The William Ris Gallery will celebrate its second anniversary on the North Fork with E S C A P E S, a group show featuring landscape and seascape art inspired by the East Coast.Included will be the works of prominent East Coast artists: Kelly Franké, Kurt Giehl, David Peikon, Jim Sabiston, Eileen Dawn Skretch, James Stewart and Amy Worth whose works range from realism to ethereal and abstract to illustrative. Curated and organized by gallery owner and director Mary Cantone, E S C A P E S will comprise of 30 works varying in medium from oil and acrylic paintings, photographs, watercolors and ink drawings.
On exhibit from June 16 through July 8 with an opening reception on Saturday, June 16th, 4 – 7 p.m.
Open daily from 12 – 5 p.m. and by appointment.


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The Batman – New Official Trailer. DC October 16
DC has presented a trailer for The Batman movie starring Robert Pattinson.
Posted: 18 Oct 2021
Updated: 18 Oct 2021 20:25
In the story, Batman is trying to find out who is behind the corruption in Gotham. In parallel, the Dark Knight enters into a deadly conflict with a serial killer under the pseudonym Riddler.
During the presentation of the trailer, Pattinson noticed that he was auditioning for a role in a Val Kilmer costume. Previously, Christian Bale wore the same costume before filming Christopher Nolan's trilogy.
The film also stars Zoë Kravitz as Cat-Woman, Colin Farrell as Penguin, Jeffrey Wright as Jim Gordon, Andy Serkis as Alfred's, John Turturro and Peter Sarsgaard.
The Batman movie starring Robert Pattinson is set to premiere on March 4, 2022.
The Witcher Season 2 – New Official Trailer. Destiny is coming close to The Christmas
Stranger Things 4 – Creel House. New Teaser
The Witcher – Road to Season 2 Trailer
Matrix 4. Warner Bros showed the first trailer
VFX Studio Rodeo FX present Breakdown for WandaVision: Vision Destruction

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Calling same action with different urls in cakepphp I am trying to call an action with two different url's
For example, following are my url's, here username is the parameter and news is the action,
if i enter the first url it works fine and executes the news action, if i enter the second url i am getting error, i need to call the news action if i enter the second url, how can i achieve this by routes.
Thanks..
http://domain.com/username/news
http://domain.com/username

Below is the route for the first url
Route:
Router::connect('/:sluguser/:action',
array('controller' => 'users', 'action' => 'news'),
array('pass' => array('sluguser')));

redpajama_set_name: RedPajamaStackExchange


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Cardi B Proves She's the Future in 3D-Printed Balenciaga
Jael Goldfine
Much like the irreverent flexes in her verses, Cardi B's style hubris rarely fails to make us do a double take.
The "Balenciaga Mama" gently broke the Internet yesterday when she shared a photo of herself looking like a plaid greek statue in a wool $3,400 Hourglass Double Breasted Coat from the brand's Fall 2018 collection, that was forged by a 3D printer.
According to Garage, when the jacket debuted at fashion week, Balenciaga's creative director Demna Gvasalia revealed that he developed the garment via a 3D scan taken of the model's body, which was then tailored, printed, molded, cast in foam and molded to the fabrics for the finished product. Gvasalia experimented with the method, which he's calling "neo-tailoring," in his architectural Spring collection.
While it's not clear if Cardi went through the same high-tech tailoring process, the rigid structure and the cartoonish yet elegant hourglass that perfectly exaggerates Cardi's shape look impossibly well-fitted, though the garment is available online.
Related | Hennessy Takes Hollywood
What we can confirm however, is that on top of Cardi's well-documented affection for those Balenciaga sneakers "the one that look like socks," she's also here for those Balenciaga jackets... the ones that look like '70s armor and were fabricated by a futuristic machine.
Photo via Instagram
The Internet Can't Stop Roasting Drake's New Haircut
Here's What the Bridgerton Cast Looks Like Without Costumes
Say Goodbye to the 'Daddy' Hat
Stella Tennant's Family Reveals Her Death Was a Suicide


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As Easter Sunday approaches menus are being finalized and grocery shopping trips are becoming frequent. Yet, it is not too late to do some last minute additions to your menu. These 5 easter desserts are ones you do not want to miss. With fresh, in season fruits, healthier alternatives, and 100% plant-based.
I am usually not a big fan of gluten-free desserts, as they are often dry, crumbly, and lack in rich flavors, however that isn't the case for these cookies.
Not only are they soft and chewy on the inside, but provide a slight crunchiness on the outer layers. Embedded with gooey chunks of chocolate chips and rich notes of peanut butter. Dunk one (or two, three…) into a glass of non-dairy milk and I think you'll have your sweet tooth satisifed for the rest of the day.
The blend of oat and almond flour provide a slightly nutty flavor with a soft and chewy texture. Almond always seems to bring out such a great flavor in chocolate chip cookies and I am always a fan of using it in a recipe even if it means using almond extract.
It may seem hard to incorporate healthy, sustainable foods into your busy schedule. Quite frankly, it is a struggle for me right now. However, prepping healthy on-the-go snacks is a great way to set yourself up for sucess. Make these Chocolate Chunk Almond Sea Salt Granola Bars and you'll definitely thank me later.
Chewy, slightly crunchy, embedded with gooey chunks of chocolate chips. Simple ingredients, affordable, easy, and 100% plant-based.
This recipe is in collaboration with Compassion in World Farming. I am so happy to be partening with them again with the goal of creating a sustainble, compassionate food system.
Eating more plants ourselves is a great way to help farmed animals and the planet of course, but helping others do the same can really multiply our impact! And anyone can become a more effective advocate if given the right set of tools.
you will have access to the Changemaker Handbook meant to help you do just that.
This hand guide is packed with resources like key scientific benefits to eating more plants, tips to having more productive conversations about plant-based eating, and practical advice to provide friends & family interested in learning how to #EatPlantsForAChange. It is true that just one person can have a big impact on our food system, but in order to end factory farming and avoid climate change it is going to take ALL of us.
I encourage you to follow the link above to sign up to be a Changemaker today!
One of my favorite desserts…since I was a little child, is cinnamon rolls. I simply can't get enough of the sweet, fluffy layers of cinnamon crusted dough. Smothered in a thick sugary icing. It truly is to die for.
If you know me, then you know I sometimes like to take a twist on traditional recipes. So, what better twist than a cinnamon roll in cupcake form? Convienent, fast, but doesn't lack any of the flavors.
A fluffy vanilla cupcake embedded with a cinnamon struesal and smothered in a simple powdered sugar icing.
Yes…I broke the cinnamon roll rules, but if it results in a mouth drooling creation than who cares, right?


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Trybanes () was a town of ancient Caria. The town was possibly the same as Tarbanes (Ταρβανῆς).

Its site is unlocated.

References

Populated places in ancient Caria
Former populated places in Turkey
Lost ancient cities and towns
Members of the Delian League

redpajama_set_name: RedPajamaWikipedia


text:
Rest In Peace – Kevin Dubrow
· Entertainment · Hard Rock Music · Hard Rock News · Music
It is a sad day at Hard Rock Hideout.com. Another great rocker that has passed way too soon. Our condolences to the Dubrow family and friends!
QUIET RIOT drummer Frankie Banali has confirmed the passing of the group's singer, Kevin DuBrow, at the age of 52. In an e-mail to Spain's The Metal Circus, which had requested a comment from Banali on the rumors of DuBrow's death, the drummer wrote, "I can't even find words to say. Please respect my privacy as I mourn the passing and honor the memory of my dearest friend Kevin DuBrow." UPDATE: The same statement can now be found on Frankie Banali's official web site.
Michael Politz of The Vegas Eye reported earlier today that DuBrow was found dead in his Las Vegas home on Sunday, November 25. According to Politz, DubRow — whom Politz calls "one of my closest friends" — recently celebrated his 52nd birthday in New Orleans "where he ate great food and even caught THE CULT at The House of Blues."
Although the band has yet to issue an official statement on the matter, bassist Kelly Garni, who was one of the founding members of QUIET RIOT, has posted the following message on the Randy Rhoads (late QUIET RIOT/OZZY OSBOURNE guitarist) web site RandyRhoads.tk:
"I ask this to all of you not only for myself but for other friends and family. I ask that no one here offer any speculation or opinions, theories or other things that could be construed as negative or, and I'm sorry for this, even sympathetic, right at this immediate time. I am already within hours of this having to deal with untrue rumors and speculation and that only adds fuel to that. There is a tendency for the subject of Kevin to incite flames on every board and now is not the time for that. I will explain to everyone here the facts and the truth in the next 24 to 48 hours as I realize this will effect us all. So please, until then, be patient. All details and other pertinate info will be passed on to you here when it becomes available to me. Thank you."
In 2004, DuBrow recorded a collection of cover versions for his first solo album, "In for the Kill".
QUIET RIOT's latest studio CD, "Rehab", was released on October 3, 2006. The lineup on the album was Kevin DuBrow, Frankie Banali, Tony Franklin and Neil Citron. Legendary singer Glenn Hughes also made a guest vocal appearance on the LP.
As reported by Blabbermouth.net
Death Kevin Dubrow
Bic & Que November 29, 2007 · Reply
Rest in Peace Kevin…..there won't be a day goes by that we won't hear your voice.
Adriamarie November 27, 2007 · Reply
Memories, the 80's Metal scene was it for Me. I remember Quiet Riot, Ratt & Motley crue and Ozzy. These guys could Rock your Soul. I will Miss Kevin, I met him in the 80's and he was nothing but a Sweet Hard Ass Rocker. Those were the days when the Groups when actually hang with fans too, A many of them I've met.
Kevin You Keep Rockn in Heaven, You will be missed very much
So "CUM on FEEL the NOISE"
bubba November 26, 2007 · Reply
I don't even know what to say. This is the first one that has hit me personally. I was too young to remember Randy when he was alive, and wasn't into rock when Freddie or Stevie passed. Quiet Riot is one of my personal favorites, even past the "Metal Health" album. I haven't fully processed this news yet.
rhodeislandrock November 26, 2007 · Reply
It's a sad day indeed. My sympathies to Kevin's family and friends and especially Frankie Banali.
Fly on Thunderbird…..
Heavy Metal Addiction
http://hardrockheavymetal.wordpress.com/
Jay November 26, 2007 · Reply
i had the opportunity to interview Kevin when I was in radio and he was a great guy, and super nice. That's terrible.


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"I wish I had done it sooner" - Talking about becoming a foster parent.
Foster Parent
About Anu
The Anu Model
Why Choose Anu?
What About The Kids
Wellbeing Tips
What Does Anu Mean?
Our Hero Foster Parents
Treatment Foster Care Services
Intensive Permanence Services
Intensive Trauma Services
Parenting Coach Certification
Community Based Flexible Support Program
FastTrack to a Family
Healing Professionals
Well-being Indicator Tool for Youth (WIT-Y)
Licensing Steps & Phases
Top 10 Ways To Know You Would Make a Great Foster Parent
Training Requirements
Treatment Foster Parents and What They Do
Join the Anu Team
We Create Permanent Connections 877-287-2441
Encouraging improved behaviors Parent Coaching
Parent coaching is a short-term intervention that provides foster, adoptive and kinship parents and other caregivers with hands-on, here-and-now tools for bringing about rapid and dramatic improvement in their child's behavior.
Using a strategy customized to each individual family, coaches work with caregivers, in the absence of the child, to teach methods and tools that promote attachment and self-esteem while feeding the development of even more positive behaviors. Caregivers are able to strengthen bonds while teaching children new and effective ways of interacting with their world.
Present Moment Parenting
Parent coaching is a customized-to-your-family opportunity to understand your intense child's behavior and learn what to do to help it improve dramatically. You'll learn Present Moment Parenting with the support of a coach who will not pass judgment on you but will stand by you as you gain new skills. There's no diagnosis from your coach, just solid, usable tools for bringing out the best in your intense child. How do we define "intense"? If you have a child with whom typical parenting methods don't help, and even make the situation worse, you've come to the right spot.
What type of needs do the kids have?
Our parent coaching clients have children of all ages who have ADHD, Oppositional Defiant Disorder, Reactive Attachment Disorder, Autism Spectrum Disorder, giftedness, adoption or foster care issues, and even those with no diagnosis at all.
For children whose lives involve the child welfare system, particular attention is paid to helping parents navigate their sometimes overwhelming grief and loss.
The aim of "trauma-informed parent coaching" at Anu is to educate parents on specific relationships between unwanted behavior and deep, enormous unexpressed feelings. Once those relationships are understood, parents will learn how to approach the child in a way that heals them, rather than re-traumatizes them. It's remarkably easy to re-traumatize a child who has suffered physical or sexual abuse, neglect, and/or abandonment. The more parents understand that typical parenting techniques fuel the fire of unexpressed grief and rage, the more they can bring out the best in these children.
Biological, adoptive, foster, and kinship parents are in the perfect position to undo some of the damage inflicted on the children in their homes. They care the most about the children, they are with them the most, and they have the most invested in their well-being. Equipped with insight into the effects of trauma on the child's body and the equally positive effects of nurturing communication on the same child's body, parents and caregivers can start to see results immediately.
When children's behavior is not only under control, but indicative of a child's sense of self in healing mode, they can stay put. They avoid the defaulting judge, blame, and punish cycle that creates havoc in their lives and causes them to overreact to what parents think is normal parenting. The result is a peaceful, permanent home.
What about the parents' past issues?
Will parents who are raising children who have been traumatized sometimes start to get in touch with the traumas they experienced as children?
It does happen. Parent coaching is not therapy, and our coaches will refer parents to appropriate services, such as psychotherapy and EMDR, even biofeedback, to learn ways to heal their own trauma. In the meantime, our coaches support the parents in interacting with their kids in the present moment, where all healing takes place. This two-pronged approach of parents receiving the help they need for their past upsets while being supported in their parenting is a powerful combination, and both processes can occur simultaneously.
Children who have been traumatized don't have months to wait for their parents to heal their childhoods. That's why parent coaches are available by phone, email, and Skype to provide weekly or bi-weekly encouragement on changing their interactions with the kids. We even have between-appointment trouble calls, where parents are free to call their coaches for on-the-spot support. When a mom or dad is about to lose it with their child, they can get help on the phone instead of yelling. This is the intensive support that's needed for creating positive behavior and strong family relationships with the outcome of permanence for children.
To make a referral call 1.877.287.2441 or complete a referral form.
Listen to what county social workers are saying about parent coaching:
"Parent coaching is the most effective service we have to offer our kids."
"I LOVE THIS SERVICE."
"All your providers are fantastic!"
"I continue to send referrals to you guys as often as possible."
"I refer all my clients to Parent Coaching!"
"Mom remains completely calm now and has told the worker that coaching "is the reason I can handle this"
"The coach not only works well with the client, but she educated the team also, allowing the team to understand what she was doing and to help support and encourage the client."
"She connects well with the family and is very creative in her approach as well as suggestions for the family to try. An asset to the team."
"I love working with her! The only suggestion is to have a parent coach work with service facilitators [case managers] too!"
"The coach was awesome and the family is communicating very well now with her help. A huge improvement from before she was involved."
"[Coaching] has been a valuable resource for our foster families and he has a special ability to connect with families and children in a non-threatening strength-based approach."
"The Parent Coach is compassionate, he boosts up the family, he gives them practical and realistic ideas. The parents are moving closer to adopting the child."
"The families feel very welcome, and don't feel judged. [The Coach] is very strengths-based."
"[Parent Coaching] allowed the foster mother to gain more confidence and this was reflected in her parenting approach."
"The family had nothing but great things to say about [the Coach]. They were at a place where they were considering giving up placement of their grandchildren, and after working with him, they say that they will never consider that."
What is Parent Coaching?
Parent Coaching involves a strengths-based approach to teaching parenting, giving clear, explicit directives to parents and caregivers, and designed to improve challenging behaviors in children and youth. The service is much like a sports coach who tells a player what to do to improve performance. It is not therapy, and the coach does not spend much time discussing the past; rather, coaching is focused on why the child is behaving this way in the present moment, and what current specific actions are needed by the caregiver/parent to alter or nourish the child's behavioral response.
This coaching is delivered only to the parents, not directly to the youth or in the presence of the youth. This is so the child doesn't have the experience of "yet another grown-up/professional" needed to come in and help because "I'm so bad;" it does not disempower parents by correcting them in the presence of the child; it strengthens the relational bond between caregiver and child; and it puts the healing where it belongs: in the hands of the caregivers/parents.
What is the "Theory of Change" for Parent Coaching?
The Theory of Change for Parent Coaching is based on the premise that communication with children affects brain development and neural pathways. Therefore, when parents give attention and energy to desired behaviors, the neural pathways for those desired responses are strengthened. In contrast, when little reaction or attention is given to undesirable behaviors, the neural pathways to those behaviors are not utilized and are pruned away.
In addition, when parents feel supported and nurtured, they can do the same for their children. It is a reciprocal healing process: when parents download messages of heartfelt appreciation to their children, the appreciation is mirrored through the children's expressions of love, affirmation, compliance, and affection. The service helps strengthen and heal the parent-child bond: benefiting both parent and child.
Who Are the Intended Recipients and Beneficiaries of Parent Coaching
Our coaches serve all parents who have the mental capacity to grasp these concepts and the desire to put them into practice. Parent Coaching is effective for foster parents, treatment foster parents, adoptive parents, kinship parents, teachers, child care providers, school social workers, residential and group home staff, and any parents or caregivers struggling with a child's challenging behaviors. If a grandfather and mother are raising the children, they come to the sessions. If a teacher or nanny wants to be involved, it's highly encouraged.
How is Parent Coaching Paid For?
Many non-child welfare involved families pay privately for Parent Coaching services. For system-involved parents, the county of residence pays for these services out of county revenue sources. Parent Coaching is not typically a medical-model or Medicaid eligible service; however, some WI counties have identified Comprehensive Community Services (CCS) as a funding source for Parent Coaching. In addition, Children's Long-Term Support Waiver funding is being used for this service.
What are the Coaches' Qualifications?
Coaches have a minimum of a four-year degree in a child-related field, and many have master's degrees. All coaches go on to complete an extensive certification process to become Certified Parent Coaches; however, coaches do not always come from a social work field and may include expertise in psychology, education, and other related fields. All coaches are supervised by a master's level supervisor.
What are the Typical Frequency and Duration of Service for Parent Coaching?
Parent Coaches generally meet with clients weekly or bi-monthly, and the focus is for the coach to meet the client where they are emotionally and cognitively, which in part, determines the duration of the service. Capacity for learning varies by parent, and some of the influencing factors include whether their basic needs are met, their own mental health, the effectiveness of their current self-care strategies, and their connection to a larger natural support system. Some clients' coaching needs are met within 4-6 sessions, and others may require 12+ months.
What Model of Parenting Does Anu Use?
Anu Family Services uses the trauma-informed "Present Moment Parenting" (PMP) to create radical change in children with challenging behaviors. This model is based on gratitude and compassion, and is written out in Present Moment Parenting: The Guide to a Peaceful Life with Your Intense Child (2017) by Tina Feigal, M.S. Ed. Tina Feigal is the Director of Family Engagement and a Parent Coach at Anu Family Services.
What is the Parent Coaching Service Delivery Model?
As our coaching sessions generally only involve the parents, this approach succeeds because it prevents parents from being "outed" in front of their children. It helps maintain authority in the family as the coach supports the parents from behind the scenes. Coaches give techniques that work and follow through with a minimum of four sessions to assure success. Parents are empowered in the healing their family experiences. After (sometimes) years of being held responsible for their children's negative behavior, they are now "held responsible" for the successes. This can also positively change the dynamics with extended families, a factor that often gets overlooked, but is vital to the emotional health of families.
What is the Evidence Base for Parent Coaching?
A significant amount of research, evidence, and practices inform Parent Coaching. "Present Moment Parenting" encourages parents to reduce negative behavior by giving attention to desired behaviors, rather than unwanted behaviors. This theory was presented in the early 1800s by Alfred Adler, a contemporary of Sigmund Freud. More recently, it has been researched by Dr. Barbara Fredrickson in her book, Positivity (2009). Dr. Fredrickson's studies revealed behavior improves at a much faster rate, and with more consistency, with positive input than with negative correction. For most parents, this requires a complete change of emphasis, for which having support over time from their coach is extremely helpful.
The survival imperative drives children's behavior. All behavior is communication: saying, "I must survive," physically, emotionally or both. When parents see behavior through the lens of the child's inner need for survival, they can let go of the need for judgment, blame, and punishment. This frees them to work with the present moment to assure the child of his or her safety, and therefore, survival.
From the work of Eckhart Tolle, author of The Power of Now, we learned the past is over: we cannot go back and make it different. The future is not yet here; we cannot jump ahead to be in it. Everything happens in the present moment. Acting on this truth can facilitate the healing needed between parent and child. Focusing on the here and now is the most effective way to bring out the best behavior and strengthen the parent-child bond. When parents acknowledge children's need to feel safe and address children from this knowing, children feel seen, and survival feels assured. The following behavior reflects the child's felt safety, and the relationship with the parent is perceived as a safe haven. The child is then free to grow and develop in "fertile soil."
"Present Moment Parenting" also taps the ideas of Dr. Daniel Siegel, whose research chronicled in The Mindful Brain (2010) and Parenting from the Inside Out (2013), reveals that children's bodies react to emotional input from others. Therefore, being mindful of the physiological effects of communication between parent and child empowers parents to obtain the results they want. We emphasize strengthening heart-to-brain neural pathways for the child's positive behaviors by repetition of positive input. The work of the Institute of Heartmath® supports this aspect of PMP, showing the heart has its own neural pathway system, communicating emotional information to the brain. Parents are taught to give positive input to children, strengthening these neural pathways for good behavior. They are thereby actually changing the physiology of their children. Further research on this topic contributes to PMP through the work of Paul Pearsall, Ph.D., whose book, The Heart's Code: Tapping the Wisdom and Power of Our Heart's Journey (1999), describes the effects of heart-to-brain communication.
Heather Forbes' ideas have also contributed to the work of PMP with her work, Beyond Consequences, Logic and Control (2006). Especially applicable to children who have experienced trauma, this work illustrates the value of eliminating the emphasis on specific negative behaviors; and focuses on the inner state of the child that prompts the behavior. So rather than seeing aggression that needs to "stop right now," parents are encouraged to explore the inner state of the child to help the child "feel felt." This is now known to be much more effective than trying to control behavior with threats and punishment. Trauma-informed parenting, where parents approach children from the perspective of "what happened to you?" rather than "what's wrong with you?" gets to the heart of behavior change much more effectively than traditional parenting methods.
"Present Moment Parenting" is proving highly effective in work with children who have experienced trauma. As it de-emphasizes the "behavior-as-bad" approach and focuses on "behavior-as-communication," parents are empowered to address the underlying message of the behavior, rather than the behavioral symptoms of grief, loss, and trauma. When parents are equipped with alternative ways to respond to children, hope is strengthened, and permanence in the home is perceived as possible.
What Makes Parent Coaching Successful?
The evidence-based methods used are not "parenting classes." Rather, specific guidance is provided to each parent to help them deal with their specific child, given their unique strengths and challenges in the present moment. This kind of coaching and directive guidance is very successful with adult learners, including those with their own mental health issues and developmental challenges.
Does Parent Coaching affect children's behavior even though the coaching is done only with the parents?
Parent Coaching teaches parents ways to show love and understanding to their children. When children feel heard, seen, and understood, their pain-based behaviors DECREASE! They don't need to move from home to home.
Is There Any Evaluation of These Services?
To evaluate the effects of coaching on child behavior and parental sense of competency and connectedness, we conduct pre- and post-surveys with parents before parent coaching begins, and after four coaching sessions. Just as important, the survey queries parents on improvements in the relationship between parent and child and on parents' sense of themselves as parents. Although the surveys are for purposes of program evaluation, and not peer-reviewed instruments, they are helpful in observing the positive effects of coaching. As shown in the survey results, while parent coaching can't eliminate behaviors, every behavior parents noted at the beginning of coaching was significantly reduced in frequency by the end of parent coaching. Not only that, but parents reported children getting along with friends more often, being more willing to cooperate and getting into trouble less frequently. Parents reported having more positive interactions with children, having more understanding and compassion for children, being more relaxed around their children and being less angry. Also according to the survey, parents are extremely satisfied with the parent coaching process and results. Parents feel more confident with parenting and are appreciative of their coaches.
Over the course of Parent Coaching, 100% of the child's behaviors, attributes, and relationships improved!
Parents see their child as more loving after they received Parent Coaching (67.7% improvement)! Parents do a better job connecting with their children when they see their children as more loving.
Parent Coaching supports permanence for children at risk of being removed from home:
95% of children who are in their parent's home when Parent Coaching begins are in their parent's home when the service ends.
N=161
My partner is in the military and is always at risk of being deployed so we can't be a foster parent.
Being a foster parent with Anu is a true TEAM effort. You will be part of a close team that will be with you every step of the way for support and help. In fact, we are currently aggressively recruiting military service members and veterans for foster care.
Become a Foster Parent Make a Referral Join the Anu Team
Copyright © 2019 Anu Family Services. All rights reserved.


Stabat Mater Dolorosa is considered one of the seven greatest Latin hymns of all time. It is based upon the prophecy of Simeon that a sword was to pierce the heart of Our Lord's mother, Mary (Lk2:35). The hymn title means "'Stood the mother, full of grief'." In Latin, the hymn consists of twenty couplets which describe the Sorrows of the Blessed Virgin at the Cross. The Stabat Mater's popularity is reflected by its use in the popular devotion of the Stations of the Cross.
The message of the Stabat Mater focuses on the spiritual and emotional bond which unites Mary and all Christians to the death of her Son on the Cross. From this bond, each Christian can recognize the incredible compassion and holiness in Mary's character. The Blessed Mother demonstrated her maternal compassion to all generations of Christians by her presence and participation with her Son Jesus in the Sacrifice of the Cross.
The Stabat Mater was introduced into the Liturgy gradually until 1727 when it was prescribed as a Sequence for Mass of the Seven Sorrows of Mary on September 15 and on Friday before Holy Week, as well as their corresponding offices. The text of the Stabat Mater has been put to different music by various composers in endless variations.
Christians of the twentieth century can truly identify with Our Lady's experience of Sorrow. The message of the Stabat Mater focuses on the spiritual and emotional bond which unites Mary and all Christians to the death of her Son on the Cross. From this bond, each Christian can recognize the incredible compassion and holiness in Mary's character. The Blessed Mother demonstrated her maternal compassion to all generations of Christians by her presence and participation with her Son Jesus in the Sacrifice of the Cross.
There is a mother-son bond that unites Mary with Christ Jesus during his experience of suffering and death. This empathetic bond indicates that Our Lady shared in her Son's suffering. Mary is Our Lady of Sorrows precisely because her Son Christ Jesus bore the sins of the world during his passion and death. As the faithful disciple, Our Blessed Mother invites us to unite our personal suffering with her own. We can share in Jesus' burden on the Cross, just as Mary did at Calvary.
As Our Lady of Sorrows, Mary also reminds us that Christians are called to expiate for his or her own sins and the sins of their neighbors, and the sins of the world. We can share in the bond between the Blessed Mother and Our Lord through fasting, prayer, and contrition for sin. Our Lady of Sorrows teaches us that the Crown of eternal life in Heaven can be reached when we each choose to share with Our Lord in His suffering and death on the Cross at Calvary.
The compassion of Mary is part of the mystery of the Church community's sharing in, and offering, the Sacrifice of Jesus for the salvation of the world. Each member of the Church has a role to play in redeeming the world. Our Lady of Sorrows is a guide who inspires and teaches us how to be compassionate.


Password protect your computer and restrict access to it with Advanced Security Administrator. It enables you to impose a variety of access restrictions to protect your privacy and stop others from tampering with your PC. You can deny access to Control Panel applets, disable boot keys, context menus, DOS windows, Registry editing, Internet and network access. Hide your desktop icons, individual drives, Start menu items, taskbar, apply password protection to Windows and restrict users to running specific applications only, control Internet usage, view statistics of computer use by your kids or employees. The program also has a built-in lock screen utility to prevent PC entry with a password while you are away. Excellent online help is available. Try our password and security tools, free downloads.


text:
L'assassinat de Louis  d'Orléans, frère du roi Charles VI, survient le  à Paris. Commis à l'instigation de son cousin Jean  de Bourgogne, dit « Jean sans Peur », cet acte déclenche la guerre civile entre Armagnacs et Bourguignons qui plonge la France dans les tumultes de la guerre civile jusqu'au traité d'Arras du 21 septembre 1435.

Contexte
La guerre de Cent Ans, qui oppose la France et l'Angleterre, connaît une trêve durable depuis la signature de la Trêve de Leulinghem le 28 avril 1393. Les Anglais, qui ont dû affronter la patiente reconquête du territoire français menée par de valeureux capitaines comme Bertrand du Guesclin sous le règne de Charles V, font face à de graves troubles politiques intérieurs. Mais la France est, elle aussi,  agitée par des dissensions internes. Depuis 1392, le roi de France Charles VI a sombré dans la folie et la France est gouvernée par un conseil de régence, présidé par  son épouse Isabeau de Bavière, mais conduit par les grands du royaume. Le caractère intermittent de la folie du roi, qu'on appelle alors ses « absences », empêche la mise en place d'une régence stable. De fait, à partir de 1401, une profonde haine oppose Philippe le Hardi, oncle du roi et duc de Bourgogne, et son neveu Louis, frère de Charles VI et duc d'Orléans. Le duc Philippe de Bourgogne est favorable à la trêve avec l'Angleterre, qui ménage les intérêts de ses sujets flamands, alors que le duc Louis  d'Orléans, digne héritier de la politique de feu son père, le roi Charles V, se fait le champion de la reprise de la lutte contre l'ennemi anglais.

Face aux intrigues de Philippe de Bourgogne, qui cherche en parallèle à réunir ses terres bourguignonnes à celles des Flandres pour former un nouvel État en cas de défaite de la France contre les Anglais, le duc Louis d'Orléans forme peu à peu des alliances au sein de l'Empire pour contrer l'ambitieux Bourguignon. En 1404, Jean sans Peur, qui succède à son père au duché de Bourgogne, jouit d'une image positive selon Christine de Pizan, mais il  a peu d'influence au Conseil du royaume de France. Le duc Louis d'Orléans, en sa qualité de frère cadet du roi, domine le conseil de régence, auquel participe Isabeau de Bavière, épouse du roi Charles VI, défaillant  par intermittences. Il obtient le contrôle du Trésor royal et manœuvre pour empêcher le duc de Bourgogne de réaliser une continuité territoriale entre ses possessions des Flandres et le duché de Bourgogne, en acquérant le duché de Luxembourg, gagé sur les deniers de la couronne. En août 1405, Jean sans Peur fait une démonstration de force en entrant dans Paris à la tête de ses armées, en obligeant la reine Isabeau de Bavière et le duc Louis  d'Orléans à s'enfuir à Melun. La guerre civile est évitée de justesse par une réconciliation, le 10 octobre 1405.

Le complot

La tension reprend en 1406 et 1407, le duc de Bourgogne tenant d'abord le duc d'Orléans responsable de son échec devant la ville de Calais, qu'il compte reprendre aux Anglais. Ensuite, par l'ordonnance du 28 avril 1407 réformant la composition du conseil du roi qui voit le nombre de Bourguignons diminuer de vingt-six représentants à seulement deux, le duc de Bourgogne, voyant le pouvoir lui échapper, décide de passer à l'acte. Dès la fin de juin 1407, il semble que le duc de Bourgogne projette l'assassinat de son cousin. Il verse le 8 août 1407 à Lille cent écus à Raoul d'Anquetonville, son homme de main désigné, et mille cinq cents écus à Lourdin de Saligny, son homme de confiance, à déposer dans un lieu secret dans Paris. Ce lieu secret est l'hôtel de l'image Notre-Dame, tenu par le couple Fouchier. Le 14 novembre 1407, un courtier de maison nommé Pierre d'Asignac se met d'accord avec l'épouse Fouchier pour louer l'hôtel jusqu'à la Saint Jean-Baptiste suivante, moyennant la somme de seize livres parisis. Ce serait la cache des meurtriers.

Le 23 novembre 1407, le duc d'Orléans va rendre visite à la reine Isabeau, qui a accouché peu de temps avant, à l'Hôtel Barbette, rue Vieille-du-Temple, à Paris. Thomas de Courteheuse, valet du roi et trempant lui-même dans la conspiration, se présente à l'hôtel de la reine Isabeau sur les coups de  et fait savoir au duc d'Orléans que le roi Charles VI le mande de toute urgence. Le duc prend donc congé de la reine et se met en route pour l'hôtel Saint-Pol, où réside le roi. Quand bien même le duc dispose selon Enguerrand de Monstrelet de près de six cents hommes d'armes dans tout Paris, son escorte ne se compose ce soir-là que d'une petite dizaine de piquiers et de valets. Le duc, perché sur son mulet et ne se doutant pas de ce qui l'attend, entreprend de remonter la rue Vieille-du-Temple. Une quinzaine d'individus, tapis dans l'ombre des bâtiments, attendent avec impatience son passage. Au signal, les malfaiteurs, menés par Raoul d'Anquetonville, se ruent sur la petite troupe du duc, disposent de la garde, et parviennent à mettre Louis à bas de son mulet avant de lui couper la main. Le duc, surpris, s'écrie : « Je suis le duc d'Orléans ! » On lui répond : « C'est lui que nous voulons ! » Un coup de hache à la tête a raison de la vie du frère du roi, qui expire au côté de son fidèle valet flamand Jacob. Son corps est déposé à l'église Notre-Dame-des-Blancs-Manteaux en attendant les funérailles.

Conséquences

Le duc de Bourgogne a le soutien de la population parisienne et de l'Université, qu'il a su séduire en promettant l'établissement d'une ordonnance proche de celle de 1357. Pouvant prendre le pouvoir, il peut donc avouer publiquement l'assassinat. Loin de s'en cacher, Jean sans Peur fait rédiger un éloge du tyrannicide par Jean Petit, théologien de la Sorbonne, auquel répond le contre-argumentaire de Thomas de Bourg, abbé de Cerisy, qui prend le parti de la veuve du duc d'Orléans, Valentine Visconti. Charles VI, pour apaiser les belligérants à la suite de cet assassinat, convoque le duc de Bourgogne et les enfants du défunt le 28 février 1409 à Chartres : il charge également le comte Guillaume IV de Hainaut, beau-frère de Jean sans Peur, d'assurer, à la tête de 400 hommes d'armes et de 100 archers, la protection de chacune des délégations lors de ses déplacements et de s'engager à se ranger aux côtés du parti éventuellement agressé. Le 15 avril 1410, à Gien, lors des noces de Charles d'Orléans, le fils du duc assassiné, et de Bonne d'Armagnac, les grands du royaume présents se liguent contre le duc de Bourgogne. La guerre civile entre Armagnacs et Bourguignons qui s'ensuit ne se terminera que 25 ans plus tard, avec la signature du traité d'Arras le 21 septembre 1435. Jean sans Peur sera lui-même assassiné par les Armagnacs en 1419.

Notes et références

Notes

Références

Bibliographie

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 .
 , , .
 , .

 .
  Eric Jager, Blood Royal : A True Tale of Crime and Detection in Medieval Paris, New York, Little, Brown and Company, 2014.
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Louis 1 Orleans
Louis 1 Orleans
Louis 1 Orleans
Louis 1 Orleans
Événement de la guerre de Cent Ans
Guerre civile entre Armagnacs et Bourguignons
1407


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James undertakes all types of wine-related work. He has vast experience as a wine & drinks trainer and educator and currently works as wine animator for a large Corporate Events company fronting tastings, quizzes and gourmet dinners throughout the UK. He also offers wine expertise and support to a large drinks company, their staff & customers, in the choice of appropriate wines & lists and wine sales development.
With almost a quarter of a century's full time experience within the Wine Trade, much of it spent travelling through the wine-making regions of Europe, James possesses dual nationality (British & French) & has his own Fine Wine Shop in Stone, Staffordshire.
He works throughout the UK as a trainer for drinks staff in top restaurants, bars & hotels and as a specialist consultant for food & wine matching & Restaurant Wine List selection. He is a Judge for the prestigious International Wine Challenge & for the English Vineyards Association & has vast experience of animating & speaking at themed Wine Tastings & Dinners throughout the UK. He writes a monthly wine column for the North Staffordshire Magazine.
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Hurtwood rebrands!
Monet, Matisse and Me
Working with great works of art is a huge privilege. But doing justice to the original is our biggest challenge.
In 2016 I visited the exhibition at the Royal Academy in London 'Painting the Modern Garden: Monet to Matisse' and, apart from being a deeply satisfying experience, it brought to mind again how hugely enjoyable it is to print these great works.
Re-creating the work of great artists
I have been very fortunate to work up close and personal with the works of two of the artists from the exhibition, Monet and Matisse.
A few years ago, having just completed a book of Joan Miró's masonite pictures for an exhibition at Art Basel we embarked on two new publications for Helly Nahmad Gallery in London. The gallery specialises in classic modern and post-war art and was mounting an exhibition of Monet to be followed by another on Matisse. Both books were to be designed by the highly creative designer Fernando Guttiérrez.
As is usual with work today, the first proofs were made from the digital files and then compared against the originals and colour corrected.
Meeting the original works for the first time is always an exciting and sobering moment. With Monet and Matisse it's a privilege to be alone in a room with some of the most beautiful and inspiring art ever made with only my notebook and a cup of tea for company. But it's daunting too because I'm attempting to discern and assess the difference between original and proof and how best to reduce this difference. Also, these are differences not just of colour in an absolute sense, but also of scale. The original painting is usually considerably larger in real life than the printed reproduction and the artist, of course, had any colour they wished at their disposal whereas I am limited to four [we can and do use special inks and non-standard process colours but that's for a different blog]. We must be able to have confidence in the changes made because from this point we're on our own. These aren't the client's corrections; they're ours.
Colour and space
One of the greatest impressionists alive and still working today is, of course, Bridget Riley. I made a series of books for Bridget and was lucky enough to be able to talk with her about colour and how we can better reproduce her work in our printing. What became clear is that it's not really about specific colours but rather the relationships between colours, just as it's not about placing ink on the paper but rather allowing the light within the work to radiate out.
Open your Mind to the Picture
Once you realise this, colour corrections become relatively simple. You will feel the changes you need. Some specifics are necessary but once you have stopped worrying about what you think you can't do you will know how to change your proof.
The effect of light, colour and the relationship of colours to each other is demonstrated very clearly in this photo. They are both reproductions of Bridget Riley's painting Clepsydra (1976) and were both printed by the same company. The one on the left was printed in 2003 and the one on the right in 1978. There are several reasons for the difference but the main one is that the 2003 version forgot about colour 'differences' and assumed that the computer must be right and the 1978 version was made by people who knew what they wanted and used their equipment to get it. This is not a 'lost' skill (as I hope our books will demonstrate) but you do need to know that it was possible, and that it still is possible. Technology has made it easier in many ways but it's the people using the technology that's important, their knowledge, their skill (both taught and innate) and their understanding of colour and its perception. Incidentally, the 2003 book won industry awards for colour reproduction.
If you want to better understand why one image seems alive and the other not, remember that impressionists don't make colour with black and then take a look at a standard Photoshop separation. But however it's done, I know which one gives me the right impression.
Words: Francis Atterbury
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\section{Introduction}
Option pricing is an important field of research in financial economics from both a theoretical and practical
point of view. The pioneering work of Black and Scholes (1973) and Merton (1973) laid the foundations
of the field and stimulated important research in option pricing theory, its mathematical models and its computational techniques.
The Black-Scholes-Merton formula has been widely adopted by traders,
analysts and investors. Despite its popularity the Black-Scholes-Merton formula has been documented in many
studies in empirical finance that the Geometric Brownian Motion (GBM) assumed in the Black-Scholes-Merton model does not provide a realistic
description for the behavior of asset price dynamics. During the past few decades many extensions to the Black-Scholes-Merton model have been
introduced in the literature to provide more realistic descriptions for asset price dynamics. In particular, many models have been introduced
to explain the empirical behavior of the implied volatility smile and smirk. Such models include the stochastic volatility models,
jump-diffusion models and models driven by L\'{e}vy process.

Recently, there has been considerable interest in applications of regime
switching models driven by a Markov chain to various financial
problems. For an overview of Markov chains, see Elliott et al. (1994). Guo (2001) investigated European option pricing
problem under a regime switching model. Buffington
and Elliott (2002a,b) considered the option pricing problems for
European and American options in a Black-Scholes market in
which the states of the economy are described by a finite state Markov process. Boyle and Draviam proposed a
numerical method to solve the system of coupled partial differential equations for the price of exotic options under regime switching.
Zhu et al. (2012) derived a closed-form solution for European options in a two-state regime switching model. There is no closed-form solution to exotic options under a regime switching model. The numerical methods to solve a system of pricing partial differential equations is complex and computational time could be substantial.

In this paper, we investigate the pricing of European-style
lookback options when the price dynamics of the underlying
risky asset are governed by a Markov-modulated Geometric Brownian motion. The Markov-modulated Geometric Brownian motion generalizes the
Geometric Brownian motion by replacing the constant market parameters with the corresponding
market parameters depending on the states of a continuous-time Markov chain model.
The Markov-modulated
model can provide a more realistic way to describe and explain the market
environment. It has been mentioned in Yao et al. (2003) that
it is of practical importance to allow the market
parameters to respond to the movements of the general market levels
since the trend of general market levels is a key factor which
governs the price movements of individual risky assets.
Markov-modulated, or regime switching, models provide one possible
way to model the situation where the market parameters depend
on a market mode which switches among a finite number
of states and reflects the state of the underlying economy,
the macro-economic condition, the general mood of the investors
in the market, business cycles and other economic factors
(See Yao et al. (2003)). By introducing the Markov-modulated
Geometric Brownian motion model, we can model the structural changes in the
volatility of the risky assets and the relationship between the
stock price and the volatility due to the change in the market regime,
in particular, economic business cycles. We derive an analytical solution for lookback options
by means of the homotopy analysis method (HAM).
HAM was initially suggested by Ortega and Rheinboldt (1970) and has been successfully used to solve a number of heat transfer problems, see Liao and Zhu (1999). Zhu (2006) proposed to adopt HAM to obtain an analytic pricing formula for American options in the Black-Scholes model. Gounden and O'Hara (2010) extended the work of Zhu to pricing an American-style Asian option of floating strike type in the Black-Scholes model. Leung (2013) used HAM to derive an analytic formula for lookback options under stochastic volatility.
The market described by the Markov-modulated Geometric Brownian motion is incomplete in
general as there is an additional source of uncertainty due to the
Markov chain. There is an infinity of equivalent
martingale measures and, hence, a range of arbitrage-free option prices.
Guo (2001) augmented the market described by a Markov-modulated
GBM by introducing a set of Arrow-Debreu's securities in order to
complete the market. Buffington and Elliott (2001a, b) followed the approach
of Guo (2001) to derive the risk-neutral dynamics of the Markov-modulated GBM.
Elliott et al. (2005) introduced the regime switching Esscher transform
to determine an equivalent martingale measure when the price dynamics of the
underlying risky asset are governed by a Markov-modulated GBM. They
justified the choice of the martingale measure by
minimizing the relative entropy. In this paper, we will adopt the regime switching Esscher transform to determine an equivalent martingale measure. We will work in this framework to price a lookback option.

This paper is organized as follows:
Section $2$ describes the dynamics of the asset price under the Markov-modulated Geometric Brownian motion.
Section $3$ describes the regime switching Esscher transform and formulates the partial differential equation system for the price of a European-style vanilla option. Section $4$ formulates a floating-strike lookback option. Section $5$ derives an exact, closed-form solution for the floating-strike lookback option. Section $6$ briefly discusses a fixed strike lookback option.  The final section draws a conclusion.
\section{Asset Price Dynamics}
Consider a complete probability space $(\Omega, {\cal F}, {\cal P})$,
where $\cal P$ is a real-world probability measure.
Let $\cal T$ denote the time index set $[0,T]$  of the model.
Write $\{W_t\}_{t \in \cal T}$ for a standard Brownian motion on
$(\Omega, {\cal F}, {\cal P})$. Suppose the states of an economy
are modelled by a finite state continuous-time Markov chain
$\{X_t\}_{t \in \cal T}$ on $(\Omega, {\cal F}, {\cal P})$. Without loss of generality, we
can identify the state space of $\{X_t\}_{t \in \cal T}$
with a finite set of unit vectors ${\cal X} := \{e_1, e_2, \dots, e_N\}$,
where $e_i = (0, \dots, 1, \dots, 0) \in \cal R^{N}$.
We suppose that $\{X_t\}_{t \in \cal T}$ and $\{W_t\}_{t \in \cal T}$
are independent.

Let $A$ be the generator $[a_{ij}]_{i,j =1, 2, \dots, N}$ of the
Markov chain process. From Elliott et al. (1994), we
have the following semi-martingale representation theorem for
$\{X_t\}_{t \in \cal T}$:
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0}
\begin{eqnarray}
X_t = X_0 + \int^{t}_{0} A X_s d s + M_t \ ,
\end{eqnarray}
where $\{M_t\}_{t \in \cal T}$ is an ${\cal R}^N$-valued
martingale increment process with respect to the filtration
generated by $\{X_t\}_{t \in \cal T}$.

We consider a financial model with two primary traded assets, namely
a money market account $B$ and a risky asset or stock $S$. Suppose
the market is frictionless; the borrowing and lending interest rates
are the same; the investors are price-takers.

The instantaneous market interest rate $\{r(t,
X_t)\}_{t \in \cal T}$ of the bank account is given by:
\begin{eqnarray}
r_t := r(t, X_t) = <r, X_t> \ ,
\end{eqnarray}
where $r := (r_1, r_2, \dots, r_N)$ with $r_i > 0$ for each $i= 1, 2, \dots, N$ and $<\cdot, \cdot>$ denotes the inner product in ${\cal R}^{N}$.

In this case, the dynamics of the price process $\{B_t\}_{t \in
\cal T}$ for the bank account are described by:
\begin{eqnarray}
d B_t = r_t B_t dt \ ,  \quad   B_0 = 1 \ .
\end{eqnarray}
Suppose the stock appreciation rate $\{\mu_t\}_{t \in \cal
T}$ and the volatility $\{\sigma_t\}_{t \in \cal T}$ of $S$
 depend on $\{X_t\}_{t \in \cal T}$ and are described by:
\begin{eqnarray}
\mu_t := \mu (t, X_t) = <\mu, X_t> \ , \quad \ \sigma_t := \sigma (t, X_t) = <\sigma,
X_t> \ ,
\end{eqnarray}
where $\mu := (\mu_1, \mu_2, \dots, \mu_N)$, $\sigma := (\sigma_1,
\sigma_2, \dots, \sigma_N)$ with $\sigma_i > 0$ for each $i = 1, 2,
\dots, N$ and $<\cdot, \cdot>$ denotes the inner product in ${\cal
R}^{N}$.

We assume that the price dynamics of the underlying risky asset
$S$ are governed by the Markov-modulated Geometric Brownian motion
:
\begin{eqnarray}
d S_t = \mu_t S_t d t + \sigma_t S_t d W_t \ ,
\quad
S_0 = s_{0}.
\end{eqnarray}
\section{Risk-Neutral Measure and a European-style Vanilla Option}
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}
In this section, we describe the
regime-switching Esscher transform introduced in Elliott et al.
(2005) in the context of a Markovian regime-switching Black-Scholes-Merton
economy. Here we employ the pricing methodology in Elliott et al. (2005) to determine an
equivalent martingale measure.

For each $t \in {\cal T}$, let $\theta_{t}$ denote a ${\cal G} (t)$-measurable
random variable, which represents a regime-switching Esscher parameter and is
defined by:
\begin{eqnarray}
\theta_{t} := \left <\mbox{\boldmath $\theta$}, {\bf X} (t) \right > \ ,
\end{eqnarray}
where $\mbox{\boldmath $\theta$} := (\theta_1, \theta_2, \dots, \theta_{N})^{\prime} \in \Re^{N}$ and
$\theta_{i}, i=1,2,\dots, N \in (-\infty, \infty)$.

Let $(\theta \cdot W) (t) := \int^{t}_{0} \theta_{u} d W_{u}$, for each $t \in {\cal T}$.
Define a $G$-adapted density process $\Lambda^{\theta} := \{ \Lambda^{\theta} (t) | t \in {\cal T} \}$
as below:
\begin{eqnarray}
\Lambda^{\theta} (t) := \frac { e^{  (\theta \cdot W) (t) }  }
{ E [ e^{ (\theta \cdot W) } (t) | {\cal F}^{\bf X} (t) ] } \ , \quad t \in {\cal T} \ .
\end{eqnarray}
Here $E [\cdot]$ represents expectation under ${\cal P}$.

Then, applying It\^o's differentiation rule on $e^{  (\theta \cdot W) (t) }$ and conditioning on ${\cal F}^{\bf X} (t)$,
\begin{eqnarray}
E [ e^{(\theta \cdot W)} (t) | {\cal F}^{\bf X} (t) ] = \exp \bigg ( \frac {1} {2} \int^{t}_{0} \theta^2_{u} d u  \bigg ) \ .
\end{eqnarray}
So,
\begin{eqnarray}
\Lambda^{\theta} (t) := \exp \bigg ( \int^{t}_{0} \theta_{u} d W_{u} - \frac {1} {2} \int^{t}_{0} \theta^2_{u} d u \bigg )  \ , \quad t \in {\cal T} \ .
\end{eqnarray}
Since $\theta_i < \infty$, for each $i = 1, 2,\dots, N$,
\begin{eqnarray*}
E \bigg [  \exp \bigg ( \frac {1} {2} \int^{T}_{0} \theta^2_{t} d t  \bigg  )  \bigg  ] < \infty \ .
\end{eqnarray*}
So, the Novikov condition is satisfied and $\Lambda^{\theta}$ is a $(G, {\cal P})$-martingale. Hence,
\begin{eqnarray*}
E [ \Lambda^{\theta} (T) ] = 1 \ .
\end{eqnarray*}
Now, we define the regime-switching Esscher transform ${\cal Q}_{\theta} \sim \cal P$ on ${\cal G} (T)$
associated with a time-indexed family of Esscher parameters $\theta := \{ \theta_{t} | t \in {\cal T}  \}$ by setting:
\begin{eqnarray}
\frac {d {\cal Q}_{\theta}} {d {\cal P}} := \Lambda (T) \ .
\end{eqnarray}
Suppose ${\tilde \theta} :=  \{ {\tilde \theta}_{t} | t \in {\cal T} \}$ denotes a time-indexed
family of risk-neutral regime-switching Esscher parameters. Let ${\tilde S} := \{ {\tilde S}_{t} | t \in {\cal T} \}$
denote the discounted price of the risky share such that ${\tilde S}_{t} := e^{ - \int_{0}^{t}r_u du} S_{t}$, for each
$t \in {\cal T}$. Then,  the martingale condition is given by considering an enlarged filtration as follows:
\begin{eqnarray}
{\tilde S}_{u} = E^{{\tilde \theta}} [ {\tilde S}_{t}  | {\cal G} (u) ] \ , \quad  \mbox {for any $t, u \in {\cal T}$ with $t \ge u$} \ , \quad \mbox{${\cal P}$-a.s.} \ ,
\end{eqnarray}
where $E^{\tilde \theta} [\cdot]$ denotes expectation with respect to ${\cal Q}_{\tilde \theta}$.

\noindent
By setting $u = 0$,
\begin{eqnarray}
S_{0} = E^{{\tilde \theta}} [ e^{-\int_{0}^{t}r_{u}du }  S_{t}  | {\cal F}^{\bf X} (t) ] \ , \quad  \mbox {for any $t \in {\cal T}$} \ , \quad \mbox{${\cal P}$-a.s.}
\end{eqnarray}
Let $E (t) := E^{{\tilde \theta}} [ e^{-\int_{0}^{t} r_{u}du}  S_{t}  | {\cal F}^{\bf X} (t) ]$, for each $t \in {\cal T}$.  Then, $E (t)$ is an almost surely constant
random variable taking value $S_{0}$ almost surely under the measure ${\cal P}$ (i.e., ${\cal P} ( E (t) = S_{0} ) = 1$, for each $t \in {\cal T}$).
When there is no regime switching,
the condition (3.7) coincides with the one presented in Gerber and Shiu (1994).

It has been shown in Elliott et al. (2005) that the martingale condition (3.7) is satisfied if and only if ${\tilde \theta}$ satisfies
\begin{eqnarray}
{\tilde \theta}_{t} = \frac {r_t - \mu_{t}} {\sigma_{t}} = \sum^{N}_{i = 1} \bigg ( \frac {r_{t} - \mu_i} {\sigma_i} \bigg ) \left < {\bf X} (t), {\bf e}_i \right > \ , \quad t \in {\cal T} \ .
\end{eqnarray}
Since $| {\tilde \theta} (t) | < \infty$, for each $t \in [0, T]$, ${\cal P}$-a.s., the Novikov condition
\begin{eqnarray*}
E \bigg [ \exp  \bigg ( \frac {1} {2} \int^{T}_{0} {\tilde \theta}^2_{t} d t  \bigg )  \bigg  ] < \infty  \ ,
\end{eqnarray*}
is satisfied. So, $\Lambda^{\tilde \theta}$ is a $(G, {\cal P})$-martingale, and
\begin{eqnarray*}
E [ \Lambda^{\tilde \theta} (T) ] = 1  \ .
\end{eqnarray*}
Then, the risk-neutral regime-switching Esscher transform ${\cal Q}_{\tilde \theta}$ is defined by setting:
\begin{eqnarray}
\frac {d {\cal Q}_{\tilde \theta}} {d {\cal P}} &=& \Lambda^{\tilde \theta} (T) \nonumber\\
&=&  \exp \bigg [\int^{t}_{0} \bigg (\frac {r_{u} - \mu_{u} } {\sigma_{u} }
\bigg ) d W_{u} - \frac{1}{2}\int^{t}_{0} \bigg (\frac {r_{u} - \mu_{u} } {\sigma_{u}}
\bigg )^2 d u \bigg] \ .
\end{eqnarray}
Then, using Girsanov's theorem,
\begin{eqnarray*}
{\tilde W}_{t} = W_{t} - \int^{t}_{0} \bigg (\frac {r_{u} - \mu_{u}} {\sigma_{u} } \bigg )  d u \ , \quad t \in {\cal T} \ ,
\end{eqnarray*}
is a standard Brownian motion with respect to $G$ under ${\cal Q}_{{\tilde \theta}}$.

We suppose that ${\tilde W}$ and ${\bf X}$ are independent under ${\cal Q}_{{\tilde \theta}}$.
Then, the probability law of the chain ${\bf X}$ is invariant under the measure change.

The price dynamics of the underlying risky share under ${\cal Q}_{{\tilde \theta}}$ are governed by
\begin{eqnarray}
d S_{t} = r_{t} S_{t} d t + \sigma_{t} S_{t} d {\tilde W}_{t} \ .
\end{eqnarray}
Write ${\tilde {\cal G}} (t)$ for the $\sigma$-field ${\cal F}^W (t) \vee {\cal F}^{\bf X} (T)$, for each
$t \in {\cal T}$. Then, given ${\tilde {\cal G}} (t)$, a conditional price of the option $V$ is:
\begin{eqnarray}
V (t) = E^{\tilde \theta} [ e^{-\int_{t}^{T} r_{u}du}  V (S_{T}) | {\tilde {\cal G}} (t) ] \ .
\end{eqnarray}
Given $S_{t} = s$ and ${\bf X} (t) = {\bf x}$, a price of the option $V$ is:
\begin{eqnarray}
V (t, s, {\bf x}) = E^{\tilde \theta} [ e^{-\int_{t}^{T} r_{u}du}  V (S_{T})  | S_{t} = s, {\bf X} (t) = {\bf x} ] \ .
\end{eqnarray}
Let $V_i := V(t, s, {\bf e}_i)$, for each $i = 1, 2,\dots, N$. Write ${\bf V} := (V_1, V_2,\dots, V_N)^{\prime}$,
so $V(t, s, {\bf x})= \left <{\bf V}, {\bf x} \right >$.  Then, Buffington and Elliott (2002a,b)
derives the following regime-switching PDE governing the
evolution of the price of the option $V := V (t, s, {\bf x})$:
\begin{eqnarray}
- r_{t} V + \frac {\partial V} {\partial t} + r_{t} s \frac {\partial V} {\partial s} + \frac {1} {2} \sigma^2_{t}
s^2 \frac {\partial^2 V} {\partial s^2} + \left <{\bf V}, {\bf A} {\bf x} \right > = 0 \ ,
\end{eqnarray}
with terminal condition:
\begin{eqnarray}
V(T, s, {\bf x}) = V(s) \ .
\end{eqnarray}
So, if ${\bf X} (t) := {\bf e}_i$ ($i = 1, 2,\dots, N $),
\begin{eqnarray}
\mu_{t} = \mu_i \ , \quad V (t, s, {\bf x})  = V (t, s, {\bf e}_i) := V_i \ ,
\end{eqnarray}
and $V_i$ ($i = 1, 2, \dots, N$) satisfy the following system of PDEs:
\begin{eqnarray}
- r_{i} V_i + \frac {\partial V_i} {\partial t} + r_{i} s \frac {\partial V_i} {\partial s} + \frac {1} {2} \sigma^2_i s^2 \frac
{\partial^2 V_i} {\partial s^2} + \left <{\bf V}, {\bf A} {\bf e}_i \right > = 0 \ ,
\end{eqnarray}
with the terminal condition:
\begin{equation}
V(T, s, {\bf e}_i)= V(s) \ ,  \quad \ i = 1, 2,\dots, N \  .
\end{equation}
\section{Lookback Option}
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
In this section, we now turn to the pricing of lookback options in a regime switching model. We shall adopt the risk-neutral price dynamics
of the risky stock under ${\cal Q}_{{\tilde \theta}}$ specified  in the last section to evaluate the price of lookback
 options. In particular, we consider a floating strike lookback option under a regime switching model. The payoff of this option is the difference between the maximum asset price over the time between initiation and expiration and the asset price at expiration. The maximum of the asset price up to time $t$ is denoted by
\begin{eqnarray}
Y(t)=\max_{0\le u\le t} S_{u}.
\end{eqnarray}
Then the payoff of the lookback option at expiration time $T$ is
\begin{eqnarray}
V(T)=Y(T)-S_{T}.
\end{eqnarray}
Given $S_{t} = s$, $Y(t)=y$ and ${\bf X} (t) = {\bf x}$, a price of the lookback option $V$ is:
\begin{eqnarray}
V (t, s,y,{\bf x}) = E^{\tilde \theta} [ e^{-\int_{t}^{T} r_{u}du}(Y(T)-S_{T})  | S_{t} = s, Y(t)=y, {\bf X} (t) = {\bf x} ] \ .
\end{eqnarray}
Applying the Feynman-Kac formula to the above equation, then $V (t, s,y,{\bf x})$ satisfies the system of partial differential equations (PDEs)
\begin{eqnarray}
\frac{\partial V}{\partial t}+r_{t}s\frac{\partial V}{\partial s}+\frac{1}{2}\sigma_{t}^{2}s^{2} \frac{\partial^{2}V}{\partial s^{2}}-r_{t}V+ \left <{\bf V}, {\bf A} {\bf x} \right > = 0 \ ,
\end{eqnarray}
in the region $\{(t,s,y); 0\le t< T, 0\le s \le y \}$ and satisfies the boundary conditions
\begin{eqnarray}
V(T,s,y,{\bf x})=f(s,y)=y-s, \quad 0\leq s\leq y \,
\end{eqnarray}
\begin{eqnarray}
\frac{\partial V}{\partial y}(t,y,y,{\bf x}) =0, \quad 0\leq t\le T,\quad y>0\,
\end{eqnarray}
Consequently, if ${\bf X} (t) := {\bf e}_i$ ($i = 1, 2,\dots, N $),
\begin{eqnarray}
\mu_{t} = \mu_i \ , \quad V (t,s,y,{\bf x})  = V (t,s,y,{\bf e}_i) := V_i \ ,
\end{eqnarray}
and $V_i$ ($i = 1, 2, \dots, N$) satisfy the following system of PDEs:
\begin{eqnarray}
- r_{i} V_i + \frac {\partial V_i} {\partial t} + r_{i} s \frac {\partial V_i} {\partial s} + \frac {1} {2} \sigma^2_i s^2 \frac
{\partial^2 V_i} {\partial s^2} + \left <{\bf V}, {\bf A} {\bf e}_i \right > = 0 \ ,
\end{eqnarray}
with the boundary conditions:
\begin{eqnarray}
V(T,s,y,{\bf e}_i )=f(s,y)=y-s,  \quad 0\leq s\leq y, \quad \ i = 1, 2,\dots, N \,
\end{eqnarray}
\begin{eqnarray}
\frac{\partial V}{\partial y}(t,y,y,{\bf e}_i) =0, \quad 0\leq t\le T,\quad y>0, \quad \ i = 1, 2,\dots, N.
\end{eqnarray}
\section{A closed-form formula}
\renewcommand{\theequation}{5.\arabic{equation}}
In this section, we restrict ourselves to a special case with the number of regimes $N$ being $2$ in order
to simplify our discussion. By means of the homotopy analysis method, we derive a closed-form solution for a floating strike lookback option under a regime switching model.
The payoff of the floating strike lookback option has a linear homogeneous property:
\begin{eqnarray}
f(s,y)=s g\bigg(\ln\big(\frac{y}{s}\big) \bigg)\
\end{eqnarray}
where
\begin{eqnarray}
g(z)=e^{z}-1, \quad z=\ln\big(\frac{y}{s}\big).
\end{eqnarray}
This linear homogeneous property along with the transformation $z=\ln(\frac{y}{s})$ and $U_{i}=\frac{V_{i}}{s}, i=1,2$ transform the system of equations (4.8)-(4.10) into
\begin{eqnarray}
\left\{\begin{array}{lll}
{\cal L}_{1} U_{1}(t,z)= a_{11}\big(U_{2}(t,z)-U_{1}(t,z)\big),\quad 0\leq t\leq T,\quad z>0
 \\
 U_{1}(T,z)=g(z)
 \\
\frac{\partial U_{1}}{\partial z}(t,z)|_{z=0}=0
\end{array} \right.
\end{eqnarray}
where
\begin{eqnarray}
 {\cal L}_{1}=\frac{\partial}{\partial t}+\frac{1}{2}\sigma_{1}^{2}\frac{\partial^{2}}{\partial z^{2}}-(r_{1}+\frac{\sigma_{1}^{2}}{2})\frac{\partial }{\partial z} \,
\end{eqnarray}
and
\begin{eqnarray}
\left\{\begin{array}{lll}
 {\cal L}_{2} U_{2}(t,z)=a_{22}\big(U_{1}(t,z)-U_{2}(t,z)\big),\quad 0\leq t\leq T,\quad z>0
 \\
 U_{2}(T,z)=g(z)
 \\
\frac{\partial U_{2}}{\partial z}(t,z)|_{z=0}=0
\end{array} \right.
\end{eqnarray}
where
\begin{eqnarray}
 {\cal L}_{2}=\frac{\partial}{\partial t}+\frac{1}{2}\sigma_{2}^{2}\frac{\partial^{2}}{\partial z^{2}}-(r_{2}+\frac{\sigma_{2}^{2}}{2})\frac{\partial }{\partial z}.
\end{eqnarray}
Following the same line as Leung (2013), the homotopy analysis method is adopted to solve $ U_{i}(t,z), i=1,2$ from equations (5.13) and (5.15).

Now we introduce an embedding parameter $p\in [0, 1]$ and construct unknown functions
${\bar{U}}_{i}(t,z,p), i=1, 2$ that satisfy the following differential systems:
\begin{eqnarray}
\left\{\begin{array}{lll}
(1-p) {\cal L}_{1}[{\bar{U}}_{1}(t,z,p)-{\bar{U}}_{1}^{0}(t,z)]=-p\bigg\{{\cal A}_{1}[{\bar{U}}_{1}(t,z,p), {\bar{U}}_{2}(t,z,p)]  \bigg\}
 \\
 {\bar{U}}_{1}(t, z, p)=g(z)
 \\
\frac {\partial {\bar{U}}_{1}} {\partial z} (t, 0, p)= (1-p) \frac {\partial {\bar{U}}_{1}^{0}} {\partial z} (t,0)
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
(1-p) {\cal L}_{2}[{\bar{U}}_{2}(t,z,p)-{\bar{U}}_{2}^{0}(t,z)]=-p\bigg\{{\cal A}_{2}[{\bar{U}}_{1}(t,z,p), {\bar{U}}_{2}(t,z,p)]  \bigg\} \,
 \\
 {\bar{U}}_{2}(t, z, p)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{2}} {\partial z} (t, 0, p)= (1-p) \frac {\partial {\bar{U}}_{2}^{0}} {\partial z} (t,0) \,
\end{array} \right.
\end{eqnarray}
Here ${\cal L}_{i}, i=1,2$ is a differential operator defined as
\begin{eqnarray}
{\cal L}_{i}=\frac{\partial}{\partial t}+\frac{1}{2}\sigma_{i}^{2}\frac{\partial^{2}}{\partial z^{2}}-(r_{i}+\frac{\sigma_{i}^{2}}{2})\frac{\partial }{\partial z} \,
\end{eqnarray}
and ${\cal A}_{i}, i=1,2$ are functionals defined as
\begin{eqnarray}
{\cal A}_{1}[{\bar{U}}_{1}(t,z,p),{\bar{U}}_{2}(t,z,p)]={\cal L}_{1}({\bar{U}}_{1})-a_{11}({\bar{U}}_{1}-{\bar{U}}_{2})
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{2}[{\bar{U}}_{1}(t,z,p),{\bar{U}}_{2}(t,z,p)]={\cal L}_{2}({\bar{U}}_{2})-a_{22} ({\bar{U}}_{2}-{\bar{U}}_{1})
\end{eqnarray}
With $p=1$, we have
\begin{eqnarray}
\left\{\begin{array}{lll}
{\cal L}_{1}({\bar{U}}_{1})=a_{11}({\bar{U}}_{1}-{\bar{U}}_{2}) \,
 \\
 {\bar{U}}_{1}(t, z, 1)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{1}} {\partial z} (t, z, 1)|_{z=0}= 0 \,
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
{\cal L}_{2}({\bar{U}}_{2})=a_{22} ({\bar{U}}_{2}-{\bar{U}}_{1}) \,
 \\
 {\bar{U}}_{2}(t, z, 1)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{2}} {\partial z} (t, z, 1)|_{z=0}= 0 \,
\end{array} \right.
\end{eqnarray}
Comparing with (5.13) and (5.15), it is obvious that ${\bar{U}}_{i}(t,z,1), i=1,2$ are equal to our searched solutions $U_{i}(t,z), i=1,2$.

Now we set $p=0$, the equations (5.17) and (5.18) become
\begin{eqnarray}
\left\{\begin{array}{lll}
 {\cal L}_{1}[{\bar{U}}_{1}(t,z,p)]={\cal L}_{1}[{\bar{U}}_{1}^{0}(t,z)] \,
 \\
 {\bar{U}}_{1}(t, z, 0)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{1}} {\partial z} (t, 0, 0)= \frac {\partial {\bar{U}}_{1}^{0}} {\partial z} (t,0) \,
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
 {\cal L}_{2}[{\bar{U}}_{2}(t,z,p)]={\cal L}_{2}[{\bar{U}}_{2}^{0}(t,z)] \,
 \\
 {\bar{U}}_{2}(t, z, 0)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{2}} {\partial z} (t, 0, 0)=  \frac {\partial {\bar{U}}_{2}^{0}} {\partial z} (t,0) \,
\end{array} \right.
\end{eqnarray}
${\bar{U}}_{i}(t,z,0), i=1,2$ will be equal to ${\bar{U}}_{i}^{0}(t,z)$ when ${\bar{U}}_{i}^{0}(T,z)=g(z), i=1,2$. ${\bar{U}}_{i}^{0}(t,z)$ is known as the initial guess of $U_{i}(t,z)$.
Following the same line as Leung (2013), ${\bar{U}}_{i}^{0}(t,z)$ is chosen as the solution of the following PDEs:
\begin{eqnarray}
\left\{\begin{array}{lll}
 {\cal L}_{1}[{\bar{U}}_{1}^{0}(t,z)]=0 \,
 \\
 {\bar{U}}_{1}^{0}(0, z)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{1}^{0}} {\partial z} (t, z)|_{z=0}= 0 \,
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
 {\cal L}_{2}[{\bar{U}}_{2}^{0}(t,z)]=0 \,
 \\
 {\bar{U}}_{2}^{0}(0, z)=g(z) \,
 \\
\frac {\partial {\bar{U}}_{2}^{0}} {\partial z} (t,z)|_{z=0}=0 \,
\end{array} \right.
\end{eqnarray}
Note that $s{\bar{U}}_{1}^{0}(t,z)$ is the price of the floating strike lookback put option under the Black-Scholes-Merton model. Its explicit, closed-form formula is given in Goldman, et al. (1979):
\begin{eqnarray}
 {\bar{U}}_{i}^{0}(t,z)=e^{z}e^{-r_{i}(T-t)}N(-d_{M}^{-})-N(-d_{M}^{+})+\frac{{\sigma}_{i}^2}{2r_{i}}\big[N(d_{M}^{+})-e^{-r_{i} (T-t)}e^{\frac{2r_{i}z}{{\sigma}_{i}^2}}N(d_{M}^{\prime})\big],
\end{eqnarray}
where
\begin{eqnarray}
d_{M}^{\pm}=\frac{-z+(r_{i}\pm\frac{{\sigma}_{i}^2}{2}(T-t)}{\sqrt{{\sigma}_{i}^2(T-t)}},
\end{eqnarray}
\begin{eqnarray}
d_{M}^{\prime}=d_{M}^{+}-\frac{2r_{i}}{{\sigma}_{i}}\sqrt{T-t}
\end{eqnarray}
and $N(.)$ is the cumulative standard normal distribution
\begin{eqnarray}
N(y)=\frac{1}{2\pi}\int_{-\infty}^{y}e^{-\frac{z^2}{2}}dz.
\end{eqnarray}
To find the values of ${\bar{U}}_{i}(t, z, 1), i=1,2$,
 we can expand the functions ${\bar{U}}_{i}(t, z, p)$ as a Taylor's series expansion of $p$
 \begin{eqnarray}
\left\{\begin{array}{lll}
{\bar{U}}_{1}(t, z, p)=\sum_{m=0}^{\infty}\frac{{\bar{U}}_{1}^{m}(t, z)}{m !}p^{m}\,
\\
{\bar{U}}_{2}(t, z, p)=\sum_{m=0}^{\infty}\frac{{\bar{U}}_{2}^{m}(t, z)}{m !}p^{m}\,
\end{array} \right.
\end{eqnarray}
 where
 \begin{eqnarray}
\left\{\begin{array}{lll}
{\bar{U}}_{1}^{m}(t, z)=\frac{\partial ^{m}}{\partial p^{m}} {\bar{U}}_{1}(t, z, p)|_{p=0}\,
\\
{\bar{U}}_{2}^{m}(t, z)=\frac{\partial ^{m}}{\partial p^{m}} {\bar{U}}_{2}(t, z, p)|_{p=0}\,
\end{array} \right.
\end{eqnarray}
To find ${\bar{U}}_{1}^{m}(t, z)$ and ${\bar{U}}_{2}^{m}(t, z)$ in equation (5.32), we put (5.32) into (5.17) and (5.18) respectively and obtain the following recursive relations:
\begin{eqnarray}
\left\{\begin{array}{lll}
{\cal L}_{1}({\bar{U}}_{1}^{m})=a_{11}({\bar{U}}_{1}^{m-1}-{\bar{U}}_{2}^{m-1})\quad m=1,2,..., \,
 \\
 {\bar{U}}_{1}^{m}(T, z)=0 \,
 \\
\frac {\partial {\bar{U}}_{1}^{m}} {\partial z} (t, z)|_{z=0}= 0 \,
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
{\cal L}_{2}({\bar{U}}_{2}^{m})=a_{22} ({\bar{U}}_{2}^{m-1}-{\bar{U}}_{1}^{m-1})\quad m=1,2,..., \,
 \\
 {\bar{U}}_{2}^{m}(T, z)=0 \,
 \\
\frac {\partial {\bar{U}}_{2}^{m}} {\partial z} (t, z)|_{z=0}= 0 \,
\end{array} \right.
\end{eqnarray}
We introduce the following transformations:
\begin{eqnarray}
\tau=T-t,\quad \alpha_{i}=\frac{2 r_{i}}{\sigma_{i}^2},
\end{eqnarray}
and
\begin{eqnarray}
{\bar{U}}_{i}^{m}(t, z)=e^{[-\frac{1}{8}\sigma_{i}^2(\alpha_{i}+1)^2 \tau + \frac{1}{2}(\alpha_{i}+1)z]}{\hat{U}}_{i}^{m}(t, z).
\end{eqnarray}
 We can rewrite equations (5.34) and (5.35) in the form of standard nonhomogeneous diffusion equations
 \begin{eqnarray}
\left\{\begin{array}{lll}
\frac{\partial {\hat{U}}_{1}^{m}}{\partial \tau}-\frac{1}{2}\sigma_{1}^2\frac{\partial^{2} {\hat{U}}_{1}^{m}}{\partial z^{2}}=a_{11} e^{[-\frac{1}{8}\sigma_{1}^2(\alpha_{1}+1)^2 \tau + \frac{1}{2}(\alpha_{1}+1)z]}\big({\bar{U}}_{1}^{m-1}-{\bar{U}}_{2}^{m-1}\big) \,
 \\
 {\hat{U}}_{1}^{m}(0, z)=0 \,
 \\
\frac {\partial {\hat{U}}_{1}^{m}} {\partial z} (\tau, 0)+\frac{1}{2}(\alpha_{1}+1){\hat{U}}_{1}^{m}(\tau, 0)= 0 \,
\end{array} \right.
\end{eqnarray}
\begin{eqnarray}
\left\{\begin{array}{lll}
\frac{\partial {\hat{U}}_{2}^{m}}{\partial \tau}-\frac{1}{2}\sigma_{2}^2\frac{\partial^{2} {\hat{U}}_{2}^{m}}{\partial z^{2}}=a_{22} e^{[-\frac{1}{8}\sigma_{2}^2(\alpha_{2}+1)^2 \tau + \frac{1}{2}(\alpha_{2}+1)z]}\big({\bar{U}}_{2}^{m-1}-{\bar{U}}_{1}^{m-1}\big) \,
 \\
 {\hat{U}}_{2}^{m}(0, z)=0 \,
 \\
\frac {\partial {\hat{U}}_{2}^{m}} {\partial z} (\tau, 0)+\frac{1}{2}(\alpha_{2}+1){\hat{U}}_{2}^{m}(\tau, 0)= 0 \,
\end{array} \right.
\end{eqnarray}
The system of PDEs (5.38) and (5.39) has a well-known closed-form solution respectively:
\begin{eqnarray}
 {\hat{U}}_{1}^{m}(\tau, z)&=& a_{11}\int_{0}^{\tau}\int_{0}^{\infty} \exp\big\{-\frac{1}{8}\sigma_{1}^2(\alpha_{1}+1)^2 u + \frac{1}{2}(\alpha_{1}+1)\xi\big\}\nonumber\\
 &\times& \big({\bar{U}}_{1}^{m-1}(T-u,\xi)-{\bar{U}}_{2}^{m-1}(T-u,\xi)\big)G_{1}(\tau-u,z,\xi)d\xi du,
\end{eqnarray}
\begin{eqnarray}
 {\hat{U}}_{2}^{m}(\tau, z)&=&a_{22}\int_{0}^{\tau}\int_{0}^{\infty} \exp\big\{-\frac{1}{8}\sigma_{2}^2(\alpha_{2}+1)^2 u + \frac{1}{2}(\alpha_{2}+1)\xi \big\}\nonumber\\
 &\times& \big({\bar{U}}_{2}^{m-1}(T-u,\xi)-{\bar{U}}_{1}^{m-1}(T-u,\xi)\big)G_{2}(\tau-u,z,\xi)d\xi du,
\end{eqnarray}
where
\begin{eqnarray}
G_{1}(t,z,\xi)&=&\frac{1}{\sqrt{2\pi\sigma_{1}^2 t}}\bigg\{\exp\bigg[\frac{-(z-\xi)^{2}}{2\sigma_{1}^2 t} \bigg]+\exp\bigg[\frac{-(z+\xi)^{2}}{2\sigma_{1}^2 t} \bigg]\nonumber\\
&+&2\kappa_{1}\int_{0}^{\infty}\exp\bigg[-\frac{(z+\xi+\eta_{1})^2}{2\sigma_{1}^2 t}+\kappa_{1}\eta_{1}\bigg]d\eta_{1}\bigg\}\nonumber\\
&=&\frac{1}{\sqrt{2\pi\sigma_{1}^2 t}}\bigg\{\exp\bigg[\frac{-(z-\xi)^{2}}{2\sigma_{1}^2 t} \bigg]+\exp\bigg[\frac{-(z+\xi)^{2}}{2\sigma_{1}^2 t} \bigg]\nonumber\\
&+&2\kappa_{1}\exp(\kappa_{1}(-z-\xi+\frac{\sigma_{1}^{2}\kappa_{1}}{2}))N\big(\frac{\sigma_{1}^{2}\kappa_{1}t-z-\xi}{\sigma_{1}\sqrt{t}}\big)\bigg\},
\end{eqnarray}
\begin{eqnarray}
G_{2}(t, z,\xi)&=&\frac{1}{\sqrt{2\pi\sigma_{2}^2 t}}\bigg\{\exp\bigg[\frac{-(z-\xi)^{2}}{2\sigma_{2}^2 t} \bigg]+\exp\bigg[\frac{-(z+\xi)^{2}}{2\sigma_{2}^2 t} \bigg]\nonumber\\
&+&2\kappa_{2}\int_{0}^{\infty}\exp\bigg[-\frac{(z+\xi+\eta_{2})^2}{2\sigma_{2}^2 t}+\kappa_{2}\eta_{2}\bigg]d\eta_{2}\bigg\}\nonumber\\
&=&\frac{1}{\sqrt{2\pi\sigma_{2}^2 t}}\bigg\{\exp\bigg[\frac{-(z-\xi)^{2}}{2\sigma_{2}^2 t} \bigg]+\exp\bigg[\frac{-(z+\xi)^{2}}{2\sigma_{2}^2 t} \bigg]\nonumber\\
&+&2\kappa_{2}\exp(\kappa_{2}(-z-\xi+\frac{\sigma_{2}^{2}\kappa_{2}}{2}))N\big(\frac{\sigma_{2}^{2}\kappa_{2}t-z-\xi}{\sigma_{2}\sqrt{t}}\big)\bigg\},
\end{eqnarray}
and $$\kappa_{i}=\frac{1}{2}(\alpha_{i}+1), \quad \eta_{i}=e^{[-\frac{1}{8}\sigma_{i}^2(\alpha_{i}+1)^2 \tau + \frac{1}{2}(\alpha_{i}+1)z]},\quad i=1,2$$
\section{Fixed strike lookback options}
\renewcommand{\theequation}{6.\arabic{equation}}
The payoff of a fixed strike lookback option does not have a linear homogeneous property. Consequently, the dimension of the corresponding PDEs cannot be reduced.
However, a model independent put-call parity for lookback options, proposed by Wong and Kwok (2003) may be used to price a fixed strike lookback option.
Denote a fixed strike lookback call by $C_{fix}(t,s,y,K,{\bf x})$, then the put-call parity is given by
\begin{eqnarray}
C_{fix}(t,s,y,K,{\bf x})=V (t, s,y,{\bf x})+Ke^{-r_{t}(T-t)}.
\end{eqnarray}
\section{Conclusion}
We consider the pricing of the floating strike lookback option in a two-state regime switching model.
The closed-form analytical pricing formulas for the floating strike lookback option is derived by the means of the homotopy analysis method.

redpajama_set_name: RedPajamaArXiv


Feb. 12 (UPI) -- U.S. Secretary of State Rex Tillerson, meeting with Egypt's foreign minister Monday in Cairo, said the United States remains committed to the peace process in the Middle East amid its controversial decision to recognize Jerusalem as Israel's capital.
Tillerson began a five-nation visit to the region Monday -- with stops later in Jordan, Turkey, Lebanon and Kuwait. A visit to Israel is not planned.
"In the announcement President Trump made regarding Jerusalem, he also made two very important points, as well, one of which was, he was not advocating for any change to the status quo as to the oversight of the holy sites, recognizing the proper role of existing authorities," Tillerson said at a joint news conference with Egyptian Foreign Minister Shoukri. "And the second is that the final boundaries of Jerusalem are yet to be determined and will be decided among the parties."
The two spoke about the presidential election next Monday in Egypt, economic and military cooperation.
"The United States, as it does in all countries, supports a transparent and credible electoral process and all citizens being given the right and the opportunity to participate freely and fairly," Tillerson said.
Incumbent President Abdel Fattah el-Sissi is running largely unopposed.
"The Egyptian people have shown their commitment and their determination and their ability to change their course and to indicate their dissatisfaction if they deemed they were dissatisfied," Tillerson said. "They have been able to change two governments in the last seven years and they have undertaken legislative elections and know how to protect their rights and advocate."
Tillerson said they also spoke about the promotion and protection of human rights and the vital role of civil society in Egypt.
Shoukri said that "development in the social and political field is an evolutionary process."
The leaders also discussed the fight against terrorism.
"We agreed that we would continue our close cooperation on counter-terrorism measures, including our joint commitment to the defeat of ISIS, and Egypt has been a very important member of the defeat-ISIS coalition from the beginning," Tilleron said. "Egypt deals with the threats of ISIS themselves and are dealing with it currently in the Sinai."
"There is a common vision and a desire to continue working together with a view to stabilizing areas of tension in the region," Shoukri added.


My brother and his partner were presenting at this past week. I am extraordinarily proud that they recieved the high honour of the "Demo God" award for their work. This is an amazing and wonderful achievement.
Congratulations big brother. This is fabulous news.