pith. sign in

arxiv: 2311.07936 · v6 · submitted 2023-11-14 · 🧮 math.PR · q-fin.MF· q-fin.PR

Occupied Processes: Going with the Flow

Valentin Tissot-Daguette This is my paper

Pith reviewed 2026-05-24 05:28 UTC · model grok-4.3

classification 🧮 math.PR q-fin.MFq-fin.PR
keywords occupied processesoccupation flowItô calculuspath-dependent PDEsMarkov processeslocal timesoptimal stoppingfinancial derivatives
0
0 comments X

The pith

Enlarging a Markov process with its occupation flow preserves the Markov property.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when a stochastic process X is Markov, the enlarged process that also tracks the occupation flow O (time spent at each level) remains Markov. It develops an Itô calculus for these occupied processes that sits between the classical version and Dupire's full functional version, yielding Itô formulae without extra path regularity. Feynman-Kac then produces path-dependent PDEs in which O acts as the time variable while the state variable stays the finite-dimensional value of X. The construction is used to treat optimal stopping problems with local times and to give Markovian lifts for exotic options and variance instruments.

Core claim

When X is Markov, the occupied process (O,X) enjoys a Markov structure as well. We develop an Itô calculus for occupied processes that lies midway between Dupire's functional Itô calculus and the classical version. We derive Itô formulae and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where O plays the role of time. The space variable, given by the current value of X, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods.

What carries the argument

The occupied process (O,X) formed by adjoining the occupation flow O, which records the time spent by the path at each level, to the original Markov process X.

If this is right

  • Optimal stopping problems that involve local times become solvable inside the Markovian framework.
  • Exotic options and variance instruments obtain unified Markovian lifts.
  • Derivative books can be priced with a single numerical solver rather than separate models for each contract.
  • Forward variance models extend to the full forward occupation surface.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite-dimensional state may allow existing elliptic PDE solvers to handle a wider range of path-dependent payoffs without functional-Itô overhead.
  • The same occupation-flow enlargement could be tested on non-Markov base processes to see whether approximate Markovianity emerges.
  • Numerical checks on Brownian motion or simple diffusions would confirm whether the new PDEs recover known prices for variance swaps.

Load-bearing premise

The technical conditions under which adding the occupation flow preserves the Markov property and permits an intermediate Itô calculus without the full functional Itô machinery or extra path regularity.

What would settle it

A concrete Markov process X together with an explicit calculation showing that (O,X) fails the Markov property at some time t or that the derived Itô formula does not hold for a simple test function.

Figures

Figures reproduced from arXiv: 2311.07936 by Valentin Tissot-Daguette.

Figure 1
Figure 1. Figure 1: Occupation flow O and its dynamics d Ot = δXtd⟨X⟩t . x 0 t δXt Xt Ot 2.3 Occupation Functional and Markov Property An occupation functional is a map f : D → R, (o, x) 7→ f(o, x). Naturally, we are interested in plugging in occupied processes and studying properties of f(Ot , Xt) or f(O˜ t , Xt). We stress that occupation functionals do not explicitly depend on the time variable. This is because occupation … view at source ↗
Figure 2
Figure 2. Figure 2: The occupation derivative ∂of(Ot , Xt) gives the sensitivity of f with respect to a Dirac impulse at the spot Xt . Proposition 3. If X is a continuous semimartingale and f ∈ C1 (M), then df(Ot) = ∂of(Ot , Xt)d⟨X⟩t , (25) df(O˜ t) = ∂of(O˜ t , Xt)dt. (26) Proof. It is a byproduct of Theorem 3.1 below. Example 1. Consider the linear functional f(o) = ϕ · o. In light of the occupation time formula (11), f(Ot)… view at source ↗
Figure 3
Figure 3. Figure 3: Unified Markovian lift for exotic options and structured products. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Unified Markovian lift for variance derivatives. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Occupation measure using calendar time ( [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Guyon’s toy volatility model with σ given in (65). One simulations using the Euler￾Maruyama scheme in Section 5.1. Parameters: x0 = 100, T = 1, κ = 12, and (α, β, γ) = (2.1, 1.2, 1.9). 0.00 0.25 0.50 0.75 1.00 t 60 70 80 90 100 110 X 1 Oκ(R) R R x Oκ(dx) 0.00 0.25 0.50 0.75 1.00 t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 σ(Oκ t , Xt) with parameters α, β, γ > 0, and functional Υ(O κ t , Xt) = Xt 1 Oκ t (R) R R … view at source ↗
Figure 7
Figure 7. Figure 7: Local occupied volatility (yellow dot) compared with local volatility (red). The [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulated price path and volatility in the Local Occupied Volatility (LOV) model, [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Calibration loss L(θ) versus epochs. volatility model. We also set r = 4.35% and q = 0.5%, aligned respectively with short-term U.S. treasury rates and SPX implied dividend yields as of 02/27/2025. All market data was sourced from Bloomberg. We sample 28 pairs of antithetic paths (hence J = 29 ) for each training iteration, which is increased to 212 pairs after 1, 000 epochs. In Step I., the occupied SDE i… view at source ↗
Figure 10
Figure 10. Figure 10: Model implied volatility skews for SPX monthly options compared to market bid [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Neural and local volatility versus spot. 2 [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Implied sensitivity function ℓ Q(t, Xt , ·) (yellow) and projected occupation measure Oˆ κ t = E Q[Oκ t |Xt ] (purple). 5.4 Joint Calibration To better pinpoint the sensitivity function, further work includes the calibration of occupied volatility to other instruments, such as VIX, exotic, or American options. Indeed, while the local volatility function guarantees perfect calibration to European vanilla o… view at source ↗
Figure 13
Figure 13. Figure 13: Forward occupation surfaces of SPX Options as of 2025/02/27, 4pm EST. Left: [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Occupation flow O and stopped spot local time L Xτ τ . Two simulations. x τ L Xτ τ x τ L Xτ τ S either mentally or visually, we can regard the stopping region as a subset of R for a given occupation measure, i.e., S(o) = {x ∈ supp(o) : (o, x) ∈ S}. For every t ∈ [0, T] , this gives the pathwise partition {Xs : s ≤ t} = supp(Ot) = S(Ot) ∪ C(Ot), with the continuation region C(o) = {x ∈ supp(o) : v(o, x) > … view at source ↗
Figure 15
Figure 15. Figure 15: Spot local time OS problem with T = {t, T}. Stopping region (red) and contin￾uation region (green) at t depending whether the intrinsic value L x t (if Xt = x) exceeds the continuation value given in (96). x t T Ot + EQ[Ot,T] Ot legitimate in the sense that the value of the OS problem (90) with φ = φ ε converges to the value of (95). If X is a simulated Brownian path on a regular time grid tn = nδt, δt = … view at source ↗
Figure 16
Figure 16. Figure 16: Verification of Proposition 7 and Corollary 1 with T = 1. Estimate of ε 7→ v ε ∗ using the inspection strategy (Section 7.3), N = 400 time intervals and 214 simulations. 0.1 0.2 0.3 0.4 ε 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Least Square Monte Carlo v ε ∗ v∗ − √ Cε v ε(T) converges to the value v∗ of (95) in the extreme cases T = {T} and T = [0, T]. We start with the former, where v ε ∗ = v ε (T) and v∗ = v(T)… view at source ↗
Figure 17
Figure 17. Figure 17: Original path ω (left), time permutation ψ defined in (101) with ξ = (−1, 1, 1, −1) and σ((1, 2, 3, 4)) = (3, 2, 4, 1) (middle) and transformed path ω ◦ ψ (right). t0 t1 t2 t3 t4 t0 t1 t2 t3 t4 t0 t1 t2 t3 t4 t0 t1 t2 t3 t4 Definition 4. A path functional f : ΩT → R is called chronology-invariant if f(ω ◦ ψ) = f(ω) for all ω ∈ ΩT , ψ ∈ ST . The next Theorem finally connects chronology invariance functiona… view at source ↗
read the original abstract

A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$ that tracks the time spent by the path at each level. When $X$ is Markov, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure as well. We develop an It\^o calculus for occupied processes that lies midway between Dupire's functional It\^o calculus and the classical version. We derive It\^o formulae and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. The framework's benefits are illustrated via an optimal stopping problem involving local times, followed by financial applications. For the latter, we show how occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price derivatives books with a single numerical solver. We finally explore an extension of forward variance models so as to leverage the entire forward occupation surface.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces occupied processes by enlarging a Markov process X with its occupation flow O that tracks time spent at each level. It claims that (O,X) inherits the Markov property, develops an intermediate Itô calculus between Dupire's functional version and the classical one, derives Itô formulae, and applies Feynman-Kac to obtain path-dependent PDEs in which O plays the role of time while the space variable (current value of X) remains finite-dimensional. Applications include an optimal stopping problem with local times and financial examples showing unified Markovian lifts for exotic options and variance instruments, plus an extension to forward variance models.

Significance. If the central claims hold with the stated derivations, the framework offers a practical intermediate calculus that keeps the state space finite-dimensional, enabling standard elliptic PDE techniques and a single numerical solver for certain path-dependent financial derivatives. This provides a novel Markovian lift via occupation flows and could facilitate analysis of problems involving local times or variance instruments.

major comments (2)
  1. [Abstract and §2 (Markov property claim)] The central claim that (O,X) is Markov when X is Markov is load-bearing, yet the abstract and introduction provide no derivations or verification of the Markov property for the pair; the technical conditions (e.g., on path regularity or the occupation flow construction) under which this holds without additional assumptions must be stated explicitly and proved in the relevant section.
  2. [Itô calculus section (around the Feynman-Kac application)] The Itô formulae for occupied processes are asserted to lie midway between functional and classical Itô calculus without invoking the full functional machinery, but no specific equations or derivations are referenced in the provided overview; the intermediate nature and avoidance of extra regularity assumptions on paths need explicit verification in the derivation section.
minor comments (2)
  1. [Introduction/Notation] Clarify the precise definition and construction of the occupation flow O at the outset to ensure readers can follow the additive structure without ambiguity.
  2. [Financial applications section] In the financial applications, expand on how the single numerical solver handles the unified pricing of the book of derivatives to make the practical benefit concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below. The revisions will consist of explicit statements of technical conditions, forward references to existing proofs and derivations, and added cross-references; these are minor textual clarifications that do not alter the results.

read point-by-point responses

  1. Referee: [Abstract and §2 (Markov property claim)] The central claim that (O,X) is Markov when X is Markov is load-bearing, yet the abstract and introduction provide no derivations or verification of the Markov property for the pair; the technical conditions (e.g., on path regularity or the occupation flow construction) under which this holds without additional assumptions must be stated explicitly and proved in the relevant section.

    Authors: We agree that the Markov property of the pair (O,X) is central. Section 2 contains the construction of the occupation flow O and the proof that (O,X) inherits the Markov property from X whenever X is a strong Markov process with càdlàg paths; the argument uses the fact that the occupation measure up to time t is a measurable functional of the path segment [0,t] together with the Markov property of X. The only standing assumptions are those already stated for X (right-continuous paths with left limits and the usual filtration). We will revise the introduction to state these conditions explicitly and add a forward reference to the proof in Section 2. This is a minor clarification. revision: yes

  2. Referee: [Itô calculus section (around the Feynman-Kac application)] The Itô formulae for occupied processes are asserted to lie midway between functional and classical Itô calculus without invoking the full functional machinery, but no specific equations or derivations are referenced in the provided overview; the intermediate nature and avoidance of extra regularity assumptions on paths need explicit verification in the derivation section.

    Authors: The intermediate Itô formula is stated and derived in Section 3 (Equation (3.2) and the subsequent proof). The formula augments the classical Itô terms for the finite-dimensional process X with an additional integral against the occupation flow O; the derivation proceeds via the chain rule applied to the joint process (O,X) and does not invoke the full functional Itô calculus of Dupire, nor does it require pathwise differentiability beyond the càdlàg assumption already imposed on X. We will insert explicit references to Equation (3.2) and the comparison paragraph in Section 3 into the introduction and the Feynman-Kac subsection, together with a short remark confirming the avoidance of extra regularity. This is a minor addition of cross-references. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the Markov property of (O,X) from the Markov property of X together with the additive definition of the occupation flow O; this is a direct structural consequence rather than a self-definition or fitted input. The intermediate Itô calculus is constructed explicitly between Dupire's functional calculus and the classical version, with Feynman-Kac applied in the standard way to produce path-dependent PDEs. No equations reduce by construction to their own inputs, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness result is imported from prior work by the same author. The derivation remains self-contained against external Markov and stochastic-calculus benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard stochastic calculus background and introduces the occupation flow as a new entity. No free parameters are evident from the abstract.

axioms (2)
  • domain assumption X is a Markov process.
    Invoked to establish that (O,X) is Markov.
  • standard math Standard conditions for Ito processes and Feynman-Kac representations hold.
    Required to derive the Ito formulae and PDE class.
invented entities (1)
  • Occupation flow O no independent evidence
    purpose: Tracks time spent by the path at each level to enlarge the process.
    Newly defined in the paper to create the occupied process.

pith-pipeline@v0.9.0 · 5721 in / 1345 out tokens · 35494 ms · 2026-05-24T05:28:10.559962+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages

  1. [1]

    Barlow and M

    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

    work page 1982

  2. [2]

    Bayer, P

    C. Bayer, P. P. Hager, S. Riedel, and J. Schoenmakers. Optimal stopping with signa- tures.The Annals of Applied Probability, 33(1):238 – 273, 2023

    work page 2023

  3. [3]

    Bayraktar, Q

    E. Bayraktar, Q. Feng, and Z. Zhang. Deep signature algorithm for multidimensional path-dependent options.SIAM Journal on Financial Mathematics, 15(1):194–214, 2024

    work page 2024

  4. [4]

    Becker, P

    S. Becker, P. Cheridito, and A. Jentzen. Deep optimal stopping.Journal of Machine Learning Research, 20(74):1–25, 2019. 53 OCCUPIED PROCESSES V. TISSOT-DAGUETTE

    work page 2019

  5. [5]

    Ben Hamida and R

    S. Ben Hamida and R. Cont. Recovering volatility from option prices by evolutionary optimization.Journal of Computational Finance, 8(4):43–76, 1997. ISSN 1755-2850

    work page 1997

  6. [6]

    Bena¨ ım and O

    M. Bena¨ ım and O. Raimond. Self-interacting diffusions II: convergence in law.Annales de l’Institut Henri Poincare (B) Probability and Statistics, 39(6):1043–1055, 2003

    work page 2003

  7. [7]

    Bena¨ ım and O

    M. Bena¨ ım and O. Raimond. Self-interacting diffusions. III. symmetric interactions. The Annals of Probability, 33(5):1716–1759, 2005

    work page 2005

  8. [8]

    Bena¨ ım and O

    M. Bena¨ ım and O. Raimond. Self-interacting diffusions IV: Rate of convergence.Elec- tronic Journal of Probability, 16:1815–1843, 2011

    work page 2011

  9. [9]

    Bena¨ ım, M

    M. Bena¨ ım, M. Ledoux, and O. Raimond. Self-interacting diffusions.Probability Theory and Related Fields, 122:1–41, 2002

    work page 2002

  10. [10]

    Bergomi.Stochastic Volatility Modeling

    L. Bergomi.Stochastic Volatility Modeling. Chapman and Hall/CRC Financial Mathe- matics Series. CRC Press, 2016

    work page 2016

  11. [11]

    Bernard and Z

    C. Bernard and Z. Cui. Pricing timer options.Journal of Computational Finance, 15, 08 2010

    work page 2010

  12. [12]

    B´ ethencourt, R

    L. B´ ethencourt, R. Catellier, and E. Tanr´ e. Brownian particles controlled by their occupation measure.SIAM Journal on Control and Optimization, 63(2):1286–1313, 2025

    work page 2025

  13. [13]

    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

    work page 1996

  14. [14]

    R. M. Blumenthal and R. K. Getoor. Local times for markov processes.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 3:50–74, 1964

    work page 1964

  15. [15]

    Bonesini, A

    O. Bonesini, A. Jacquier, and C. Lacombe. A theoretical analysis of Guyon’s toy volatil- ity model.SIAM Journal on Financial Mathematics, 16(2):271–309, 2025

    work page 2025

  16. [16]

    G. P. Boswell and F. A. Davidson. Modelling hyphal networks.Fungal Biology Reviews, 26(1):30–38, 2012. ISSN 1749-4613. Hyphal networks: mechanisms, modelling and ecology

    work page 2012

  17. [17]

    Bouchard and X

    B. Bouchard and X. Tan. On the regularity of solutions of some linear parabolic path- dependent PDEs.arXiv:2310.04308, 2023

    work page arXiv 2023

  18. [18]

    Bouveret, R

    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

    work page 2020

  19. [19]

    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

    work page 1978

  20. [20]

    Brunick and S

    G. Brunick and S. Shreve. Mimicking an Itˆ o process by a solution of a stochastic differential equation.The Annals of Applied Probability, 23(4):1584 – 1628, 2013. 54 OCCUPIED PROCESSES V. TISSOT-DAGUETTE

    work page 2013

  21. [21]

    Cardaliaguet, F

    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

    work page 2019

  22. [22]

    Carmona and F

    R. Carmona and F. Delarue.Probabilistic Theory of Mean Field Games with Applica- tions I-II. Probability Theory and Stochastic Modelling. Springer International Pub- lishing, 2018

    work page 2018

  23. [23]

    Chassagneux and G

    J.-F. Chassagneux and G. Pag` es. Computing the invariant distribution of mckean-vlasov sdes by ergodic simulation.arXiv:2406.13370, 2025

    work page arXiv 2025

  24. [24]

    Chesney, M

    M. Chesney, M. Jeanblanc-Picqu´ e, and M. Yor. Brownian excursions and Parisian barrier options.Advances in Applied Probability, 29(1):165–184, 1997

    work page 1997

  25. [25]

    R. Cont. Functional Itˆ o calculus and functional Kolmogorov equations. In V. Bally, L. Caramellino, and R. Cont, editors,Stochastic Integration by Parts and Functional Itˆ o Calculus. Advanced Courses in Mathematics - CRM Barcelona, Birkauser, 2016

    work page 2016

  26. [26]

    Cont and D.-A

    R. Cont and D.-A. Fourni´ e. Change of variable formulas for non-anticipative functionals on path space.Journal of Functional Analysis, 259(4):1043–1072, 2010

    work page 2010

  27. [27]

    A. M. G. Cox and S. K¨ allblad. Model-independent bounds for asian options: A dynamic programming approach.SIAM Journal on Control and Optimization, 55(6):3409–3436, 2017

    work page 2017

  28. [28]

    A. M. G. Cox, S. K¨ allblad, M. Larsson, and S. Svaluto-Ferro. Controlled measure-valued martingales: a viscosity solution approach.arXiv:2109.00064, 2021

    work page arXiv 2021

  29. [29]

    Cranston and Y

    M. Cranston and Y. Le Jan. Self attracting diffusions: Two case studies.Mathematische Annalen, 303:87–93, 1995. doi: 10.1007/BF01460980

    work page doi:10.1007/bf01460980 1995

  30. [30]

    Cuchiero, M

    C. Cuchiero, M. Larsson, and S. Svaluto-Ferro. Probability measure-valued polynomial diffusions.Electronic Journal of Probability, 24:1 – 32, 2019

    work page 2019

  31. [31]

    Cuchiero, F

    C. Cuchiero, F. Guida, L. di Persio, and S. Svaluto-Ferro. Measure-valued affine and polynomial diffusions.arXiv:2112.15129, 2021

    work page arXiv 2021

  32. [32]

    D. A. Dawson. Stochastic evolution equations and related measure processes.Journal of Multivariate Analysis, 5(1):1–52, 1975

    work page 1975

  33. [33]

    D. A. Dawson. Measure-valued Markov processes. In ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XXI—1991, volume 1541 ofLecture Notes in Math., pages 1–260. Springer, Berlin, 1993

    work page 1991

  34. [34]

    Detemple.American-Style Derivatives: Valuation and Computation

    J. Detemple.American-Style Derivatives: Valuation and Computation. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, 2005

    work page 2005

  35. [35]

    Di Nunno, B

    G. Di Nunno, B. Øksendal, and F. Proske.Malliavin Calculus for L´ evy Processes with Application to Finance. Springer, 2009

    work page 2009

  36. [36]

    Dixon, I

    M. Dixon, I. Halperin, and P. Bilokon.Machine Learning in Finance: From Theory to Practice. Springer International Publishing, 2020. ISBN 9783030410674. 55 OCCUPIED PROCESSES V. TISSOT-DAGUETTE

    work page 2020

  37. [37]

    K. Du, Y. Jiang, and J. Li. Empirical approximation to invariant measures for McKean– Vlasov processes: Mean-field interaction vs self-interaction.Bernoulli, 29(3):2492–2518, Aug. 2023

    work page 2023

  38. [38]

    B. Dupire. Pricing with a smile.Risk, 7(1), 1994

    work page 1994

  39. [39]

    B. Dupire. A unified theory of volatility. In P. Carr, editor,Derivatives Pricing: The Classic Collection. Risk Books, London, 2004. Reprint of a 1996 working paper from Paribas Capital Markets

    work page 2004

  40. [40]

    B. Dupire. Functional Itˆ o calculus.SSRN, 2009. Republished inQuantitative Finance, 19(5):721–729, 2019

    work page 2009

  41. [41]

    Durrett and L

    R. Durrett and L. Rogers. Asymptotic behavior of Brownian polymers.Probability Theory and Related Fields, 92(3):337–349, 1992

    work page 1992

  42. [42]

    W. E, J. Han, and A. Jentzen. Deep learning-based numerical methods for high- dimensional parabolic partial differential equations and backward stochastic differential equations.Commun. Math. Stat., 5, 2017

    work page 2017

  43. [43]

    W. E, J. Han, and A. Jentzen. Solving high-dimensional partial differential equations using deep learning.Proc. Natl. Acad. Sci., 115(34), 2018

    work page 2018

  44. [44]

    Ekren, C

    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

    work page 2014

  45. [45]

    W. H. Fleming and H. M. Soner.Controlled Markov Processes and Viscosity Solutions, volume 25 ofStochastic Modelling and Applied Probability. Springer, 2 edition, 2006

    work page 2006

  46. [46]

    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

    work page 1986

  47. [47]

    W. H. Fleming and M. Viot. Some measure-valued Markov processes in population genetics theory.Indiana University Mathematics Journal, 28(5):817–843, 1979

    work page 1979

  48. [48]

    P. K. Friz and J. Gatheral. The forest expansion of forward variance models. In C. Bayer, P. K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, and M. Rosenbaum, editors,Rough Volatility, pages 183–198. SIAM, 2023. doi: 10.1137/1.9781611977783.ch9. URLhttps: //epubs.siam.org/doi/10.1137/1.9781611977783.ch9

    work page doi:10.1137/1.9781611977783.ch9 2023

  49. [49]

    Gatheral and M

    J. Gatheral and M. Keller-Ressel. Affine forward variance models.Finance and Stochas- tics, 23(1):501 – 533, 2019

    work page 2019

  50. [50]

    Geman and J

    D. Geman and J. Horowitz. Occupation Densities.The Annals of Probability, 8(1):1 – 67, 1980

    work page 1980

  51. [51]

    Glasserman.Discretization Methods, pages 339–376

    P. Glasserman.Discretization Methods, pages 339–376. Springer New York, New York, NY, 2003

    work page 2003

  52. [52]

    J. Guyon. Path-dependent volatility.Risk, 2014. 56 OCCUPIED PROCESSES V. TISSOT-DAGUETTE

    work page 2014

  53. [53]

    J. Guyon. Path-dependent volatility: practical examples.Global Derivatives Conference, 2017

    work page 2017

  54. [54]

    Guyon and P

    J. Guyon and P. Henry-Labord` ere. Being particular about calibration.Risk, 25:88–93, January 2012

    work page 2012

  55. [55]

    Guyon and P

    J. Guyon and P. Henry-Labord` ere.Nonlinear Option Pricing. Chapman and Hall/CRC Financial Mathematics Series. CRC Press, 2013

    work page 2013

  56. [56]

    Guyon and J

    J. Guyon and J. Lekeufack. Volatility is (mostly) path-dependent.Quantitative Finance, 23(9):1221–1258, 2023

    work page 2023

  57. [57]

    Haber, P

    S. Haber, P. J. Sch¨ onbucher, and P. Wilmott. Pricing parisian options.The Journal of Derivatives, 6(3):71–79, 1999

    work page 1999

  58. [58]

    L. G. Hanin. An extension of the kantorovich norm.Contemporary Mathematics, 226: 113–130, 1999. ISSN 0271-4132

    work page 1999

  59. [59]

    D. G. Hobson and L. C. G. Rogers. Complete Models with Stochastic Volatility.Math- ematical Finance, 8(1):27–48, 1998

    work page 1998

  60. [60]

    Hu and M

    R. Hu and M. Lauri` ere. Recent developments in machine learning methods for stochastic control and games.Numerical Algebra, Control and Optimization, 14(3):435–525, 2024

    work page 2024

  61. [61]

    Huang, V

    Y. Huang, V. Tissot-Daguette, and X. Zhang. Deep BSDE solver for occupied pricing PDEs.In preparation

    work page

  62. [62]

    Hugonnier

    J.-N. Hugonnier. The Feynman–Kac formula and pricing occupation time derivatives. International Journal of Theoretical and Applied Finance, 02(02):153–178, 1999

    work page 1999

  63. [63]

    Hur´ e, H

    C. Hur´ e, H. Pham, and X. Warin. Deep backward schemes for high-dimensional nonlin- ear pdes.Mathematics of Computation, 89(324):1785–1821, 2020. doi: 10.1090/mcom/ 3514

    work page doi:10.1090/mcom/ 2020

  64. [64]

    Jacquier and M

    A. Jacquier and M. Oumgari. Deep curve-dependent PDEs for affine rough volatility. SIAM Journal on Financial Mathematics, 14(2):353–382, 2023

    work page 2023

  65. [65]

    Kloeden and E

    P. Kloeden and E. Platen.Numerical Solution of Stochastic Differential Equations. Springer Berlin, 1992

    work page 1992

  66. [66]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Mean field games.Japanese Journal of Mathematics, 2: 229–260, 03 2007

    work page 2007

  67. [67]

    R. Lee. Weighted variance swap.Encyclopedia of Quantitative Finance, 2010

    work page 2010

  68. [68]

    Lieberman and A

    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

    work page 2007

  69. [69]

    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

    work page 2001

  70. [70]

    R. C. Merton. Theory of rational option pricing.The Bell Journal of Economics and Management Science, 4(1):141–183, 1973. 57 OCCUPIED PROCESSES V. TISSOT-DAGUETTE

    work page 1973

  71. [71]

    Peng and F

    S. Peng and F. Wang. BSDE, path-dependent PDE and nonlinear Feynman-Kac for- mula.Science China Mathematics, 08 2011

    work page 2011

  72. [72]

    Peskir and A

    G. Peskir and A. Shiryaev. Optimal stopping and free-boundary problems.Basel: Birkh¨ auser Verlag, 2006

    work page 2006

  73. [73]

    O. Raimond. Self-attracting diffusions: Case of the constant interaction.Probability Theory and Related Fields, 107:177–196, 1997

    work page 1997

  74. [74]

    Reppen, H

    A. Reppen, H. Soner, and Tissot-Daguette. Deep stochastic optimization in finance. Digital Finance, 5, 2023

    work page 2023

  75. [75]

    Revuz and M

    D. Revuz and M. Yor.Continuous Martingales and Brownian Motion. Springer Berlin, Heidelberg, 1999

    work page 1999

  76. [76]

    Y. F. Saporito and Z. Zhang. Path-dependent deep Galerkin method: A neural net- work approach to solve path-dependent partial differential equations.SIAM Journal on Financial Mathematics, 12(3):912–940, 2021

    work page 2021

  77. [77]

    N. Sawyer. SG CIB launches timer options.https://www.risk.net/derivatives/ structured-products/1506473/sg-cib-launches-timer-options, 2007.Risk. Ac- cessed 10/25/2023

    work page arXiv 2007

  78. [78]

    H. M. Soner, V. Tissot-Daguette, and J. Zhang. Controlled occupied processes and viscosity solutions.arXiv:2411.12080, 2024

    work page arXiv 2024

  79. [79]

    Talbi, N

    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, 2023

    work page 2023

  80. [80]

    Talbi, N

    M. Talbi, N. Touzi, and J. Zhang. Viscosity solutions for obstacle problems on wasser- stein space.SIAM Journal on Control and Optimization, 61(3):1712–1736, 2023

    work page 2023

Showing first 80 references.