[\frac{\partial C}{\partial t}+\frac{\partial C}{\partial Z}\big[A(t,W_t)-\frac{\partial C}{\partial Z}A(t,W_t)\big]+\frac{1}{2}\frac{\partial^2 C}{\partial Z^2}B_t^2\Big]dt.
\end{align*}
On the other hand by absence of arbitrage we have $dV_t=rV_tdt$ \cite{13}. Therefore, we have:
\begin{align}
&r\Big[-C+\frac{\partial C}{\partial Z}Z\Big]dt\notag\\
=&-\Big[\frac{\partial C}{\partial t}+\frac{\partial C}{\partial Z}\big[A(t,W_t)-\frac{\partial C}{\partial Z}A(t,W_t)\big]+\frac{1}{2}\frac{\partial^2 C}{\partial Z^2}B_t^2\Big]dt\notag.
\end{align}
Simplifying, we get the equation:
\begin{align*}
&\frac{\partial C}{\partial t}+(A(t,W_t)+r)\frac{\partial C}{\partial Z}+\big(\frac{1}{2}B_t^2-A(t,W_t)\big)\frac{\partial^2 C}{\partial Z^2}-rC=0.
\end{align*}
Where
\begin{align*}
&A(t,W_t)=\frac{1}{2}\big[\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}\big]-\beta t Z_t+\frac{qW_t}{t^{\beta-2}},\\
&B_t=\frac{q}{t^{\beta-1}}+\frac{\sigma}{t^{\beta}}, \quad q=\mu-\frac{1}{2}\sigma^2.
\end{align*}
The European call option is exercised when $\lvert Z\rvert>\ln(K)$. Therefore, the final condition for $t=T$ is $C(Z,T)=\max\big[\lvert Z\rvert-\ln(K),0\big]$.
Also about the boundary conditions we have:
\begin{align*}
&C(0,t)=0, \quad 0<t<T,\\
&C(Z,t)\sim\lvert Z\rvert-\ln(K)e^{-r(T-t)}, \quad \text{as} \quad \lvert Z\rvert\rightarrow \infty.
\end{align*}
\end{proof}

\section{Conclusion and future research} \label{se8}
In this paper, we provided a method to classify positive processes in terms of ergodic behavior using the Birkhoff ergodic theorem. We defined the new notion of ergodic maker operator, which gives a scale of control in financial market using the inhibition degree parameter. We studied the basic properties of this operator and gave some examples of log-ergodic processes.

Financial models that exhibit ergodic behavior in the mean can be studied using log-ergodic processes.

By means of log-ergodic processes, we will model futures trading strategy that include long and short positions in upcoming studies. Utilizing mean-recurrence property, we will be able to determine the ideal moment to take a trading position. Additionally, we investigate the pricing of financial derivatives using a model under ergodic circumstances, broadening the use of ergodic theory in mathematical finance.

\section*{Data Availability Statement}
All stock data used during this study are openly available from Yahoo Finance and Trading View websites as mentioned in the context.
\section*{Declaration of interest}
The authors have no conflicts of interest to declare.
Both authors have seen and agree with the contents of the manuscript and
there is no financial interest to report.

\bibliographystyle{plain}

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Tyler Cowen (Economics professor, George Mason U.) joins Dave to discuss his new book "Stubborn Attachments: A Vision for a Society of Free, Prosperous, and Responsible Individuals" covering topics like government regulation, why he identifies as a "small L" libertarian, and economic ideas like Universal Basic Income, climate change, the cryptocurrency revolution, his sensible plan for immigration etc.
There has been, not too surprisingly, much discussion of the government's idea of a compulsory 'gun buyback' scheme, see for example, Peter Cresswell at the Not PC blog. This legislation the Deputy PM says will cost somewhere around $300 million. An obvious question is what will this money buy us?
"The 1996-97 National Firearms Agreement (NFA) in Australia introduced strict gun laws, primarily as a reaction to the mass shooting in Port Arthur, Tasmania in 1996, where 35 people were killed. Despite the fact that several researchers using the same data have examined the impact of the NFA on firearm deaths, a consensus does not appear to have been reached. In this paper, we re-analyze the same data on firearm deaths used in previous research, using tests for unknown structural breaks as a means to identifying impacts of the NFA. The results of these tests suggest that the NFA did not have any large effects on reducing firearm homicide or suicide rates."
"In 1997, Australia implemented a gun buyback program that reduced the stock of firearms by around one-fifth. Using differences across states in the number of firearms withdrawn, we test whether the reduction in firearms availability affected firearm homicide and suicide rates. We find that the buyback led to a drop in the firearm suicide rates of almost 80 per cent, with no statistically significant effect on non-firearm death rates. The estimated effect on firearm homicides is of similar magnitude, but is less precise. The results are robust to a variety of specification checks, and to instrumenting the state-level buyback rate".
"Perhaps a more likely explanation of the strength of the relationship found is that the NFA led states with relatively weak legislation or enforcement relating to sale, ownership and storage of firearms to strengthen their regimes relative to states with initially stronger standards. There is evidence that states with relatively high firearm ownership and therefore high gun buyback rates also had relatively weak regulation prior to 1996. Then, our estimates need to be interpreted as reflecting a combination of both the removal of firearms and the relative strengthening of legislation and enforcement. We might expect to see smaller effects in the case of a buyback that was not accompanied by stricter firearm legislation".
Thus there was more going on in the Australian case than just a buyback, and it's difficult to know which bits of the reforms drive the results.
At the very least such results should highlight the need to very clear as to what we are talking about when discussing the likely effects of the government's proposed legislation. Are we talking about the whole package of reforms that the government wishes to introduce or are we just talking about the buyback scheme. It is possible that the whole package could have worthwhile effects, while the buyback scheme by itself would not. It is also possible that the whole package may not be worthwhile.
One reason for not rushing into any new legislation is to give time for a proper valuation of the empirical evidence to be done.
Lee, Wang-Sheng and Sandy Suardi (2008). "The Australian Firearms Buyback and Its Effect on Gun Deaths", Melbourne Institute Working Paper Series Working Paper No. 17/08 August.
Leigh, Andrew and Christine Neill (2010). "Do Gun Buybacks Save Lives? Evidence from Panel Data", IZA Discussion Paper No. 4995, June.
At the New Zealand Initiative, Matt Burgess makes an important point about the advantages of using price rather than policy to reduce emissions. Prices act like information signals that allocate resources to their best uses in the economy. This, as Matt points out, means that prices can do easily what policymakers find almost impossible, dealing with the trade-offs involved with choosing between different methods of reducing emissions. When should we stop investing in one particular method of emissions reduction and invest in other methods instead? Policymakers find this an almost insurmountable problem.
When governments want to reduce emissions, they have a choice between using policy or price.
Policy includes rules – for example, 100% of electricity must be generated from renewables – as well as incentive payments, such as electric vehicle subsidies.
Alternatively, governments can price carbon using cap-and-trade, or tax carbon directly.
The fact that emissions occur in millions of places in the economy strongly affects the relative performance of policy and price.
Consider the following question: At what share of renewable electricity does further investment in renewable electricity cease to be competitive with other ways of reducing emissions?
For policymakers, this is an astonishingly difficult question.
It is not just a matter of working out how the per tonne carbon abatement cost rises as the share of renewables approaches 100%. That is hard enough.
It is also about understanding the consequences for downstream users of electricity, who comprise the rest of the economy.
At a very high share of renewables, the cost of electricity will tend to increase. For downstream users, that affects emissions: If electricity costs more, they will be less willing to switch from petrol to electric vehicles, or to switch their industrial processes from coal or gas to electricity.
For policymakers, working out how the share of renewables affects overall emissions is impossibly complicated.
But for a carbon price, whether through cap-and-trade or a tax, discovery of the 'right' share of renewable electricity is easy.
Confronted with the relative cost of emissions-intensive coal and gas generation against green alternatives, buyers of electricity decide their willingness to pay.
For some users, green energy is attractive. For other users, coal and gas has real advantages and means a high willingness to pay.
For a problem like emissions, price enables discovery of the answer to non-obvious questions like how much coal and gas generation to retain. Price can access information that is lost to top-down policy.
Policy's disadvantage is measurable. A survey of the literature on the performance of government emissions reduction programmes reveals governments spend perhaps $5 to avoid harm from emissions worth $1, on average.
Under cap-and-trade like an Emissions Trading Scheme (ETS), retaining coal and gas generation does not increase overall emissions. These high-emissions generators stay in business only by outcompeting alternative emissions sources for the right to emit.
The government recently calling the ETS its "main tool" for achieving its emissions targets is a step in the right direction.
Moderating content in a polarized political climate while also respecting the value of free speech is a challenge still vexing social media companies. Thomas Kadri of the Yale Information Society Project comments.
A massive new plan unveiled by Democrats is a wish list of restrictions on free political speech. If this kind of attack on free speech can occur in the US why not New Zealand?
An interesting discussion of many of the issues to do with the effects of minimum wage increases and how to work them out and estimate how big they are. Well worth the hour to listen to.
A point worth making about the modern models of the theory of the firm is that they illustrate a general issue to do with post-1970 microeconomics, namely, the retreat from the use of general equilibrium (GE) models.
"[ ...] there were problems that could not be tackled within the Arrow-Debreu framework. These include money (attempts were made to develop a general-equilibrium theory of money, but they failed), information, and imperfect competition. In order to tackle such problems, economists were forced to use less general models, often dealing only with a specific part of the economy or with a particular problem. The search for ever more general models of general competitive equilibrium, that culminated in Theory of Value, was over" (Backhouse 2002: 262).
"[e]conomics not only requires a framework for organizing our ideas [which Walras provides], it requires also ideas to be organized. We need the right kind of language; we also need something to say. Substantive hypotheses about economic phenomena of the kind that were the goal of Cournot are an essential ingredient of a fruitful and meaningful economic theory. Walras has little to contribute in this direction; for this we must turn to other economists, notably, of course, to Alfred Marshall" (Friedman 1955: 908).
By the mid-1970s microeconomic theorists had largely turned away from Walras and back to Marshall, at least insofar as they returned to using partial equilibrium analysis to investigate economic phenomena such as strategic interaction, asymmetric information and economic institutions.
All the models considered in this book are partial equilibrium models, but in this regard, the theory of the firm is no different from most of the microeconomic theory developed since the 1970s. Microeconomics such as incentive theory, incomplete contract theory, game theory, industrial organisation, organisational economics etc, has largely turned its back, presumably temporarily, on GE theory and has worked almost exclusively within a partial equilibrium framework. This illustrates the point that there is a close relationship between the economic mainstream and the theory of the firm; when the mainstream forgoes general equilibrium, so does the theory of the firm.
"[t]he theory of contracts has evolved from the failures of general equilibrium theory. In the 1970s several economists settled on a new way to study economic relationships. The idea was to turn away temporarily from general equilibrium models, whose description of the economy is consistent but not realistic enough, and to focus on necessarily partial models that take into account the full complexity of strategic interactions between privately informed agents in well defined institutional settings".
The Foss, Lando and Thomsen classification scheme for the theory of the firm clearly illustrates the movement of the current theory of the firm literature away from GE towards partial equilibrium analysis. The scheme divides the contemporary theory into two groups based on which of the standard assumptions of GE theory is violated when modelling issues to do with the firm. The theories are divided into either a principal-agent group, based on violating the 'symmetric information' assumption, or an incomplete contracts group, based on the violation of the 'complete contracts' assumption. The reference point approach extends the incomplete contracts grouping to situations where ex-post trade is only partially contractible.
The introduction of the entrepreneur, as in the models proposed by Silver, Spulber and by Foss and Klein, also challenges, albeit in a different way, the standard GE model since, as William Baumol noted more than 40 years ago, the entrepreneur has no place in formal neoclassical theory.
"Contrast all this with the entrepreneur's place in the formal theory. Look for him in the index of some of the most noted of recent writings on value theory, in neoclassical or activity analysis models of the firm. The references are scanty and more often they are totally absent. The theoretical firm is entrepreneurless−the Prince of Denmark has been expunged from the discussion of Hamlet" (Baumol 1968: 66).
The reasons for this are not hard to find. Within the formal model, the 'firm' is a production function or production possibilities set, it is simply a means of creating outputs from inputs. Given input prices, technology and demand, the firm maximises profits subject to its production plan being technologically feasible. The firm is modelled as a single agent who faces a set of relatively uncomplicated decisions, e.g. what level of output to produce, how much of each input to utilise etc. Such 'decisions' are not decisions at all, they are simple mathematical calculations, implicit in the given conditions. The 'firm' can be seen as a set of cost curves and the 'theory of the firm' as little more than a calculus problem. In such a world there is a role for a 'decision maker' (manager) but no role for an entrepreneur.
"[ ...] the organization of the many institutions that govern economic relationships is entirely absent from these [GE] models. This is particularly striking in the case of firms, which are modeled as a production set. This makes the very existence of firms difficult to justify in the context of general equilibrium models, since all interactions are expected to take place through the price system in these models" (Salanie 2005: 1).
This would suggest that to make GE models a ubiquitous tool of microeconomic analysis - including the analysis of issues to do with non-market organisations such as the firm - developing models which can account for information asymmetries, contractual incompleteness, strategic interaction, the existence of institutions and the like is not so much desirable as essential. One catalyst for the development of such a new approach to GE is that partial equilibrium models can obscure the importance of the theory of the firm for overall resource allocation, a point which is more easily appreciated in a GE framework.


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Will Smith opens up on impact of Oscars slap incident on his next film Emancipation! Wed November 30th, 2022
Hollywood biggie Will Smith is all set for his big 2022 release. The actor who won the Best Actor Academy Award left audiences stunned after he slapped presenter Chris Rock on stage at the Oscars ceremony over a joke cracked by the latter about his wife Jada Pinkett Smith.
While it was stated that Emancipation was facing delays following the Oscars scandal, the movie is now slated to come out soon and ahead of the same, Smith has now begun his press tour to promote the same. The actor doesn't seem to be shying away from the Oscars controversy and lately talked about the same and the impact it will have on the film's reception.
In a media chat, when Smith was questioned about what he wants to say to the viewers who aren't yet ready for his comeback following the slap incident, the actor told, "I completely understand — if someone is not ready, I would absolutely respect that and allow them their space to not be ready. My deepest concern is my team – Antoine has done what I think is the greatest work of his entire career. The people on this team have done some of the best work of their entire careers, and my deepest hope is that my actions don't penalize my team. At this point, that's what I'm working for."
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Oregon Governor Kate Brown Signs "Fair Shot" Legislation into Law—Employers Beware!
Aaron Goldstein
Oregon Governor Kate Brown signed a package of four bills into law on Monday, three of which will directly impact Oregon employers. Specifically, it requires even small employers to give up to 40 hours of paid sick time, "bans the box" to make it illegal to inquire into criminal records before extending an offer of employment, and sets up a voluntary retirement savings program for employees of private employers. The laws are effective on January 1, 2016 (June 16, 2017, for the retirement savings program).
SB 454 requires employers to provide employees with up to 40 hours of paid sick time per year. Oregon joins Connecticut, California and Massachusetts as the only states to mandate paid sick time for employees—though several cities, including Seattle and Portland, have already adopted ordinances requiring paid sick leave. The act goes into effect for requests for paid sick time made on or after January 1, 2016.
HB 3025 is Oregon's "ban the box" legislation—making it illegal for companies to inquire into an applicant's criminal record prior to an initial interview or to exclude an applicant from an initial interview based upon the applicant's criminal record. Employers that do not conduct interviews may not consider an applicant's criminal record before making a conditional offer of employment to the applicant. The law goes into effect on January 1, 2016. This law goes farther than many pieces of "ban the box" legislation in prohibiting employers from even considering criminal records prior to an initial interview, even if the employer did not solicit criminal background information.
HB 2960 establishes a seven-member Oregon Retirement Savings Board, which is directed to develop a contribution retirement plan for employees whose employers don't provide one. The law sets a target date of June 16, 2017 for employees to begin making contributions to the retirement plans. Employers aren't required to make contributions to the plans under the law and employees may opt out. However employers should keep any eye out for additional administrative requirements as the law mandates, "whenever possible, use [of] existing employer and public infrastructure to facilitate contributions to the plan, recordkeeping and outreach."
Stay tuned as these laws come into effect!
Blogs Multistate Non-solicitation Agreements: Does One Size Fit All?
Client Alerts/eUpdates/Alerts Obesity Is A Disability, Washington Supreme Court Rules
Recognition and Awards Super Lawyers Recognizes Dorsey Lawyers in Des Moines


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(Arddu/Wikimedia Commons/CC BY2.0)
One of The World's Rarest Tigers Was Just Killed in a Horrible, Preventable Accident
There are just a few hundred of them left in the wild - and now the population of critically endangered Sumatran tigers in the Indonesian jungle has suffered a blow after a magnificent animal was killed in a senseless accident.
A hunter hoping to snare a wild pig on the island of Sumatra found not a porcine bounty in his trap, but a large female Sumatran tiger (Panthera tigris sumatrae).
He notified conservation officials, AFP reported, who travelled to the site only to find the tiger missing. They later found her dead near a ravine, rope from the trap wrapped tightly around her middle. They suspect that this was the cause of the animals death.
As if that wasn't terrible enough, the tiger was pregnant, due to give birth to a litter of cubs in the near future.
Exact numbers are difficult to gauge, but estimates put the population of Sumatran tigers as low as under 400 individuals, with the population still in decline.
A Sumatran tiger in Adelaide Zoo (Kevin1234/Wikimedia Commons/CC BY-SA 3.0)
According to the IUCN Red List of Endangered Species, threats to the species are primarily human, including rapid habitat loss to farming, the local illegal trade of tiger parts, and declines in the populations of its prey.
Every individual, and every cub, is therefore incredibly precious. Unfortunately, their habitat loss is also causing more conflict with humans, and loss of lives on both sides.
"Habitat destruction forces tigers into settled areas in search of food and where they are more likely to come into conflict with people," states the World Wide Fund for Nature.
"People have been killed or wounded, and livestock fall prey to tigers. Retaliatory action by villagers can result in the killing of tigers."
Under Indonesia's Conservation Act of 1990, killing a protected species such as a Sumatran tiger can incur a prison sentence of up to 5 years and a fine of 100 million rupiah (around US$6,700).


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Sunday Chatter
United States New Mexico Albuquerque
All Albuquerque
By Aileen O'Catherine
Welcome to Sunday Chatter
Photo © Aileen O'Catherine
Albuquerque has a unique musical program, Sunday Chatter, that takes place most Sunday mornings in the Wells Park neighborhood. Started by the late Felix Wurman, who was a cellist with the New Mexico Symphony Orchestra, the weekly program features chamber music and poetry/spoken word, mixed with a few minutes of silence. It's an alternative program that is nothing short of extraordinary.
Sunday Chatter was once performed at the Kosmos on 4th Street NW. Its new home is Las Puertas, located at 1512 First NW.
Tickets can be purchased at the door. Online tickets are available only for Sunday Frequent attendees; buy ten and get one free. Admission is $15, or $9 for anyone 30 and under/full time students with ID; $5 for kids 12 and under. Frequent attendees get to sit in a special section in the front which is saved for them until 10 a.m.
Once inside, visitors take their seats. Each program lasts one hour, and is made up of ensemble music and poetry/spoken word. The music changes, with special guests featured, or special groups, such as the Santa Fe Desert Chorale. Many weeks, the music is played by local classical musicians. David Felberg and James Shields are co-artistic directors of Sunday Chatter.
The weekly program changes, with a variety of musicians and poets in addition to the two minutes of silence. Sunday Chatter has been described as a church-like experience for non-churchgoers. The large warehouse has a high, curved wooden beamed ceiling, warm red brick walls and even a few stained glass windows. Chandeliers add a touch of quirky elegance. Vintage lamps dot tabletops. Couches and chairs line one wall, giving the space a homey and comfortable atmosphere.
Baristas Make Coffee Drinks
The program begins at 10:30 a.m., but it's best to get there early enough to enjoy the coffee. Coffee at the complimentary espresso bar gets served up before the show. Baristas will make audience members espresso and other drinks, asking only for tips.
There are also baked goods to enjoy. Volunteers offer up everything from cheesecake to chocolate chip cookies.
Music with Sybarite 5
On this particular day, the music was performed by Sybarite 5. These classically trained musicians perform crossover music, performing everything from Mozart to Radiohead. The quintet features Sami Merdinian and Sarah Whitney on violin; Angela Pickett on viola; Laura Metcalf on cello; and Louis Levitt on bass. The well known quintet has made an annual visit to Sunday Chatter for years. In November 2012, they make their Carnegie Hall debut.
Music in the first half of their set included pieces by Samuel Barber and Piotr Szewczyk. Their second set included remixes of Radiohead songs, to include 15 Step. For an encore, they performed Led Zeppelin's Stairway to Heaven. Their performance was eclectic and electric, and a great way to start the day.
Each week, the musicians change. Sometimes musicians are local, sometimes they are visiting or specially asked to perform. The first performance of Sunday Chatter, then called Church of Beethoven, was February 10, 2008. Felix Wurman was a cellist and composer and founded of Church of Beethoven, now known as Sunday Chatter, as a place to feel spiritual and to celebrate music.
Poets Perform at Sunday Chatter
Between musical sets, representatives from the ABQ Slam Poetry team performed. Here, Damien Flores and NJ Austin act out a piece.
Each week, a poet is invited to perform between musical sets. Past performers have included Albuquerque poet laureate Hakim Bellamy, Lisa Gill, Erin Adair-Hodges, Demetria Martinez, Tony Mares, Margaret Randall and V. B. Price. Sometimes talks or performances are given in lieu of poetry. Past non-poetry events have included the talk Dark Matter by Dr. Dinesh Loomba, a performance by the Tricklock Company actors, and essayists.
After the performances, there is a two minute period of silence, a kind of celebration of silence. Then the second set of music comes on, with a substantial chamber work.
After the hour is up, visitors have the opportunity to visit the art gallery next door, the 5G Gallery. The Far Flung show was there on this Sunday, featuring art by Cedra Wood, Lauren Greenwald and Brooke Steiger.
Sunday Chatter is a great way to hear music on a Sunday morning, and enjoy the company of others. In addition to Sunday Chatter, there Chatter 20-21, which features compositions from the 20th and 21st century, and Chatter Cabaret, an evening program held at the Hotel Andaluz.
Visit Chatter online.
Tour the Albuquerque Isotopes Stadium, New Mexico's Minor League Team
See ABQ on a Tour by Bike, Boat and More
Discover Tingley Beach in Albuquerque
Tie on Your Ice Skates in Albuquerque
Find the Haunted Places of Albuquerque
Strike It Big By Bowling
Discover Albuquerque's Dinner Theaters
Find Puzzle Rooms and Houses and Make Your Escape
Hear Classical Music with the New Mexico Philharmonic
Enjoy Winter Sports in Albuquerque
Find a Balloon and Take Flight in Albuquerque
These are the best Places to Propose in Albuquerque
Find a Horse and Ride the Trails in Albuquerque, New Mexico
Where Can I Find the Best Water Parks in Albuquerque?
Where to Find Indoor Playgrounds and Amusement Centers in Albuquerque
Great Places to Throw a Kid's Birthday Party in Albuquerque


This is exciting! Not only can we change the aged appearance, but the FYBBL also helps our skin act younger at the molecular level!! So what does this mean and how is it different than the BBL we've been doing for 11 years?
The protocol is different using more than one wavelength of light at each treatment which addresses underlying reds, pigmented lesions (brownies:), and skin firming/texture. You're essentially getting the benefit of three treatments in one!
Just as it is often needed (and recommended) to have more than one BBL to treat browns or reds, the number of treatments of Forever Young BBL is dependent on each individual's skin condition and desired outcome. For aged-damaged skin, a plan of three treatments every 2-4 weeks to bring correction, followed by annual maintenance of 3-4 treatments over 12 months is recommended. For younger, less damaged, skin an annual treatment may be enough. For those clients desiring a faster correction of damaged skin, paring with a Halo Pro is an option as well.
The outer appearance diminishing the damaged look is not the only response. There are changes bringing your skin to a more youthful state at the molecular level too!
FYBBL is an easy treatment to receive with little to no downtime and only a topical anesthetic application 30 minutes ahead of the treatment.
How is the treatment similar to the standard BBL?
It is the bright light treatment as before.
It requires sun screening and moisturizer before and following the treatment.
There can be minimal redness post-treatment for up to a few days as well as darkening of "brownies" until they come off in the first 7-10 days. It is possible to get a bit of swelling below the eyes which reduces in a couple of days.
So, your options are to treat damaged skin and improve your appearance OR do the same AS WELL AS reverse the aging clock at the cellular level. With diligent maintenance treatments and protective support products, you can absolutely have younger looking skin down the road!! Certainly an amazing option and well worth consideration!
If you're ready to embark on this mission, let's get started in the month of February at a promotional price! Check out our specials page!
Tags: Aesthetic trends in 2019, beauty Chillicothe, beauty clinics Chillicothe, laser clinics Chillicothe, What's new at New Image?


In 2011, whilst working as a freelance graphic designer, I did a community college evening course in porcelain jewellery and was hooked! After a couple of years of balancing graphic design work and ceramic exploration, I decided to focus on ceramics completely. I still feel like a relative newcomer to clay and will never stop experimenting. I have always loved figurative sculpture so I can't help giving everyday functional items a personality whether it is human or animal or a mixture of the 2. I also love colour and porcelain and stoneware are the ideal materials to add vibrant colour to. I use a simple, decorative and graphic approach to the pieces and I purpose glaze selected elements to highlight the contrast between matt and gloss. I am driven to make things that make me happy and hope that in doing so I bring a little joy into people's homes and make them smile.


Sugar has been getting a lot of negative attention in the media over the last few years. That's because not only is it one of the main causes of our countries obesity epidemic, but it also can cause a lot of preventable diseases such as Type 2 Diabetes, Heart Disease, and cavities just to name a few. Cutting sugar out completely can seem impossible but here are some simple things to look for to avoid consuming more added sugars.
Read the Labels: There are products that you know sugar is already in such as fruit juice, sweetened cereals and candy. The problem is, sugar is hidden in a lot of foods we typically wouldn't expect. Everything from tomato sauce, to ketchup, to granola bars and even some spices contain sugar. Make sure to read the label fully and look for the amount of sugar per serving size. Ideally, you should aim for sugar to be no more than 48 grams. There are about 56 grams of sugar in a bottle of Gatorade… just to emphasize the importance of reading the labels first before consuming.
Buy Fresh Foods over Canned: It is recommended by the USDA food guidelines to aim for 5 servings of fruits and vegetables each day and a colorful assortment of them. Although fruit does have sugar in it, these are natural sugars and not artificially added sugars like the ones we find in our foods. Canned foods, especially fruits, are swimming in syrupy sugar water in order to preserve their freshness and shelf life. Always choose fresh over canned and just shop for the amount of fresh produce that you're actually going to eat that week, since it will go bad faster than the canned versions.
Try Healthy Substitutions: There are a number of baking substitutions you can use to reduce your sugar intake. Applesauce is a great natural sweetener to use in place of large amounts of sugar. You can also use extracts to bring sweetness to baked goods and cinnamon and nutmeg that can cut down on the added sugars in a dish. When you bake at home, you can even cut the sugar called in recipes down and not notice the difference in taste.
Make Your Own Food: The best way to make sure you don't consume any extra sugars is to cook your own food at home. You get to choose the ingredients, you know exactly how much sugar does/doesn't go into a recipe, and you can control your portion sizes. This is a great way to take control of your health and really give your family the best possible food and nutrition. If you need some inspiration on less sugar dishes, check out some of the healthy recipes below.