# Virtual Takeovers in the Metaverse: Interrogating Power in Our Past and
Future(s) with Multi-Layered Narratives

Heather Snyder Quinn<EMAIL_ADDRESS>DePaul UniversityChicagoILUSA and
Jessa Dickinson<EMAIL_ADDRESS>DePaul UniversityChicagoILUSA

###### Abstract.

Mariah is an augmented reality (AR) mobile application that exposes power
structures (e.g., capitalism, patriarchy, white supremacy) through
storytelling and celebrates acts of resistance against them. People can use
Mariah to “legally trespass” the metaverse as a form of protest. Mariah
provides historical context to the user’s physical surroundings by
superimposing images and playing stories about people who have experienced,
and resisted, injustice. We share two implementations of Mariah that raise
questions about free speech and property rights in the metaverse: a protest
against museums accepting “dirty money” from the opioid epidemic, and a
commemoration of sites where people have resisted power structures. Mariah is
a case study for how experimenting with a technology in non-sanctioned ways
(i.e., “hacking”) can expose ways that it might interact with, and potentially
amplify, existing power structures.

Augmented reality, social justice, capitalism, speculative design, narrative,
opioid epidemic

Figure 1. Mariah augments an artifact in the Metropolitan Museum of Art’s
Dendur Wing with a statistic about the opioid crisis.

Image of Mariah overlay at the New York Met, statistic reads: To date, 300,000
Americans have died from overdoses related to OxyContin.

##  1\. Introduction

People create and use technologies in the context of interconnected structures
of power (e.g., capitalism, white supremacy, patriarchy (Collins, 2019;
Crenshaw, 1990)). Traditional companies are governed by capitalist rules, such
as legally requiring board members to protect and maximize the company’s
profits. For instance, board members of Purdue Pharma cited their feduciary
duties in defense of their production and aggressive marketing of a highly
addictive pain killer and opioid called OxyContin (Hakim et al., 2019). The
Sackler family and their company, Purdue Pharma, caused the opioid epidemic in
the United States with their perverse strategies to addict people to their
drug and then sell them addiction treatment (Hakim et al., 2019; Alpert et
al., 2022). From 1997 (when OxyContin was released) to 2017, deaths caused by
drug overdoses in the United States increased by 3.6% (Hedegaard et al.,
2018). The epidemic has claimed hundreds of thousands of lives (Totenburg,
2023), and has exasperated preexisting inequities along the fault lines of
race, class, gender, and region (Hansen and Netherland, 2016; Singh et al.,
2019).

The first author and her collaborators designed Mariah, an augmented reality
(AR) application (app), to expose the power structures behind the opioid
crisis. They used Mariah to critique the Metropolitan Museum of Art (the
“Met”) and the Louvre for accepting “dirty” donations from the Sackler family
(Pogrebin, 2021). The team also implemented Mariah in San Francisco as part of
an art installation to tell the stories of historic protests at different
sites across the city. Mariah creates a guerilla-style AR experience in which
users “legally trespass” the metaverse, viewing and listening to content that
tells a narrative that adds complexity and historical context to their
physical surroundings. We use Mariah as a case through which to explore how
using technologies in subversive ways can reveal potential future impacts the
technology could have on society. We ask:

  1. (1)

How might augmented reality be used to create counter-narratives that expose
and challenge power structures?

  2. (2)

What are the legal challenges and questions that arise from exploring the
metaverse as a site of protest?

We contribute to literature in human-computer interaction (HCI) that considers
how social and institutional contexts shape how technologies are designed,
governed, and used—as well as the impacts they have on society (Toyama, 2015;
Thebault-Spieker et al., 2015). Drawing from afrofuturist speculative design
(Harrington et al., 2022; Harrington and Dillahunt, 2021; Dosono et al., 2022;
Benjamin, 2019; Bray et al., 2022), we frame Mariah as a tool with which to
explore how power structures (and technologies) might shape our potential
futures (Collins, 2019). For instance, a collaborative team of community
members, designers, lawyers, and policymakers could use Mariah as a probe
(Mattelmäki et al., 2006; Tomlinson et al., 2020) to consider preposterous,
plausible, and probable futures. Or, perhaps more importantly, liberatory
futures. This continues the work in HCI that interrogates structures such as
capitalism (e.g., (Feltwell et al., 2018; Tomlinson et al., 2020)), white
supremacy (e.g., (Ogbonnaya-Ogburu et al., 2020; Erete et al., 2021a; To et
al., 2022)), and colonialism (e.g., (Irani et al., 2010; Das and Semaan,
2022)).

##  2\. Related Work

Mariah uses storytelling as a vehicle to personalize injustice and encourage
reflection in ways that traditional media and statistics may not, as found by
Zhang, et al. (2021) (Zhang et al., 2021). Drawing from (Solórzano and Yosso,
2002), we explore how we can use counter-narratives in our work as a way to
examine (Silva et al., 2022; Rankin et al., 2021), cope with (To et al.,
2022), and resist (Das and Semaan, 2022; Dickinson et al., 2021) dominating
power structures. Our focus on structural power follows work that takes a
transformative justice approach to expand our HCI approaches to issues to
include the structural factors that create them (Erete et al., 2021b; Rabaan,
2021; Musgrave et al., 2022). Continuing the agenda set out by Silva, et al.
(2022) (Silva et al., 2022), we raise concerns about how free speech, property
rights, and privacy in AR will be addressed in the context of capitalism and
legislative structures that are rooted in racial capitalism (Gilmore, 2007;
Robinson, 2020; Melamed, 2015).

##  3\. Augmented Reality: The Basics

Augmented reality (AR) is an emerging technology that superimposes a computer-
generated image on a user’s view of the real world, thus providing a composite
view. AR is one of the ways in which users see and access the metaverse (also
called the AR cloud and the spatial web). Currently, AR can only be viewed via
a limited number of applications (apps), and there is a risk that the spatial
web will become dominated by a centralized authority. The potential for
monopoly control of the metaverse has raised alarm in the HCI community (Silva
et al., 2022). In order to discuss the social implications of AR, we share a
basic description of how the technology works. AR can currently be implemented
in three ways:

  1. (1)

Trigger Image: An AR application uses a phone’s camera to scan for a
particular image (e.g., a QR code). When the application recognizes the image,
it will reveal an image, text, video, or sound to the user.

  2. (2)

Geolocation: A phone’s latitude and longitude coordinates can be used as an AR
trigger. When a user enters the trigger area, the app will augment the
physical environment.

  3. (3)

Ground/Plane: AR can virtually place an object in a user’s environment using
their camera display.

Mariah uses all of these technologies to superimpose images and text on the
user’s display, and to play audio of stories relevant to the surroundings. The
first author and her collaborators created Mariah using public photographs as
the trigger images. They did not have to set foot in the museum to create an
application that could augment it, which raised ethical and legal questions we
will explore in this paper.

##  4\. Mariah

Mariah is a collaboratively developed initiative that challenges institutional
power, including big tech and big pharma, by “hacking the metaverse” as an act
of protest. The application is named after Mariah Lotti, who lost her life to
opioids at 19 years old. The first author created the initiative in
partnership with Mariah’s mother, Rhonda Lotti, and Adam DelMarcelle, who lost
his brother to opioids. They are designers and artists and approached Mariah
as a protest and a way to personally process the premature deaths of their
loved ones. Their second implementation (“Strikethrough”) of Mariah expanded
its scope to include the commemoration of sites of protest. In this section,
we describe both implementations of Mariah. For more information, see the
short documentary film Mariah (which premiered in 2023 at The New Media Film
Festival and won Best AR Film), a video demonstration of the Mariah at the
Met, and the Strikethrough Instagram reel, and press articles (Blakemore,
2020; Bishara, 2020).

###  4.1. Exposing “Dirty Money” in Museums

The team initially designed Mariah for use in the Temple of Dendur in the Met.
At the time, the wing of the museum that holds the iconic Dendur Temple was
named for the Sackler family, in honor of their financial contribution to the
museum (their name has since been removed) (Pogrebin, 2021). Launched in 2020,
Mariah transformed the Met’s Sackler wing into a virtual memorial for Mariah
Lotti and others who had lost their lives to the opioid epidemic. Exploring
AR’s potential to revise historical narratives and its ethical implications,
the app augments Sackler donated art with “virtual memorials” (audio and video
of the lives of opioid victims). Mariah is a publicly available mobile app.
The first author and her team did not gather data on app usage because they
created it as a protest and memorial, rather than a research study. People
learned about Mariah through word of mouth and media coverage.

The augmented experience includes audio, video, and graphics that correspond
with sixteen (16) Sackler-donated artifacts in the Dendur Wing. Each sculpture
is augmented visually with a global statistic about the opioid epidemic. The
app also plays audio of Mariah’s mother telling stories about Mariah’s life to
humanize the opioid epidemic. The experience concludes in the Temple of
Dendur, where the viewer is able to place a virtual memorial in the room that
renames the room “The Mariah Lotti Memorial Gallery” (see figure 2).

Following the “installation” of Mariah at the Met, the artists expanded the
project to The Pyramid du Louvre in Paris, France (which also received Sackler
donations (Pogrebin, 2021)) with a geolocative installation. Titled “Funded by
Loss: Death in Real-Time,” the installation is a large-scale, 3-dimensional
number that increases every 5 minutes to represent real-time opioid fatalities
(see figure 2). This installation also tells Joseph DelMarelle’s story, who
lost his son to an epidemic that was created by, and enriched, the Sackler
family and Purdue Pharma (Alpert et al., 2022; Pogrebin, 2021).

Figure 2. Screenshots of the museum installations of Mariah. The middle screen
shows the virtual memorial for Mariah Lotti “placed” in the Temple of Dendur
in the Met. The right screen shows the opioid “death clock” superimposed over
the Louvre.

###  4.2. Commemorating Sites of Protest with Mariah

In 2022, the Letterform Archive commissioned the first author and her team to
expand the Mariah app as a vehicle of protest for the Strikethrough:
Typographic Messages of Protest exhibition. They used Mariah to augment seven
sites of historic protest with historic audio and video throughout the San
Francisco Bay area. The locations were marked on a map and with removable
spray chalk on sidewalks (see figure 3). The sites included:

  * •

BLM: Black Lives Matter march across the Golden Gate Bridge, 2020

  * •

IOAT: Indians of all tribes, Occupation of Alcatrez, 1969-71

  * •

MARIAH: War on drugs at Civic Center and the Tenderloin, 2022

  * •

MILK: Harvey Milk memorial march on Castro and Market Streets, 1979

  * •

STRIKE: Shipyard worker strikes near the Bethlehem Steel building, 1941

  * •

FREE SPEECH: Student protests and poster printing on the Berkeley campus,
1960s

  * •

PANTHERS: Central HQ of the Black Panthers in West Oakland, 1967-71

  * •

OCCUPY: Shutdown of Oakland Port by Occupy movement for economic equality,
2011

Figure 3. Three locations where Mariah was implemented for Strikethrough. From
left to right: Strike, commemorating labor protests; Mariah, protesting the
war on drugs and the opioid epidemic; and Panthers, commemorating the Black
Panther’s headquarters.

##  5\. Discussion

###  5.1. Complicating Narratives and Exposing Power with AR

The museum and Strikethrough installations of Mariah addressed different
topics, but both used narratives to reveal power structures and complicate
narratives about societal issues. Mariah shifts the narrative focus of the the
opioid epidemic and the “war on drugs” from the individuals suffering from
addition, to the people and institutions that caused these crises (Aizen et
al., 2018; Hakim et al., 2019). The personal stories told through Mariah also
serve to humanize the people lost in this epidemic, which resists narratives
that dehumanize opioid victims (e.g., referring to people as “addicts,” “drug
users”). In this way, Mariah seeks to build a counter-story (Solórzano and
Yosso, 2002) about the opioid epidemic and the war on drugs. Strikethrough
also leveraged Mariah as a tool to celebrate collective action against
capitalism, white supremacy, and cis-hetero-patriarchy by creating virtual
memorials and sharing historical stories.

Mariah was a project that was rooted in personal experience and loss. The
people who created Mariah worked from their experiences of the opioid crisis
as white people, and shared the personal stories of their loved ones from that
standpoint. In the museum installations of Mariah, they did not take on the
intersectional (Collins, 2019) dimensions of the opioid epidemic (Hansen and
Netherland, 2016; Singh et al., 2019) . The Strikethrough installation gave
them the opportunity to examine and protest the institutions and power
structures that created the racist war on drugs beginning in the 1970s and
subsequent concentrated incarceration of Black and Brown neighborhoods
(Alexander, 2012; Lugalia-Hollon and Cooper, 2018; Travis et al., 2014). The
opioid epidemic that began in the 1990s has been treated more so as a public
health issue than a criminal issue, largely due to the racial and class
differences between the populations most impacted (Hansen and Netherland,
2016).

Mariah is an example of how AR can be leveraged as a tool for examining power
and building counter-narratives. Although the Met was not directly involved in
the opioid crisis, they benefited from it in the form of large donations from
the Sacklers. The Met is a powerful cultural institution held in high regard
by the public, and Mariah complicated the public narrative about the museum
while prompting reflection on the impacts of capitalism on our society. Power
structures such as capitalism are codified and implemented through our legal
system, which will be tasked with governing the metaverse in the future.

###  5.2. Legal Questions in the Metaverse

The artists who created Mariah did not receive permission to augment the Met,
but consulted with an attorney and received a letter of support regarding its
legality under the First Amendment. The experience of “installing” Mariah
prompted the first author to consider future implications of our human
freedoms in the metaverse, including interpretations of free speech and
property rights (both physical and intellectual). Who owns our virtual space
and what can be placed there? According to lawyer Brian Wassom, “of particular
concern is how AR will challenge traditional property rights and stretch
interpretations of free speech.” Wassom asserts that AR content is an
expression akin to digital speech and is protected by the First Amendment. If
this is correct, then the digital space around (maybe even inside) your
property could become “a kind of common land making every street, every
surface, every portion of space with a GPS coordinate, free real estate for
those with the ability and the will to augment.” The implications are
concerning, especially considering that the same institutions and power
structures that created the war on drugs and the opioid epidemic will be
tasked with governing the metaverse. We pose that the method the first author
employed with Mariah (i.e., using AR in an experimental, subversive way) can
be used to as a speculative design probe (Harrington et al., 2022; Dosono et
al., 2022; Bray et al., 2022; Mattelmäki et al., 2006) to explore
technologies’ potential societal impacts and collectively envision more just
futures.

##  6\. Conclusion

The Met never commented on Mariah, though it replaced some artifacts in the
Dendur Wing, which the artists re-augmented. Following litigation against the
Sacklers and Purdue Pharma (Hakim et al., 2019), the Met removed the Sackler
name (Pogrebin, 2021). Although this is a positive step, simply removing the
Sackler name obscures the complex history of funding and harm instead of
engaging with it. In future work, the first author will use AR to explore how
museums have come to possess certain cultural artifacts, and the role that
colonization and other forms of power played in the artifacts’ journeys.

##  7\. Acknowledgements

Artists: Adam DelMarcelle, Heather Snyder Quinn. Curators: Silas Munro,
Stephen Coles. Contributors: Mariah Lotti, Rhonda Lotti, Westfield Productions
Design and tech: Zishou Wang, Valerie Shur, Leslie Ramirez, Les Garcia, Jake
Juraka, Flor Salatino. Funders: The Letterform Archive, Washington University
in St Louis, DePaul University. The Mariah app and website use Redaction—a
typeface that charts the history and abuses of the US criminal justice system.

## References

  * (1)
  * Aizen et al. (2018) Roy Aizen, Gabriela Marcu, Anjali Misra, Gregory Sieber, David G. Schwartz, Alexis Roth, and Stephen Lankenau. 2018.  Designing an Emergency Response Community for Opioid Overdoses in Philadelphia. In _Extended Abstracts of the 2018 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Montreal QC¡/city¿, ¡country¿Canada¡/country¿, ¡/conf-loc¿) _(CHI EA ’18)_. Association for Computing Machinery, New York, NY, USA, 1–6.  https://doi.org/10.1145/3170427.3188581
  * Alexander (2012) Michelle Alexander. 2012.  _The New Jim Crow: Mass Incarceration in the Age of Colorblindness_.  The New Press.
  * Alpert et al. (2022) Abby Alpert, William N Evans, Ethan MJ Lieber, and David Powell. 2022.  Origins of the opioid crisis and its enduring impacts.  _The Quarterly Journal of Economics_ 137, 2 (2022), 1139–1179.
  * Benjamin (2019) Ruha Benjamin. 2019.  _Captivating technology: Race, carceral technoscience, and liberatory imagination in everyday life_.  Duke University Press.
  * Bishara (2020) Hakim Bishara. 2020.  An Augmented Reality App Connects the Life of an Opioid Victim to the Met Museum’s Sackler Wing.  https://hyperallergic.com/592739/mariah-app-metropolitan-museum/ Accessed: April 17, 2024.
  * Blakemore (2020) Erin Blakemore. 2020.  Mariah App, an augmented reality program, reveals haunting link of Met’s Sackler Wing to opioid crisis.  https://www.washingtonpost.com/health/sackler-opioid-victim-oxycontin-mariah-lotti/2020/11/06/e99207f8-1ed2-11eb-ba21-f2f001f0554b_story.html Accessed: April 17, 2024.
  * Bray et al. (2022) Kirsten E Bray, Christina Harrington, Andrea G Parker, N’Deye Diakhate, and Jennifer Roberts. 2022.  Radical futures: Supporting community-led design engagements through an afrofuturist speculative design toolkit. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_. 1–13.
  * Collins (2019) Patricia Hill Collins. 2019.  _Intersectionality as critical social theory_.  Duke University Press.
  * Crenshaw (1990) Kimberle Crenshaw. 1990.  Mapping the margins: Intersectionality, identity politics, and violence against women of color.  _Stan. L. Rev._ 43 (1990), 1241.
  * Das and Semaan (2022) Dipto Das and Bryan Semaan. 2022.  Collaborative Identity Decolonization as Reclaiming Narrative Agency: Identity Work of Bengali Communities on Quora. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿New Orleans¡/city¿, ¡state¿LA¡/state¿, ¡country¿USA¡/country¿, ¡/conf-loc¿) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 236, 23 pages.  https://doi.org/10.1145/3491102.3517600
  * Dickinson et al. (2021) Jessa Dickinson, Jalon Arthur, Maddie Shiparski, Angalia Bianca, Alejandra Gonzalez, and Sheena Erete. 2021.  Amplifying Community-led Violence Prevention as a Counter to Structural Oppression.  _Proc. ACM Hum.-Comput. Interact._ 5, CSCW1 (2021), 28.
  * Dosono et al. (2022) Bryan Dosono, Ihudiya Finda Ogbonnaya-Ogburu, Yolanda A. Rankin, Angela D. R. Smith, and Kentaro Toyama. 2022.  Anti-Racism in Action: A Speculative Design Approach to Reimagining SIGCHI. In _Extended Abstracts of the 2022 CHI Conference on Human Factors in Computing Systems_ (New Orleans, LA, USA) _(CHI EA ’22)_. Association for Computing Machinery, New York, NY, USA, Article 156, 5 pages.  https://doi.org/10.1145/3491101.3516504
  * Erete et al. (2021a) Sheena Erete, Yolanda A. Rankin, and Jakita O. Thomas. 2021a.  I Can’t Breathe: Reflections from Black Women in CSCW and HCI.  _Proc. ACM Hum.-Comput. Interact._ 4, CSCW3, Article 234 (Jan. 2021), 23 pages.  https://doi.org/10.1145/3432933
  * Erete et al. (2021b) Sheena Erete, Karla Thomas, Denise Nacu, Jessa Dickinson, Naomi Thompson, and Nichole Pinkard. 2021b.  Applying a transformative justice approach to encourage the participation of Black and Latina Girls in computing.  _ACM Transactions on Computing Education (TOCE)_ 21, 4 (2021), 1–24.
  * Feltwell et al. (2018) Tom Feltwell, Shaun Lawson, Enrique Encinas, Conor Linehan, Ben Kirman, Deborah Maxwell, Tom Jenkins, and Stacey Kuznetsov. 2018.  ”Grand Visions” for Post-Capitalist Human-Computer Interaction. In _Extended Abstracts of the 2018 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Montreal QC¡/city¿, ¡country¿Canada¡/country¿, ¡/conf-loc¿) _(CHI EA ’18)_. Association for Computing Machinery, New York, NY, USA, 1–8.  https://doi.org/10.1145/3170427.3170609
  * Gilmore (2007) Ruth Wilson Gilmore. 2007.  _Golden gulag: Prisons, surplus, crisis, and opposition in globalizing California_. Vol. 21.  Univ of California Press.
  * Hakim et al. (2019) Danny Hakim, Roni Caryn Rabin, and William K. Rashbaum. 2019.  Met Museum Removes Sackler Name From Wing Over Opioid Ties.  https://www.nytimes.com/2019/04/01/health/sacklers-oxycontin-lawsuits.html Accessed: April 17, 2024.
  * Hansen and Netherland (2016) Helena Hansen and Julie Netherland. 2016.  Is the prescription opioid epidemic a white problem?  , 2127–2129 pages.
  * Harrington and Dillahunt (2021) Christina Harrington and Tawanna R Dillahunt. 2021.  Eliciting Tech Futures Among Black Young Adults: A Case Study of Remote Speculative Co-Design. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Yokohama¡/city¿, ¡country¿Japan¡/country¿, ¡/conf-loc¿) _(CHI ’21)_. Association for Computing Machinery, New York, NY, USA, Article 397, 15 pages.  https://doi.org/10.1145/3411764.3445723
  * Harrington et al. (2022) Christina N. Harrington, Shamika Klassen, and Yolanda A. Rankin. 2022.  “All that You Touch, You Change”: Expanding the Canon of Speculative Design Towards Black Futuring. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (, New Orleans, LA, USA,) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 450, 10 pages.  https://doi.org/10.1145/3491102.3502118
  * Hedegaard et al. (2018) Holly Hedegaard, Arialdi Miniño, and Margaret warner. 2018.  Drug Overdose Deaths in the United States, 1999–2017.  _International Journal of Maternal and Child Health and AIDS_ 329 (2018), 1–8.
  * Irani et al. (2010) Lilly Irani, Janet Vertesi, Paul Dourish, Kavita Philip, and Rebecca E. Grinter. 2010.  Postcolonial Computing: A Lens on Design and Development. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ (Atlanta, Georgia, USA) _(CHI ’10)_. Association for Computing Machinery, New York, NY, USA, 1311–1320.  https://doi.org/10.1145/1753326.1753522
  * Lugalia-Hollon and Cooper (2018) Ryan Lugalia-Hollon and Daniel Cooper. 2018.  _The War on Neighborhoods: Policing, Prison, and Punishment in a Divided City_.  Beacon Press.
  * Mattelmäki et al. (2006) Tuuli Mattelmäki et al. 2006\.  _Design probes_.  Aalto University.
  * Melamed (2015) Jodi Melamed. 2015.  Racial Capitalism.  _Critical Ethnic Studies_ 1, 1 (2015), 76–85.  http://www.jstor.org/stable/10.5749/jcritethnstud.1.1.0076
  * Musgrave et al. (2022) Tyler Musgrave, Alia Cummings, and Sarita Schoenebeck. 2022.  Experiences of Harm, Healing, and Joy among Black Women and Femmes on Social Media. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (New Orleans, LA, USA) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 240, 17 pages.  https://doi.org/10.1145/3491102.3517608
  * Ogbonnaya-Ogburu et al. (2020) Ihudiya Finda Ogbonnaya-Ogburu, Angela DR Smith, Alexandra To, and Kentaro Toyama. 2020.  Critical Race Theory for HCI. In _Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems_. 1–16.
  * Pogrebin (2021) Robin Pogrebin. 2021.  Met Museum Removes Sackler Name From Wing Over Opioid Ties.  https://www.nytimes.com/2021/12/09/arts/design/met-museum-sackler-wing.html?pgtype=Article&action=click&module=RelatedLinks Accessed: April 17, 2024.
  * Rabaan (2021) Hawra Rabaan. 2021.  Exploring Transformative Justice Principles to Inform Survivor-Centered Design for Muslim Women in the United States. In _Companion Publication of the 2021 Conference on Computer Supported Cooperative Work and Social Computing_ (Virtual Event, USA) _(CSCW ’21)_. Association for Computing Machinery, New York, NY, USA, 291–294.  https://doi.org/10.1145/3462204.3481797
  * Rankin et al. (2021) Yolanda A Rankin, Jakita O Thomas, and Sheena Erete. 2021.  Real talk: saturated sites of violence in CS education.  _ACM Inroads_ 12, 2 (2021), 30–37.
  * Robinson (2020) Cedric J Robinson. 2020.  _Black Marxism, revised and updated third edition: The making of the black radical tradition_.  UNC press Books.
  * Silva et al. (2022) Rafael M. L. Silva, Erica Principe Cruz, Daniela K. Rosner, Dayton Kelly, Andrés Monroy-Hernández, and Fannie Liu. 2022.  Understanding AR Activism: An Interview Study with Creators of Augmented Reality Experiences for Social Change. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿New Orleans¡/city¿, ¡state¿LA¡/state¿, ¡country¿USA¡/country¿, ¡/conf-loc¿) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 630, 15 pages.  https://doi.org/10.1145/3491102.3517605
  * Singh et al. (2019) Gopal K Singh, Isaac E Kim, Mehrete Girmay, Chrisp Perry, Gem P Daus, Ivy P Vedamuthu, A Andrew, Christine T Ramey, Elijah K Martin, and Michelle Allender. 2019.  Opioid epidemic in the United States: empirical trends, and a literature review of social determinants and epidemiological, pain management, and treatment patterns.  _International Journal of Maternal and Child Health and AIDS_ 8, 2 (2019), 89.
  * Solórzano and Yosso (2002) Daniel G Solórzano and Tara J Yosso. 2002.  Critical race methodology: Counter-storytelling as an analytical framework for education research.  _Qualitative inquiry_ 8, 1 (2002), 23–44.
  * Thebault-Spieker et al. (2015) Jacob Thebault-Spieker, Loren G. Terveen, and Brent Hecht. 2015.  Avoiding the South Side and the Suburbs: The Geography of Mobile Crowdsourcing Markets. In _Proc. of CSCW_ (Vancouver, BC, Canada) _(CSCW ’15)_. ACM, New York, NY, USA, 265–275.  https://doi.org/10.1145/2675133.2675278
  * To et al. (2022) Alexandra To, Hillary Carey, Riya Shrivastava, Jessica Hammer, and Geoff Kaufman. 2022.  Interactive Fiction Provotypes for Coping with Interpersonal Racism. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿New Orleans¡/city¿, ¡state¿LA¡/state¿, ¡country¿USA¡/country¿, ¡/conf-loc¿) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 453, 14 pages.  https://doi.org/10.1145/3491102.3502044
  * Tomlinson et al. (2020) Bill Tomlinson, M. Six Silberman, Andrew W. Torrance, Kurt Squire, Paramdeep S. Atwal, Ameya N. Mandalik, Sahil Railkar, and Rebecca W. Black. 2020.  A Participatory Simulation of the Accountable Capitalism Act. In _Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Honolulu¡/city¿, ¡state¿HI¡/state¿, ¡country¿USA¡/country¿, ¡/conf-loc¿) _(CHI ’20)_. Association for Computing Machinery, New York, NY, USA, 1–13.  https://doi.org/10.1145/3313831.3376326
  * Totenburg (2023) Nina Totenburg. 2023.  Purdue Pharma, Sacklers’ OxyContin settlement lands at the Supreme Court.  https://www.npr.org/2023/12/04/1215717223/purdue-sacklers-oxycontin-supreme-court Accessed: April 17, 2024.
  * Toyama (2015) Kentaro Toyama. 2015.  _Geek Heresy: Rescuing Social Change from the Cult of Technology_.  PublicAffairs.
  * Travis et al. (2014) Jeremy Travis, Bruce Western, and F Stevens Redburn. 2014.  The growth of incarceration in the United States: Exploring causes and consequences.  (2014).
  * Zhang et al. (2021) Weiyu Zhang, Tian Yang, and Simon Tangi Perrault. 2021.  Nudge for Reflection: More Than Just a Channel to Political Knowledge. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Yokohama¡/city¿, ¡country¿Japan¡/country¿, ¡/conf-loc¿) _(CHI ’21)_. Association for Computing Machinery, New York, NY, USA, Article 705, 10 pages.  https://doi.org/10.1145/3411764.3445274


$\sigma(o,x)\mathrel{\mathop{\ordinarycolon}}=\sigma_{\text{loc}}(o(\operatorname{\mathbb{R}}),x)\;\Longrightarrow\;\sigma(\tilde{\operatorname{\mathcal{O}}}_{t},X_{t})=\sigma_{\text{loc}}(t,X_{t}).$
(82)

Hence (81) gives a sensible generalization of the local volatility model as
long as $\sigma$ satisfies the smile calibration condition
$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\sigma(\tilde{\operatorname{\mathcal{O}}}_{t},X_{t})^{2}\
|\ X_{t}]=\sigma_{\text{loc}}(t,X_{t})^{2}$; see, e.g., [36].

On the other hand, occupied volatility is a particular example of path-
dependent volatility [36, 37, 39, 41] and thus enjoys similar benefits.
Notably, market completeness is preserved as the added path-dependent
feature$-$here the occupation measure$-$is purely endogenous. Importantly, the
volatility (or drift if $b\neq 0$) may depend on the trend of the asset,
defined as $\Upsilon(o,x)=x-\frac{1}{o(\operatorname{\mathbb{R}})}\int
yo(dy)$, e.g. to incorporate mean-reversion, or in relative terms,

$\quad\Upsilon(o,x)=\frac{x}{\frac{1}{o(\operatorname{\mathbb{R}})}\int_{\operatorname{\mathbb{R}}}xo(dx)},\quad\Longrightarrow\;\Upsilon(\tilde{\operatorname{\mathcal{O}}}_{t},X_{t})=\frac{X_{t}}{\frac{1}{t}\int_{0}^{s}X_{s}ds}.$
(83)

To give more importance to recent values of the path, one can also replace
$\tilde{\operatorname{\mathcal{O}}}$ by

$\mathcal{O}^{\kappa}_{t}(A)=\int_{0}^{t}\mathds{1}_{A}(X_{u})e^{\kappa
u}du,\quad\kappa>0.$ (84)

Hence, the average appearing in the trend functional $\Upsilon$ reads

$\frac{1}{\mathcal{O}^{\kappa}_{t}(\mathbb{R})}\int x\
\mathcal{O}^{\kappa}_{t}(dx)=\frac{1}{1-e^{-\kappa t}}\int_{0}^{t}X_{u}\kappa
e^{-\kappa(t-u)}du,$ (85)

giving the exponential moving average of $X$ with weight parameter $\kappa$.
The relevance of $\operatorname{\mathcal{O}}^{\kappa}$ is demonstrated in the
following example.

Figure 9: Guyon’s toy volatility model
$dX_{t}=\sigma(\operatorname{\mathcal{O}}_{t}^{\kappa},X_{t})dW_{t}$ with
$\sigma$ given in (86). Two simulations using the Euler-Maruyama scheme in
Section B.1. Parameters: $x_{0}=100,T=1,\kappa=12$, and
$(\alpha,\beta,\gamma)=(2.1,1.2,1.9)$ taken from [37, page 25].

######  Example 10.

(Guyon’s toy volatility model) Consider the occupied volatility model,

$\frac{1}{x}\sigma(o,x)=\Sigma(\Upsilon(o,x)),\quad\Sigma(y)=-\frac{\alpha}{\beta}+\gamma
y^{-\beta},$ (86)

with parameters $\alpha,\beta,\gamma>0$ and trend functional $\Upsilon$
defined in (83). Plugging in the exponential decay occupation measure (84)
into (86), then the resulting specification
$dX_{t}=\sigma(\operatorname{\mathcal{O}}^{\kappa}_{t},X_{t})dW_{t}$ is
precisely Guyon’s toy volatility model, introduced in [37] and analyzed in
[10]. As the function $\Sigma$ in (86) is decreasing, the volatility
functional $\sigma$ is inversely proportional to the trend of the asset. This
is in line with the leverage effect, a stylized fact in equity markets stating
that volatility tends to increase when the asset price drops.

Fig. 9 displays two realizations of
$(X,\sigma(\operatorname{\mathcal{O}}^{\kappa},X))$ where $X_{0}=100$, $T=1$
(in years), $\kappa=12$ (to reflect a one-month window for the weighted
average (85)) and parameters $(\alpha,\beta,\gamma)=(2.1,1.2,1.9)$ taken from
[37, page 25]. We indeed observe that the volatility shoots up when the asset
falls far below the running average. The trajectories in Fig. 9 were simulated
according to the Euler-Maruyama scheme in Section B.1 using $N=1,000$ time
steps and levels $x_{m}=50+m$, $m=0,\ldots,100$.

##  6 Futher Work

We here gather some research areas where this work opens into.

Stochastic control and viscosity theory. We have seen in Theorem 3.4 a
Feynman-Kac theorem linking Dirichlet problems to the solution of occupied
stochastic differential equations (OSDEs). More generally, it would be
relevant to introduce controlled OSDEs and, as hinted at by the optimal
stopping problem in Section 4.1, to establish a viscosity theory for the
associated dynamic programming equation as in [53, 54].

Itô-Tanaka’s formula. We proved in Theorem 3.1 the usual Itô’s formula where
the occupation functional $f(o,x)$ is twice differentiable with respect to the
space variable. It would be interesting to extend Itô-Tanaka’s formula [51,
Chapter VI] (or more generally, Bouleau-Yor’s formula [12]) to our context,
especially given the interplay between the occupation measure (or equivalently
the local time) arising from the lack of regularity in $x$ and inside the
functional itself.

Volatility Modeling. It would be worthwhile to further investigate occupied
volatility models (Section 5.4) by incorporating other statistics of the
occupation measure, e.g., the range of the asset which is available from the
support of $\tilde{\operatorname{\mathcal{O}}}$ (or
$\operatorname{\mathcal{O}}$). Alternatively, one could use neural SDEs as in
[21, 40] where the occupied volatility is parameterized by a feedforward
neural network, taking the (discretized) occupation measure and spot $X_{t}$
as input.

##  Appendix A Proofs

###  A.1 Theorem 2.1

###### Proof.

It is clear that II. $\Longrightarrow$ I. since the occupation measure is
chronology invariant. Let us outline a proof for completeness. Fix
$X\in\Lambda_{t}$, $\psi\in\operatorname{\mathfrak{S}}_{t}^{N}$ with
permutation $\sigma$ and sign vector $\xi\in\\{\pm 1\\}^{N}$, and write
$Y=X\circ\psi$. First, we may assume that $\xi_{n}=1$ for all $n$ as clearly
$\tilde{\operatorname{\mathcal{O}}}(\overleftarrow{Z})=\tilde{\operatorname{\mathcal{O}}}(Z)$
for all $Z\in\Lambda$. From the additivity of
$\tilde{\operatorname{\mathcal{O}}}$, we conclude that

$\displaystyle\tilde{\operatorname{\mathcal{O}}}(Y)(A)=\sum_{n=1}^{N}\int_{I_{n}}\mathds{1}_{A}(Y_{s})ds=\sum_{n=1}^{N}\int_{I_{\sigma(n)}}\\!\mathds{1}_{A}(X_{s})ds=\tilde{\operatorname{\mathcal{O}}}(X)(A),\quad\forall\
A\in\operatorname{\mathcal{B}}(\operatorname{\mathbb{R}}).$

To prove I. $\Longrightarrow$ II. , introduce the equivalence relation $X\sim
Y\ \Longleftrightarrow\
\tilde{\operatorname{\mathcal{O}}}(X)=\tilde{\operatorname{\mathcal{O}}}(Y)$
and construct a representative for the equivalence classes in
$\Lambda/\\!\sim$ by characterizing the pre-image of
$\tilde{\operatorname{\mathcal{O}}}\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathcal{M}}$.
Given $o=\tilde{\operatorname{\mathcal{O}}}(X)\in\operatorname{\mathcal{M}}$
for some $X\in\Lambda_{t}$, let $Z^{o}\in\Lambda_{t}$ defined as

$Z_{s}^{o}=\sup\\{x\in\operatorname{\mathbb{R}}\
\mathrel{\mathop{\ordinarycolon}}\ o((-\infty,x])\leq s\\},\quad s\leq t.$
(87)

Since $Z^{o}_{s}\leq x\ \Longleftrightarrow\ o((-\infty,x])\geq s$ by
construction, it is then immediate that
$\tilde{\operatorname{\mathcal{O}}}(Z^{o})=o$. Moreover, $Z^{o}$ is clearly
càdlàg and non-decreasing. See Fig. 10. We can therefore define the
representation map
$\operatorname{\mathcal{R}}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{M}}\to\Lambda_{t}/\\!\sim$,
$\operatorname{\mathcal{R}}(o)=Z^{o}$ and projection
$\pi\mathrel{\mathop{\ordinarycolon}}\Lambda_{t}\to\Lambda_{t}/\\!\sim$ as
$\pi=\operatorname{\mathcal{R}}\circ\ \tilde{\operatorname{\mathcal{O}}}$. In
short,

$X\ \mapsto\
\tilde{\operatorname{\mathcal{O}}}_{t}=\tilde{\operatorname{\mathcal{O}}}(X)\
\mapsto\
\operatorname{\mathcal{R}}(\tilde{\operatorname{\mathcal{O}}}_{t})=Z^{\tilde{\operatorname{\mathcal{O}}}_{t}}\in\Lambda_{t}/\\!\sim.$

Next, fix a chronology-invariant functional
$F\mathrel{\mathop{\ordinarycolon}}\Lambda\to\operatorname{\mathbb{R}}$ and
$X\in\Lambda_{t}$. If $X$ is constant in each subinterval of the partition
$\\{\frac{nt}{N}\mathrel{\mathop{\ordinarycolon}}n\leq N\\}$ for some
$N\in\operatorname{\mathbb{N}}$, then
$\tilde{\operatorname{\mathcal{O}}}_{t}=\tilde{\operatorname{\mathcal{O}}}(X)$
is a discrete measure so the representative
$Z^{\tilde{\operatorname{\mathcal{O}}}_{t}}$ is piecewise constant as well.
Thus, there exists a ”sorting” permutation
$\psi\in\operatorname{\mathfrak{S}}_{t}^{N}$ such that $X\circ\psi$ is non-
decreasing and as
$\operatorname{\mathcal{O}}(X\circ\psi)=\operatorname{\mathcal{O}}_{t}$, we
gather that $Z^{\tilde{\operatorname{\mathcal{O}}}_{t}}=X\circ\psi$. For
general càdlàg $X$, the same reasoning can be applied by first taking a
piecewise constant approximation of $X$ as seen in Section 2.3 and a passage
to the limit. To conclude, set
$f\mathrel{\mathop{\ordinarycolon}}=F\circ\operatorname{\mathcal{R}}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{M}}\to\operatorname{\mathbb{R}}$
and $\psi\in\operatorname{\mathfrak{S}}_{t}$ such that
$Z^{\tilde{\operatorname{\mathcal{O}}}_{t}}=X\circ\psi$. Hence,

$(f\circ\tilde{\operatorname{\mathcal{O}}})(X)=f(\tilde{\operatorname{\mathcal{O}}}_{t})=(F\circ\operatorname{\mathcal{R}})(\tilde{\operatorname{\mathcal{O}}}_{t})=F(Z^{\tilde{\operatorname{\mathcal{O}}}_{t}})=F(X\circ\psi)=F(X).$

∎

Figure 10: Path $X\in\Lambda_{t}$, its standard time occupation measure
$\tilde{\operatorname{\mathcal{O}}}_{t}$, and representative path
$Z_{t}^{\tilde{\operatorname{\mathcal{O}}}_{t}}$ in the proof of Theorem 2.1.
Two simulations.

###  A.2 Theorem 3.1

###### Proof.

We first show the result when $X$ is Brownian motion. Let
$\Pi^{N}=\\{s=t_{0}<...<t_{N}=t\\}$ be a sequence of partitions with mesh size
decreasing to zero as $N\to\infty$. If
$f\in\operatorname{\mathcal{C}}^{1,2}(\operatorname{\mathcal{D}})$ and
$Y_{t}=f(\operatorname{\mathcal{O}}_{t},X_{t})$, then,

$\displaystyle Y_{t}-Y_{s}$
$\displaystyle=\sum_{t_{n}\in\Pi^{N}}(Y_{t_{n+1}}-Y_{t_{n}})\quad\qquad(t_{N+1}\mathrel{\mathop{\ordinarycolon}}=t)$
$\displaystyle=\sum_{t_{n}\in\Pi^{N}}(Y_{t_{n+1}}-f(\operatorname{\mathcal{O}}_{t_{n}},X_{t_{n+1}}))$
(88)
$\displaystyle+\sum_{t_{n}\in\Pi^{N}}(f(\operatorname{\mathcal{O}}_{t_{n}},X_{t_{n+1}})-Y_{t_{n}})$
(89)

The telescopic sum (89) generates the terms
$\int_{s}^{t}\partial_{x}f(\operatorname{\mathcal{O}}_{u},X_{u})dX_{u}$ and
$\int_{s}^{t}\frac{1}{2}\partial_{xx}f(\operatorname{\mathcal{O}}_{u},X_{u})d\langle
X\rangle_{u}$ (for $\operatorname{\mathbb{Q}}-$almost all $\omega\in\Omega$
and a suitable sequence $(\Pi^{N})$) as the occupation measure appearing in
the summand of (89) is ”frozen” at time $t_{n}$ for each $n$. Concretely,
defining the predictable processes
$H^{(1)}_{t}=\partial_{x}f(\operatorname{\mathcal{O}}_{t},X_{t})$,
$H_{t}^{(2)}=\partial_{xx}f(\operatorname{\mathcal{O}}_{t},X_{t})$, a second
order Taylor expansion in the space variable yields

$\sum_{t_{n}\in\Pi^{N}}(f(\operatorname{\mathcal{O}}_{t_{n}},X_{t_{n+1}})-Y_{t_{n}})=\sum_{t_{n}\in\Pi^{N}}H_{t_{n}}^{(1)}(X_{t_{n+1}}-X_{t_{n}})+\frac{1}{2}\sum_{t_{n}\in\Pi^{N}}H_{t_{n}}^{(2)}(X_{t_{n+1}}-X_{t_{n}})^{2}+o(1),$

so the result follows from the standard Itô’s formula. We now compute the
limit of (88) as $N\to\infty$. In light of the definition of $\delta_{\mu}$
(Definition 2), we set
$\Delta_{n}=\operatorname{\mathcal{O}}_{t_{n},t_{n+1}}=\operatorname{\mathcal{O}}_{t_{n+1}}-\operatorname{\mathcal{O}}_{t_{n}}$
for simplicity (note that $\Delta_{n}$ depends on $N$) and rewrite the
summands in (88) as

$\displaystyle
Y_{t_{n+1}}-f(\operatorname{\mathcal{O}}_{t_{n}},X_{t_{n+1}})=\int_{0}^{1}\delta_{\mu}f(\operatorname{\mathcal{O}}_{t_{n}}+\eta\Delta_{n},X_{t_{n+1}})\cdot\Delta_{n}d\eta=\delta_{\mu}f(\operatorname{\mathcal{O}}_{t_{n}}+\eta_{n}\Delta_{n},X_{t_{n+1}})\cdot\Delta_{n},$

for some $\eta_{n}\in[0,1]$ by the mean value theorem. Next, the occupation
time formula (10) yields

$\displaystyle\sum_{t_{n}\in\Pi^{N}}(Y_{t_{n+1}}-f(\operatorname{\mathcal{O}}_{t_{n}},X_{t_{n+1}}))$
$\displaystyle=\sum_{t_{n}\in\Pi^{N}}\delta_{\mu}f(\operatorname{\mathcal{O}}_{t_{n}}+\eta_{n}\Delta_{n},X_{t_{n+1}})\cdot\Delta_{n}$
$\displaystyle=\sum_{t_{n}\in\Pi^{N}}\int_{t_{n}}^{t_{n+1}}\delta_{\mu}f(\operatorname{\mathcal{O}}_{t_{n}}+\eta_{n}\Delta_{n},X_{t_{n+1}})(X_{u})d\langle
X\rangle_{u}$ $\displaystyle=\int_{s}^{t}Z^{N}_{u}d\langle X\rangle_{u},$

with the step process

$Z^{N}_{u}=\sum_{t_{n}\in\Pi^{N}}\delta_{\mu}f(\operatorname{\mathcal{O}}_{t_{n}}+\eta_{n}\Delta_{n},X_{t_{n+1}})(X_{u})\
\mathds{1}_{[t_{n},t_{n}+1)}(u),\qquad[t_{N},t_{N+1})=\\{t\\}.$

For every $n$ and $u\in[t_{n},t_{n+1})$, observe that
$\operatorname{\mathfrak{m}}_{1}(\operatorname{\mathcal{O}}_{u},\operatorname{\mathcal{O}}_{t_{n}}+\eta_{n}\Delta_{n})=\eta_{n}\Delta_{n}(\operatorname{\mathbb{R}})\leq\frac{T}{N}$
and $|X_{u}-X_{t_{n+1}}|\leq\rho(\frac{T}{N})$, where $\rho=\rho(\omega)$ is
the modulus of continuity of $t\mapsto X_{t}$ on the compact $[s,t]$. As
$(o,x,y)\mapsto\delta_{\mu}f(o,x)(y)$ is jointly continuous, we have

$Z^{N}_{u}\to
Z_{u}^{\infty}=\delta_{\mu}f(\operatorname{\mathcal{O}}_{u},X_{u})(X_{u})=\partial_{o}f(\operatorname{\mathcal{O}}_{u},X_{u}),$

uniformly in $u\in[s,t]$, $\operatorname{\mathbb{Q}}-$almost surely. We
finally conclude that

$\lim_{N\to\infty}\int_{s}^{t}Z^{N}_{u}d\langle
X\rangle_{u}=\int_{s}^{t}\partial_{o}f(\operatorname{\mathcal{O}}_{u},X_{u})d\langle
X\rangle_{u},\quad\operatorname{\mathbb{Q}}-a.s.$

∎

###  A.3 Proposition 4

###### Proof.

First, the additivity property of the occupation measure gives

$\operatorname{\mathcal{O}}(X+h\mathds{1}_{[t,T]})(A)=\operatorname{\mathcal{O}}_{t}(A)+\int_{t}^{T}\mathds{1}_{A}(X_{s}+h)ds=\operatorname{\mathcal{O}}_{t}(A)+\operatorname{\mathcal{O}}_{t,T}^{h}(A),\quad\forall
A\in\operatorname{\mathcal{B}}(\operatorname{\mathbb{R}}),$

where $\operatorname{\mathcal{O}}^{h}(A)=\operatorname{\mathcal{O}}(A-h)$.
Hence, a parallel shift of $X$ in $[t,T]$ induces a translation of the
increment occupation measure $\operatorname{\mathcal{O}}_{t,T}$. Writing
$\Delta_{t,T}^{h}=\operatorname{\mathcal{O}}_{t,T}^{h}-\operatorname{\mathcal{O}}_{t,T}$,
then from the definition of $\delta_{\mu}$ and $F$, we have

$\displaystyle F(X+h\mathds{1}_{[t,T]})-F(X)$
$\displaystyle=\int_{0}^{1}\delta_{\mu}f(\operatorname{\mathcal{O}}_{T}+\eta\Delta_{T}^{t,h})\cdot\Delta_{t,T}^{h}\
d\eta=\delta_{\mu}f(\operatorname{\mathcal{O}}_{T}+\eta^{h}\Delta_{T}^{t,h})\cdot\Delta_{t,T}^{h},$

for some $\eta^{h}\in[0,1]$. Formally, $\lim\limits_{h\to
0}\frac{\Delta_{t,T}^{h}(A)}{h}=\lim\limits_{h\to
0}\int_{A}\frac{L^{x-h}_{t,T}-L^{x}_{t,T}}{h}dx=-\int_{A}\partial_{x}L_{t,T}^{x}dx,$
hence

$D_{t}f(\operatorname{\mathcal{O}}_{T})=-\int_{\operatorname{\mathbb{R}}}\delta_{\mu}f(\operatorname{\mathcal{O}}_{T})(x)\partial_{x}L_{t,T}^{x}dx=\int_{\operatorname{\mathbb{R}}}\partial_{xo}f(\operatorname{\mathcal{O}}_{T},x)L_{t,T}^{x}dx=\int_{t}^{T}\partial_{xo}f(\operatorname{\mathcal{O}}_{T},X_{u})du,$

using integration by parts and the occupation time formula. Although local
time of Brownian motion is $\operatorname{\mathbb{Q}}-$a.s. nowhere
differentiable (e.g. as a consequence of the first Ray-Knight theorem [51,
Chapter XI]), this instructive heuristics turns out to be correct. Indeed, for
all $\phi\in\operatorname{\mathcal{C}}^{1}(\operatorname{\mathbb{R}})$, we
have using the occupation time formula that

$\frac{1}{h}(\phi\cdot\Delta_{t,T}^{h})=\frac{1}{h}\int_{t}^{T}\phi(X_{s}+h)ds\longrightarrow\int_{t}^{T}\partial_{x}\phi(X_{s})ds.$

Noting that
$\delta_{\mu}f(\operatorname{\mathcal{O}}_{T}+\eta^{h}\Delta_{t,T}^{h})=\delta_{\mu}f(\operatorname{\mathcal{O}}_{T})+\varepsilon_{h}$,
$\lim_{h\to 0}\varepsilon_{h}=0$, we can apply the above limit to
$\phi=\delta_{\mu}f(\operatorname{\mathcal{O}}_{T})\in\operatorname{\mathcal{C}}^{1}(\operatorname{\mathbb{R}})$
and obtain (41). ∎

###  A.4 Theorem 3.2

###### Proof.

The first assertion follows from the fact that
$v(\operatorname{\mathcal{O}},X)$ is an
$(\operatorname{\mathbb{F}},\operatorname{\mathbb{Q}})-$martingale, as a
consequence of (42a) and Itô’s formula (Theorem 3.1) applied to
$v\in\operatorname{\mathcal{C}}^{1,2}(\operatorname{\mathcal{D}})$. Hence
$v(o,x)=\operatorname{\mathbb{E}}_{o,x}^{\operatorname{\mathbb{Q}}}[v(\operatorname{\mathcal{O}}_{T},X_{T})]$,
which is (43) in light of the terminal condition (42b).

For the second part, it is enough to verify that
$v\in\operatorname{\mathcal{C}}^{2}_{2}(\operatorname{\mathcal{D}})$ as the
uniqueness then follows from the previous assertion. For all
$o,o^{\prime}\in\operatorname{\mathcal{M}}$, $\Delta=o^{\prime}-o$, and
condition (25), observe that

$\displaystyle v(o^{\prime},x)-v(o,x)$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(o^{\prime}+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})-\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})]$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{0}^{1}\delta_{\mu}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x}+\eta\Delta,x+X_{T-t})\cdot\Delta\
d\eta\right]$ $\displaystyle=\int_{0}^{1}w(o+\eta\Delta,x,\cdot)\cdot\Delta\
d\eta,$ $\displaystyle w(o,x,y)$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\delta_{\mu}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})(y)].$

Clearly, the joint continuity of $(o,x,y)\mapsto\delta_{\mu}\varphi(o,x)(y)$
carries over to $w$. Moreover, defining
$C\mathrel{\mathop{\ordinarycolon}}=\sup_{(o,x)\in\operatorname{\mathcal{D}},y\in\operatorname{\mathbb{R}}}\frac{|\delta_{\mu}\varphi(o,x)(y)|}{1+|y|^{2}}$
which is finite as
$\varphi\in\operatorname{\mathcal{C}}_{2}^{2}(\operatorname{\mathcal{D}})$, we
have from Jensen’s inequality that $|w(o,x,y)|\leq C(1+|y|^{2})$, uniformly in
$(o,x)$. Hence $\delta_{\mu}v=w$ in light of Definition 2, which implies that
$v\in\operatorname{\mathcal{C}}^{1,0}(\operatorname{\mathcal{D}})$. Next, it
is easily seen that

$\delta_{\mu\mu}v(o,x)(y,y^{\prime})=\delta_{\mu}w(o,x,y)(y^{\prime})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\delta_{\mu\mu}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})(y,y^{\prime})],$

which is, similar to $\varphi$, continuous in $(o,x)$, twice differentiable
with respect to $(y,y^{\prime})$ and satisfies the quadratic growth condition
II. in Definition 2 (with respect to $y^{\prime}$).

Moving to the space derivatives, observe that for every
$f\in\operatorname{\mathcal{C}}^{1}(\operatorname{\mathcal{M}})$ such that
$y\mapsto\delta_{\mu}f(o)(y)\in\operatorname{\mathcal{C}}^{1}(\operatorname{\mathbb{R}})$,
we have
$\frac{d}{dh}f(o+\operatorname{\mathcal{O}}_{s}^{x+h})=\partial_{y}\delta_{\mu}f(o+\operatorname{\mathcal{O}}_{s}^{x})\cdot\operatorname{\mathcal{O}}_{s}^{x}$
using similar arguments as in the proof of Proposition 4; see Section A.3.
Thus, the chain rule and bounded convergence theorem yields

$\displaystyle\partial_{x}v(o,x)$
$\displaystyle=\underbrace{\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\partial_{x}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})]}_{=\mathrel{\mathop{\ordinarycolon}}\
w_{1}^{\varphi}(o,x)}+\underbrace{\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\partial_{y}\delta_{\mu}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})\cdot\operatorname{\mathcal{O}}_{T-t}^{x}]}_{=\mathrel{\mathop{\ordinarycolon}}\
w_{2}^{\varphi}(o,x)},$

which is jointly continuous. Next, it is immediate that
$\partial_{x}w_{1}^{\varphi}=w_{1}^{\partial_{x}\varphi}+w_{2}^{\partial_{x}\varphi}\in\operatorname{\mathcal{C}}(\operatorname{\mathcal{D}})$.
For $w_{2}^{\varphi}$, we compute

$\displaystyle\partial_{x}w_{2}^{\varphi}(o,x)$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[(\partial_{x}+\partial_{y})\partial_{y}\delta_{\mu}\varphi(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})\cdot\operatorname{\mathcal{O}}_{T-t}^{x}],$
$\displaystyle+\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\partial_{y^{\prime}y}\delta_{\mu\mu}(o+\operatorname{\mathcal{O}}_{T-t}^{x},x+X_{T-t})\cdot(\operatorname{\mathcal{O}}_{T-t}^{x}\otimes\operatorname{\mathcal{O}}_{T-t}^{x})]$

where $\phi\cdot(o\otimes
o^{\prime})=\int_{\operatorname{\mathbb{R}}^{2}}\phi(y,y^{\prime})o(dy)o^{\prime}(dy^{\prime})$,
$\phi\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}^{2}\to\operatorname{\mathbb{R}}$,
and using [15, Lemma 2.2.4.] to rewrite
$\partial_{y^{\prime}}\delta_{\mu}\partial_{y}\delta_{\mu}$ more compactly as
$\partial_{y^{\prime}y}\delta_{\mu\mu}$. Given our assumptions on $\varphi$,
we conclude that
$\partial_{x}w_{2}^{\varphi}\in\operatorname{\mathcal{C}}(\operatorname{\mathcal{D}})$
and thus $v\in\operatorname{\mathcal{C}}^{2}_{2}(\operatorname{\mathcal{D}})$.
∎

###  A.5 Proposition 6

###### Proof.

Let $\overleftarrow{X}_{s}=X_{T-s}-X_{T}$ and denote its flow of occupation
measures by $\overleftarrow{\operatorname{\mathcal{O}}}$ and local time field
by $\overleftarrow{L}$. Then, $L_{T}^{X_{T}+z}=\overleftarrow{L_{T}^{z}}$ for
all $z\in\operatorname{\mathbb{R}}$, which implies that

$\varphi^{\varepsilon}(\operatorname{\mathcal{O}}_{T},X_{T})=\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}L_{T}^{X_{T}+z}dz=\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}\overleftarrow{L_{T}}\
d\lambda=\varphi^{\varepsilon}(\overleftarrow{\operatorname{\mathcal{O}}}_{T},0).$

Thus,

$\displaystyle v^{\varepsilon}(T)$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi^{\varepsilon}(\overleftarrow{\operatorname{\mathcal{O}}}_{T},0)]=\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\overleftarrow{L}_{T}]d\lambda=\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{[0,\varepsilon]}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[L_{T}^{x}]dx\
\uparrow\ \operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[L_{T}^{0}],$

where we used that
$\text{Law}_{\operatorname{\mathbb{Q}}}(\overleftarrow{L}_{T}^{x})=\text{Law}_{\operatorname{\mathbb{Q}}}(L_{T}^{x})$
and
$\partial_{x}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[L_{T}^{x}]<0$
$\forall x>0$ to prove the monotone convergence. Finally, recall from
Proposition 5 that
$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[L_{T}^{0}]=v(T)$. For
the second assertion, notice from Tanaka’s formula that
$\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[L_{T}^{x}]=2c(T,x)$,
where
$c(T,x)=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[(X_{T}-x)^{+}]$
is the price of a call option struck at $x$ with maturity $T$ in a Bachelier
model with unit volatility. Similar to [14], we have
$\partial_{x}c(T,x)=\operatorname{\mathbb{Q}}(X_{T}\leq x)-1=\Phi_{T}(x)-1$
and $\partial_{xx}c(T,x)=\phi_{T}(x)$, where $\Phi_{T},\phi_{T}$ is the CDF
and PDF of $\operatorname{\mathcal{N}}(0,T)$, respectively. Hence, a second
order Taylor expansion of $x\mapsto c(T,x)$ at $x=0$ yields

$v^{\varepsilon}(T)=\frac{2}{\varepsilon}\int_{0}^{\varepsilon}c(T,x)dx=v(T)-\frac{\varepsilon}{2}+\frac{\varepsilon^{2}/3}{\sqrt{2\pi
T}}(1+r_{\varepsilon}),\quad\lim_{\varepsilon\downarrow 0}r_{\varepsilon}=0.$

∎

###  A.6 Proposition 7

###### Proof.

For every stopping time $\tau\in\vartheta(\operatorname{\mathcal{T}})$,
$x\in\operatorname{\mathbb{R}}$, we have by Jensen’s inequality that

$|\varphi^{\varepsilon}(\operatorname{\mathcal{O}}_{\tau},x)-L_{\tau}^{x}|^{2}=\left|\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}(L_{\tau}^{x+z}-L_{\tau}^{x})\
dz\right|^{2}\leq\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}|L_{\tau}^{x+z}-L_{\tau}^{x}|^{2}\
dz.$

Moreover, there exists $C>0$ (to be determined) such that

$\left\lVert\sup_{(t,x)\in[0,T]\times\operatorname{\mathbb{R}}}|L_{t}^{x+z}-L_{t}^{x}|\right\rVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}\leq
C|z|,\quad|z|\leq 1.$

Indeed, writing $L^{x}(Y)$ for the local time of some process $Y$ at $x$ and
${|\\!|\\!|Y|\\!|\\!|}\mathrel{\mathop{\ordinarycolon}}=\lVert\sup_{t\in[0,T]}Y_{t}\rVert_{L^{2}(\operatorname{\mathbb{Q}})}$,
we can apply Theorem 6.5 in Barlow and Yor [2] with $\gamma=2$ to the
continuous martingales $M=X-z$, $N=X$. This yields

$\displaystyle\left\lVert\sup_{(t,x)\in[0,T]\times\operatorname{\mathbb{R}}}|L_{t}^{x+z}-L_{t}^{x}|\right\rVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}$
$\displaystyle=\left\lVert\sup_{(t,x)\in[0,T]\times\operatorname{\mathbb{R}}}|L_{t}^{x}(M)-L_{t}^{x}(N)|\right\rVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}$
$\displaystyle\leq C_{2}^{2}\
({|\\!|\\!|M|\\!|\\!|}+{|\\!|\\!|N|\\!|\\!|})\underbrace{{|\\!|\\!|M-N|\\!|\\!|}}_{=\
|z|}\
\max\left(\log\frac{{|\\!|\\!|M|\\!|\\!|}+{|\\!|\\!|N|\\!|\\!|}}{{|\\!|\\!|M-N|\\!|\\!|}},1\right)$
$\displaystyle\leq 2C|z|,$

with $C=C_{2}^{2}\ \sqrt{2(T+1)}\max(\log 2\sqrt{2(T+1)}/|z|,1)$888note that
$\frac{1}{2}{|\\!|\\!|N|\\!|\\!|}^{2}\leq\frac{1}{2}{|\\!|\\!|M|\\!|\\!|}^{2}\leq\
\rVert X_{T}-z\lVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}=T+|z|^{2}\leq T+1$
using Doob’s inequality and $|z|\leq 1$. Hence
${|\\!|\\!|M|\\!|\\!|}+{|\\!|\\!|N|\\!|\\!|}\leq 2\sqrt{2(T+1)}$. and $C_{2}$
as in [2, Theorem 6.5]. Thus,

$\displaystyle\left\lVert\varphi^{\varepsilon}(X_{\tau},\operatorname{\mathcal{O}}_{\tau})-L_{\tau}^{X_{\tau}}\right\rVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}$
$\displaystyle\leq\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}\left\lVert
L_{\tau}^{X_{\tau}+z}-L_{\tau}^{X_{\tau}}\right\rVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}\
dz\leq 2C\mathop{\,\,\hbox
to0.0pt{--\hss}\int}\nolimits_{B_{\varepsilon}}|z|dz=C\varepsilon,\quad\forall\
\varepsilon\leq 1.$

The second assertion simply follows from Jensen’s inequality. ∎

##  Appendix B Algorithm

###  B.1 Numerical Integration of OSDEs

This section provides practical details towards the simulation of processes
$X$ governed by OSDEs, that is
$dX_{t}=b(\operatorname{\mathcal{O}}_{t},X_{t})dt+\sigma(\operatorname{\mathcal{O}}_{t},X_{t})dW_{t},$
$X_{0}\in\operatorname{\mathbb{R}}$, and $\operatorname{\mathcal{O}}$ evolving
as in (46b). An almost identical procedure can be carried out for OSDEs
depending on the standard time or exponential decay occupation measure as in
Example 10.

First, fix $N\in\operatorname{\mathbb{N}}$ and the regular time partition
$t_{n}=n\delta t$, $\delta t=\frac{T}{N}$. To make the simulation procedure
feasible, the occupation measure must be inevitably discretized. Let
$x_{1}<\ldots<x_{M}$, $M\in\operatorname{\mathbb{N}}$, be prescribed levels
and consider the partition

$C^{m}=[x_{m}-\varepsilon_{m-1},x_{m}+\varepsilon_{m}),\quad\varepsilon_{m}=\frac{x_{m+1}-x_{m}}{2},\quad(x_{M+1}=-x_{0}=\infty).$

The choice for $x_{1}<\ldots<x_{M}$ is natural once the initial value $X_{0}$
and scale of the coefficients $b,\sigma$ are set; see Example 10. Next, define
the discrete occupation measure,

$\operatorname{\mathcal{O}}_{t_{n}}^{M}=\sum_{m=1}^{M}\operatorname{\mathcal{O}}_{t_{n}}(C_{m})\delta_{x_{m}},$

which is in one-to-one correspondence with
$O_{t_{n}}\mathrel{\mathop{\ordinarycolon}}=(O^{1}_{t_{n}},\ldots,O^{M}_{t_{n}})\in\operatorname{\mathbb{R}}^{M}$,
$O^{m}_{t}=\operatorname{\mathcal{O}}_{t}(C_{m})$. We can then introduce the
function
$b^{M}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathbb{R}}^{M}\times\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}$,
$b^{M}(O_{t},X_{t})=b(\operatorname{\mathcal{O}}_{t}^{M},X_{t})$, and
similarly for $\sigma^{M}$.

If $X_{0}\in\operatorname{\mathbb{R}}$, $O^{\text{\tiny
tmp}}=0\in\operatorname{\mathbb{R}}^{M}$, and $e_{m}$ is the $m$-th row of the
$M\times M$ identity matrix, we can approximate the solution of (46a)$-$(46b)
according to the Euler-Maruyama scheme (see, e.g., [43]),

$\displaystyle X_{t_{n}}\\!$ = $X_{t_{n-1}}+b^{M}(O^{\text{\tiny
tmp}},X_{t_{n-1}})\delta t+\sigma^{M}(O^{\text{\tiny
tmp}},X_{t_{n-1}})\sqrt{\delta t}Z_{n},$ (90a) $\displaystyle O^{\text{\tiny
tmp}}\\!$ $\leftarrow\ O^{\text{\tiny tmp}}+e_{m}\sigma^{M}(O^{\text{\tiny
tmp}},X_{t_{n}})^{2}\delta t,\quad X_{t_{n}}\ \in\ C_{m}$, (90b)

where $Z_{n}\overset{i.i.d.}{\sim}\operatorname{\mathcal{N}}(0,1)$,
$n=1,\ldots,N$. For OSDEs entailing the standard time occupation measure, then
(90b) is replaced by $O^{\text{\tiny tmp}}\leftarrow O^{\text{\tiny
tmp}}+e_{m}\delta t$. Note that the occupation measure is represented by the
temporary variable $O^{\text{\tiny tmp}}$ updated at each iteration. Indeed,
it is not necessary (and costly) to store the occupation measure
$\tilde{\operatorname{\mathcal{O}}}_{t_{n}}$ for each $n$. If one is
interested in the flow $\operatorname{\mathcal{O}}$, it can be retrieved from
the simulated path anyway.

Note that for fully path-dependent SDEs $dX_{t}=b((X_{s})_{s\leq
t})dt+\sigma((X_{s})_{s\leq t})dW_{t},$ one has to store all the values so far
$X_{t_{1}},\ldots,X_{t_{n}}$ to generate the next one. This becomes highly
problematic when the time grid gets finer. In the case of OSDEs, we can, in
contrast, decouple the dimensionality of the space and time discretization.
Hence, the number of levels to approximate $\operatorname{\mathcal{O}}$ ($M$)
can be set to fairly large values as demonstrated in Example 10.

###  B.2 Least Square Monte Carlo for Spot Local Time

Algorithm 1 summarizes the Least Square Monte Carlo approach seen in Section
4.3. Notice the exponential transformation of the discrete Brownian in step
IV.2. so that all variables are nonnegative. We choose a family of functions
on $\operatorname{\mathbb{R}}_{+}$, namely the Laguerre polynomials $(p_{k})$
(up to degree $3$) which are orthonormal in the weighted Hilbert space
$L^{2}(\operatorname{\mathbb{R}}_{+},wd\lambda)$, $w(x)=e^{-x}$. The basis
functions $\phi_{k}$ in Algorithm 1 then corresponds to individual evaluations
of the feature variables
$\operatorname{\mathcal{O}}_{t_{n}}^{\bar{m}},\bar{X}_{t_{n}}$ through the
polynomials. Note that
$p_{1}(\operatorname{\mathcal{O}}_{t_{n}}^{\bar{m}})\mathrel{\mathop{\ordinarycolon}}=(p_{1}(L_{t_{n}}^{x_{m-\bar{m}}}),\ldots,p_{1}(L_{t_{n}}^{x_{m+\bar{m}}}))$
contains the intrinsic value $L_{t_{n}}^{X_{t_{n}}}$ itself (up to a constant)
which is of course valuable information. One can also choose instead, as in
[48], the better-behaved Laguerre polynomials weighted by $\sqrt{w}$, yielding
similar results.

In both phases, we use antithetic sampling, i.e., we simulate $X(\omega_{j})$
for $j=1,\ldots,J/2$ (provided that $J$ is even) and set
$X(\omega_{j})=-X(\omega_{j-J/2})$ for $j=J/2+1,\ldots,J$,
$J\in\\{J_{\text{off}},J_{\text{on}}\\}$.

Algorithm 1 (Least Square Monte Carlo, Spot Local Time)

$T>0$, $N\in\operatorname{\mathbb{N}}$, $\bar{m}\ll N$ (truncation level),
$J_{\text{off}}$ (# simulations, offline phase), $J_{\text{on}}$ (#
simulations, online phase),
$\\{\phi_{k}\mathrel{\mathop{\ordinarycolon}}k=1,\ldots,K\\}$ (basis
functions)
OFFLINE

  * I.

Space-time grid $\begin{cases}t_{n}\ =n\delta t,\;\ &n=0,\ldots,N,\quad\delta
t=\frac{T}{N},\\\ x_{m}=m\sqrt{3\delta t},\;\ &m=-N,\ldots,N\end{cases}$

  * II.

Generate paths $X(\omega_{j})$, $j=1,\ldots,J_{\text{off}}$, where

$\displaystyle X_{t_{n+1}}=X_{t_{n}}+Z_{n+1}\sqrt{\delta t},\quad
X_{0}=0,\quad
Z_{1},\ldots,Z_{N}\overset{i.i.d.}{\sim}\nu=\frac{1}{6}\left(\delta_{-\sqrt{3}}+4\delta_{0}+\delta_{\sqrt{3}}\right)$

  * III.

Terminal value
$Y_{T}=L_{T}^{X_{T}}=\frac{1}{2\varepsilon}\sum_{n=1}^{N}\mathds{1}_{\\{X_{t_{n}}=X_{T}\\}}\delta
t$, $\ \varepsilon=\frac{x_{m+1}-x_{m}}{2}=\frac{\sqrt{3\delta t}}{2}$.

  * IV.

For $n=N-1,\ldots,\bar{m}$:

    1. 1.

Let $\operatorname{\mathcal{J}}=\\{j=1,\ldots,J_{\text{off}}\
\mathrel{\mathop{\ordinarycolon}}\ X_{t_{n}}(\omega_{j})=x_{m},\ |m|\leq
n-\bar{m}\\}$

    2. 2.

Feature variables: $\bar{X}_{t_{n}}=e^{X_{t_{n}}}$,
$\operatorname{\mathcal{O}}_{t_{n}}^{\bar{m}}=(L_{t_{n}}^{x_{m-\bar{m}}},\ldots,L_{t_{n}}^{x_{m+\bar{m}}})$,
$x_{m}=X_{t_{n}}$.

    3. 3.

Regression:
$\hat{\beta}^{n}=\text{argmin}_{\beta}\sum_{j\in\operatorname{\mathcal{J}}}\left|Y_{t_{n+1}}-\sum_{k=1}^{K}\beta_{k}\
\phi_{k}(\operatorname{\mathcal{O}}_{t_{n}}^{\bar{m}},\bar{X}_{t_{n}})\right|^{2}(\omega_{j})$.

    4. 4.

Continuation value: $C_{t_{n}}=\sum_{k=1}^{K}\hat{\beta}_{k}^{n}\
\phi_{k}(\operatorname{\mathcal{O}}_{t_{n}}^{\bar{m}},\bar{X}_{t_{n}})$.

    5. 5.

Update:
$Y_{t_{n}}(\omega_{j})=\begin{cases}Y_{t_{n+1}}(\omega_{j}),&C_{t_{n}}(\omega_{j})>L_{t_{n}}^{X_{t_{n}}}(\omega_{j})\text{
or }j\notin\operatorname{\mathcal{J}},\\\
L_{t_{n}}^{X_{t_{n}}}(\omega_{j}),&\text{otherwise}.\end{cases}$

ONLINE

  * I’.

Generate paths $X(\omega_{j})$, $j=1,\ldots,J_{\text{on}}$ as in II.

  * II’.

Repeat steps III., IV.1, 2, 4, 5 using
$\hat{\beta}^{N-1},\ldots,\hat{\beta}^{\bar{m}}$ from IV.3.

  * III’.

Return
$Y_{0}=\frac{1}{J_{\text{on}}}\sum_{j=1}^{J_{\text{on}}}Y_{t_{1}}(\omega_{j})$.

## References

  * Arous [1989] G. B. Arous.  Flots et séries de Taylor stochastiques.  _Probability Theory and Related Fields_ , 81:29–77, 1989.
  * Barlow and Yor [1982] M. Barlow and M. Yor.  Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times.  _Journal of Functional Analysis_ , 49(2):198–229, 1982.
  * Bayer et al. [2023] C. Bayer, P. P. Hager, S. Riedel, and J. Schoenmakers.  Optimal stopping with signatures.  _The Annals of Applied Probability_ , 33(1):238 – 273, 2023.
  * Bayraktar et al. [2023] E. Bayraktar, Q. Feng, and Z. Zhang.  Deep signature algorithm for multi-dimensional path-dependent options.  _arXiv:2211.11691_ , 2023.
  * Becker et al. [2019] S. Becker, P. Cheridito, and A. Jentzen.  Deep optimal stopping.  _Journal of Machine Learning Research_ , 20(74):1–25, 2019.
  * Bergomi [2016] L. Bergomi.  _Stochastic Volatility Modeling_.  Chapman and Hall/CRC Financial Mathematics Series. CRC Press, 2016.
  * Bernard and Cui [2010] C. Bernard and Z. Cui.  Pricing timer options.  _Journal of Computational Finance_ , 15, 08 2010.
  * Bhatt and Borkar [1996] A. G. Bhatt and V. S. Borkar.  Occupation measures for controlled markov processes: Characterization and optimality.  _The Annals of Probability_ , 24(3):1531–1562, 1996.
  * Blumenthal and Getoor [1964] R. M. Blumenthal and R. K. Getoor.  Local times for markov processes.  _Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete_ , 3:50–74, 1964.
  * Bonesini et al. [2020] O. Bonesini, A. Jacquier, and C. Lacombe.  A theoretical analysis of Guyon’s toy volatility model.  _arXiv:2001.05248_ , 2020.
  * Bouchard and Tan [2023] B. Bouchard and X. Tan.  On the regularity of solutions of some linear parabolic path-dependent PDEs.  _arXiv:2310.04308_ , 2023.
  * Bouleau and Yor [1981] N. Bouleau and M. Yor.  Sur la variation quadratique des temps locaux de certaines semimartingales.  _Comptes rendus de l’Académie des Sciences_ , 292:491–494, 1981.
  * Bouveret et al. [2020] G. Bouveret, R. Dumitrescu, and P. Tankov.  Mean-field games of optimal stopping: A relaxed solution approach.  _SIAM Journal on Control and Optimization_ , 58(4):1795–1821, 2020.
  * Breeden and Litzenberger [1978] D. T. Breeden and R. H. Litzenberger.  Prices of state-contingent claims implicit in option prices.  _The Journal of Business_ , 51(4):621–651, 1978.
  * Cardaliaguet et al. [2019] P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions.  _The Master Equation and the Convergence Problem in Mean Field Games:(AMS-201)_ , volume 201.  Princeton University Press, 2019.
  * Carmona and Delarue [2018] R. Carmona and F. Delarue.  _Probabilistic Theory of Mean Field Games with Applications I-II_.  Probability Theory and Stochastic Modelling. Springer International Publishing, 2018.
  * Chesney et al. [1997] M. Chesney, M. Jeanblanc-Picqué, and M. Yor.  Brownian excursions and parisian barrier options.  _Advances in Applied Probability_ , 29(1):165–184, 1997.
  * Cox and Källblad [2017] A. M. G. Cox and S. Källblad.  Model-independent bounds for asian options: A dynamic programming approach.  _SIAM Journal on Control and Optimization_ , 55(6):3409–3436, 2017.
  * Cox et al. [2021] A. M. G. Cox, S. Källblad, M. Larsson, and S. Svaluto-Ferro.  Controlled measure-valued martingales: a viscosity solution approach.  _arXiv:2109.00064_ , 2021.
  * Cuchiero et al. [2019] C. Cuchiero, M. Larsson, and S. Svaluto-Ferro.  Probability measure-valued polynomial diffusions.  _Electronic Journal of Probability_ , 24:1 – 32, 2019.
  * Cuchiero et al. [2020] C. Cuchiero, W. Khosrawi, and J. Teichmann.  A generative adversarial network approach to calibration of local stochastic volatility models.  _Risks_ , 8(4), 2020.
  * Cuchiero et al. [2021] C. Cuchiero, F. Guida, L. di Persio, and S. Svaluto-Ferro.  Measure-valued affine and polynomial diffusions.  _arXiv:2112.15129_ , 2021.
  * Dawson [1975] D. A. Dawson.  Stochastic evolution equations and related measure processes.  _Journal of Multivariate Analysis_ , 5(1):1–52, 1975.
  * Dawson [1993] D. A. Dawson.  Measure-valued Markov processes.  In _École d’Été de Probabilités de Saint-Flour XXI—1991_ , volume 1541 of _Lecture Notes in Math._ , pages 1–260. Springer, Berlin, 1993.
  * Detemple [2005] J. Detemple.  _American-Style Derivatives: Valuation and Computation_.  Chapman and Hall/CRC Financial Mathematics Series. CRC Press, 2005.
  * Di Nunno et al. [2009] G. Di Nunno, B. Øksendal, and F. Proske.  _Malliavin Calculus for Lévy Processes with Application to Finance_.  Springer, 2009.
  * Dupire [1994] B. Dupire.  Pricing with a smile.  _Risk_ , 7(1), 1994.
  * Dupire [2009] B. Dupire.  Functional Itô calculus.  _SSRN_ , 2009.  Republished in Quantitative Finance, 19(5):721–729, 2019.
  * Dupire and Tissot-Daguette [2022] B. Dupire and V. Tissot-Daguette.  Functional Expansions.  _arXiv:2212.13628_ , 2022.
  * Ekren et al. [2014] I. Ekren, C. Keller, N. Touzi, and J. Zhang.  On viscosity solutions of path dependent PDEs.  _The Annals of Probability_ , 42(1):204 – 236, 2014.
  * Fleming and Soner [2006] W. H. Fleming and H. M. Soner.  _Controlled Markov processes and viscosity solutions_ , volume 25.  Springer Science & Business Media, 2006.
  * Fleming and Vermes [1988] W. H. Fleming and D. Vermes.  Generalized solutions in the optimal control of diffusions.  _stochastic control theory and applications (Minneapolis, Minn., 1986)_ , pages 119–127, 1988.
  * Fleming and Viot [1979] W. H. Fleming and M. Viot.  Some measure-valued Markov processes in population genetics theory.  _Indiana University Mathematics Journal_ , 28(5):817–843, 1979.
  * Geman and Horowitz [1980] D. Geman and J. Horowitz.  Occupation Densities.  _The Annals of Probability_ , 8(1):1 – 67, 1980.
  * Guo et al. [2023] X. Guo, H. Pham, and X. Wei.  Itô’s formula for flows of measures on semimartingales.  _Stochastic Processes and their Applications_ , 159:350–390, 2023.
  * Guyon [2014] J. Guyon.  Path-dependent volatility.  _Risk_ , 2014.
  * Guyon [2017] J. Guyon.  Path-dependent volatility: practical examples.  _Global Derivatives Conference_ , 2017.
  * Guyon and Henry-Labordère [2013] J. Guyon and P. Henry-Labordère.  _Nonlinear Option Pricing_.  Chapman and Hall/CRC Financial Mathematics Series. CRC Press, 2013.
  * Guyon and Lekeufack [2023] J. Guyon and J. Lekeufack.  Volatility is (mostly) path-dependent.  _Quantitative Finance_ , 23(9):1221–1258, 2023.
  * Guyon and Mustapha [2023] J. Guyon and S. Mustapha.  Neural joint S&P 500/VIX smile calibration.  _Risk_ , 2023.
  * Hobson and Rogers [1998] D. G. Hobson and L. C. G. Rogers.  Complete Models with Stochastic Volatility.  _Mathematical Finance_ , 8(1):27–48, 1998.
  * Hugonnier [1999] J.-N. Hugonnier.  The Feynman–Kac formula and pricing occupation time derivatives.  _International Journal of Theoretical and Applied Finance_ , 02(02):153–178, 1999.
  * Kloeden and Platen [1992] P. Kloeden and E. Platen.  _Numerical Solution of Stochastic Differential Equations_.  Springer Berlin, 1992.
  * Lasry and Lions [2007] J.-M. Lasry and P.-L. Lions.  Mean field games.  _Japanese Journal of Mathematics_ , 2:229–260, 03 2007.
  * Lee [2010] R. Lee.  Weighted variance swap.  _Encyclopedia of Quantitative Finance_ , 2010.
  * Lieberman and Omprakash [2007] P. Lieberman and A. Omprakash.  Corridor variance swaps: A cheaper way to buy volatility?  _BNP Paribas, U.S. Equities and Derivatives Strategy_ , 2007.  White paper.
  * Litterer and Oberhauser [2014] C. Litterer and H. Oberhauser.  On a Chen-Fliess approximation for diffusion functionals.  _Monatshefte fur Mathematik_ , 175(4):577–593, 2014.
  * Longstaff and Schwartz [2001] F. A. Longstaff and E. S. Schwartz.  Valuing American options by simulation: a simple least-squares approach.  _The Review of Financial Studies_ , 14(1):113–147, 2001.
  * Peng and Wang [2011] S. Peng and F. Wang.  BSDE, path-dependent PDE and nonlinear Feynman-Kac formula.  _Science China Mathematics_ , 08 2011.
  * Peskir and Shiryaev [2006] G. Peskir and A. Shiryaev.  Optimal stopping and free-boundary problems.  _Basel: Birkhäuser Verlag_ , 2006.
  * Revuz and Yor [1999] D. Revuz and M. Yor.  _Continuous Martingales and Brownian Motion_.  Springer Berlin, Heidelberg, 1999.
  * Sawyer [2007] N. Sawyer.  SG CIB launches timer options.  https://www.risk.net/derivatives/structured-products/1506473/sg-cib-launches-timer-options, 2007.  Risk. Accessed 10/25/2023.
  * Talbi et al. [2023a] M. Talbi, N. Touzi, and J. Zhang.  Dynamic programming equation for the mean field optimal stopping problem.  _SIAM Journal on Control and Optimization_ , 61(4):2140–2164, 2023a.
  * Talbi et al. [2023b] M. Talbi, N. Touzi, and J. Zhang.  Viscosity solutions for obstacle problems on wasserstein space.  _SIAM Journal on Control and Optimization_ , 61(3):1712–1736, 2023b.
  * Viens and Zhang [2019] F. Viens and J. Zhang.  A martingale approach for fractional Brownian motions and related path dependent PDEs.  _The Annals of Applied Probability_ , 29(6):3489 – 3540, 2019.
  * Zhang [2017] J. Zhang.  _Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory_.  Probability Theory and Stochastic Modelling. Springer New York, 2017.