Pith. sign in

REVIEW 3 major objections 5 minor 37 references

Two-barriers-reflected BSDE with Rank-based Data

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper establishes that the solution of a two-barriers-reflected backward stochastic differential equation whose data depend on the ranks of an underlying diffusion is a viscosity solution of a double-obstacle parabolic PDE, and that…

desk verdict Solid two-barrier extension with a real gap in the uniqueness proof: Lemma 7's artificial boundary condition is asserted, not proved, and Theorem 6 leans on it. read the letter →

arxiv 2411.17104 v1 pith:R757ICEN submitted 2024-11-26 math.PR

Xinwei Feng , Lu Wang This is my paper · ORCID
classification math.PR MSC 60H1060H3035D4091G20
keywords backwardstochasticdifferentialequationstwo-reflecting-barriersBSDErank-basedviscositysolutiondouble-obstacleparabolicPDEAmericangameoptionDynkinrankedBrownianmotions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's aim is to build a bridge between rank-based diffusion models and nonlinear obstacle problems. It proves that the two-barriers-reflected BSDE (8), whose generator, terminal condition, and barriers all depend on the ordered vector $\widetilde{X}^{t,x}$, has a unique solution, and that the deterministic function $u(t,x)=Y^{t,x}(t)$ is a viscosity solution of the double-obstacle parabolic PDE (14). That PDE carries a collision boundary condition on the faces where two ranks coincide. The paper also proves uniqueness of this viscosity solution under a modulus-continuity condition on $G$ and a growth condition of the form $e^{A\log^2|x|}$. If the construction is correct, the backward component of the BSDE is the value of a Dynkin game, and in a market of rank-based stocks the fair price of an American game option is $e^{r_0 t}\widetilde{Y}(t)$.

What carries the argument

The key mechanism is the reduction of the rank-based BSDE to a standard two-barriers-reflected BSDE. Because the ranked processes satisfy $dX_{(l)}=\delta_l\,ds+\sigma_l\,d\beta_l+\tfrac12 d\Lambda_{l,l+1}-\tfrac12 d\Lambda_{l-1,l}$, with $\beta_l$ independent Brownian motions formed from the original noise by observing which component holds rank $l$, the stochastic integral $\int Z\cdot dW$ equals $\int \widehat{Z}\cdot d\beta$. This turns (8) into the classical two-barrier BSDE (10) driven by $\beta$, whose existence and uniqueness are imported from [19]. The same ranked-semimartingale structure supplies the Itô formula with local times used in the viscosity proof, and the obstacle problem (14) carries the boundary condition that the value function's derivatives agree at collision faces, enforced through the local-time terms.

What would settle it

Take n=3 with diffusion coefficients satisfying the variance condition, set L, U, and g so that the double-obstacle PDE (14) has an explicit solution, and compare that closed form to Monte Carlo simulation of the BSDE (8) on a fine grid; if the simulated $Y^{t,x}(t)$ differs from the PDE value at a point where $L<u<U$, Theorem 5 is false. A second test uses n=3 with $\sigma_1=\sigma_3=1$ and $\sigma_2=0.8$, which violates $\tfrac12(\sigma_1^2+\sigma_3^2)\le\sigma_2^2$, and checks whether the ranked noise processes defined by (3) are independent Brownian motions.

Watch

Extended reading notes

Core claim

The central discovery is that the map $(t,x)\mapsto Y^{t,x}(t)$, where $(Y,Z,K^+,K^-)$ is the unique solution of the two-barriers-reflected BSDE (8) with rank-based data, is a viscosity solution of the double-obstacle parabolic PDE (14). The obstacle problem is posed on the ordered region $\Pi_n$ with the boundary condition $\partial u/\partial x_{k+1}=\partial u/\partial x_k$ on each collision face $F_k$. Under a mild modulus-continuity assumption on the generator and a growth bound of the form $e^{A\log^2|x|}$, this viscosity solution is unique. In the financial application, the paper proves that the value $V(t)$ of an American game option on rank-based stocks equals $e^{r_0 t}\widetilde{Y}(t)$, with explicit saddle-point stopping times given by the hitting times of the two barriers.

Load-bearing premise

The load-bearing premise is that the rank-based stock process has a unique strong solution and never develops triple collisions, so the ranked processes are driven by independent Brownian motions; if triple collisions or dependence among the ranked noises appear, the reduction of the reflected BSDE to a standard two-barrier BSDE fails.

Editorial extensions

If this is right

  • Because $u(t,x)=Y^{t,x}(t)$ is a viscosity solution of the double-obstacle PDE (14), the BSDE can be evaluated by solving a deterministic obstacle problem, and convergent numerical PDE schemes give approximations to the reflected BSDE solution.
  • Under the modulus condition (27) and growth condition (28), the viscosity solution is unique, so the BSDE representation identifies the PDE solution unambiguously and vice versa.
  • For the Dynkin game with rank-based payoffs, the backward component $Y$ is the value of the game, and the hitting times of the upper and lower barriers form a saddle point.
  • In the $n$-stock rank-based market, the American game option has fair value $e^{r_0 t}\widetilde{Y}(t)$, and the paper constructs an explicit hedging strategy whose cancellation time is the hitting time of the upper barrier.
  • The discounted reflected BSDE used for the option is itself a two-barrier reflected BSDE with rank-based data, so Theorem 5 applies and the option price also solves an obstacle PDE.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the collision boundary condition $\partial u/\partial x_{k+1}=\partial u/\partial x_k$ on each $F_k$ means the value function is indifferent to which label holds which rank; numerical schemes could therefore be posed directly on the ordered simplex $\Pi_n$, reducing the effective dimension of the state space.
  • A natural extension not pursued in the paper is to use the BSDE representation as a Monte Carlo pricer for American game options in rank-based markets: simulate the ranked process and the reflected BSDE with penalization, then compare against a finite-difference solution of the deterministic obstacle PDE.
  • If the variance condition fails, I would expect the correspondence to degrade precisely when triple collisions occur; checking how the $L^2$ error between the penalized BSDE and the PDE grows as $\sigma_2^2$ approaches $\tfrac12(\sigma_1^2+\sigma_3^2)$ from above would quantify how load-bearing the no-triple-collision assumption is.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies two-barrier reflected backward stochastic differential equations (BSDEs) whose data are generated by a rank-based SDE (1). It defines u(t,x) as the initial value Y^{t,x}(t) of the reflected BSDE (8), proves via the standard penalization and Ishii-lemma template that u is a viscosity solution of the double-obstacle parabolic PDE (14) (Theorem 5), and claims uniqueness of such viscosity solutions under a modulus condition on G and a growth bound at infinity (Theorem 6). The final section applies the BSDE representation to the fair value of an American game option with rank-based stock prices (Theorem 13). The paper's main technical device is the reduction of (8) to a standard two-barrier BSDE (10) driven by the ranked Brownian motions β_l, using the strong uniqueness results of [22,35].

Significance. If the main theorems are correct, the paper supplies a probabilistic representation for a double-obstacle PDE with oblique-derivative boundary conditions generated by rank-based coefficients, and it converts the Dynkin-game value into an explicit hedging interpretation for an American game option. The argument is constructive and contains no fitted parameters; the PDE statement is falsifiable and the financial application is concrete. The main novelty is an extension of the existing one-barrier rank-based BSDE theory in [6,7] to the two-barrier and game-option setting. The significance is moderate rather than high: the proof structure follows established templates, but the boundary conditions at tie hyperplanes and the artificial comparison boundary O_j require genuine additional work.

major comments (3)
  1. [§4.2, Lemma 7 and Theorem 6] Lemma 7 states that w = u - v is a viscosity subsolution of (30), including the boundary condition sum_{l=1}^j (∂w/∂x_{k_l} - ∂w/∂x_{k_l+1}) = 0 on O_j. The proof of Lemma 7 only treats a maximum point in the interior In(Π^α_n) and derives (31); it never addresses the case (t,x) ∈ O_j. This is not a cosmetic omission: O_j is an artificial boundary at positive distance α from the original tie hyperplanes, so the boundary condition of (14) on F_k gives no information at O_j. In the proof of Theorem 6, the paragraph around (34) explicitly invokes the Lemma 7 boundary condition to discard the first term of the min when x* ∈ O_j. Without a proof that w inherits the stated zero-normal-derivative condition on O_j, the comparison argument does not close. The theorem may still be true, but as written the uniqueness result is not established.
  2. [§3, Theorem 1] The proof of Theorem 1 consists of the assertion that (10) "can be obtained from [19]" after noting (9). The paper does not verify the hypotheses of [19] for (10): one needs that the transformed driver G(u,~X(u),y,\hat z) is Lipschitz in z, that g(~X(T)) is square integrable, that L(u) < U(u) for all u almost surely, and that β is a Brownian motion with respect to the filtration used in [19]. The identity (9) itself is quoted from [7, Section 2.3] without proof. Since all later results, including the flow property Y^{t,x}(s) = u(s,~X^{t,x}(s)) used in Theorem 5, rest on Theorem 1, these verifications should be spelled out.
  3. [§4.1, Theorem 5, Case 2] In the boundary case x ∈ F_k, the proof assumes there is ε > 0 with inf_{|z-x|≤α, t≤r≤t+α} (∂u/∂x_{k+1} - ∂u/∂x_k)(r,z) ≥ ε. However, u is only known to be continuous (Proposition 2); it need not be differentiable on F_k. The later local-time computation (26) uses derivatives of the test function φ, not of u, so the contradiction hypothesis should be phrased in terms of φ. As written this part of the proof of the subsolution property at F_k is not justified; it must be reworked with a rigorous viscosity formulation, for instance by considering the superjet at the boundary rather than pointwise derivatives of u.
minor comments (5)
  1. [§3, Proposition 2] The semicontinuity assertions are reversed: a decreasing sequence of continuous functions converges to an upper semicontinuous limit, and an increasing sequence converges to a lower semicontinuous limit. Since both directions are proved, the conclusion u ∈ C([0,T] × Γ_n) is unaffected, but the text should be corrected.
  2. [§2, Assumption (A2)] Assumption (A2) introduces a function h : [0,T] × Γ_n → R, but h is never used anywhere in the paper. Either remove h or state its role.
  3. [§4.2, Lemma 8 and Theorem 6] The proof of Theorem 6 applies Lemma 8 successively on intervals [t_i, t_{i-1}] with t_{i+1} = (t_i - A/C)_+. On the later intervals the terminal data are only known to satisfy w(t_i, ·) ≤ 0, not w(t_i, ·) = 0. The argument should state explicitly that the comparison is valid with terminal data bounded above by zero, which the proof of N(ρ,T) already implicitly allows.
  4. [§5, Theorem 13] The notation ~Y(t) is used for both the solution process in (41) and the initial value of the discounted game value; please clarify the distinction between the process and its time-t value.
  5. [Throughout] There are numerous typographical errors, including "Funcition" in Definition 4, "Futhermore" in Theorem 9, and inconsistent use of the spaces Γ^n and Γ^{n,+}. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the BSDE-defined u is proven, not assumed, to solve the obstacle PDE; the main correctness risk is an unproved boundary condition in the uniqueness proof, not a circular reduction.

full rationale

The central claim is not circular. The function u(t,x)=Y^{t,x}(t) is defined directly from the two-barriers-reflected BSDE (8), and Theorem 5 proves that this u is a viscosity solution of the obstacle PDE (14) via the flow property Y^{t,x}(s)=u(s,\tilde X^{t,x}(s)) (from the uniqueness in Theorem 1), the Itô expansion of the rank-based dynamics (1), and the Skorokhod conditions on K^{t,x,+} and K^{t,x,-}; no parameter is fitted and later renamed as a prediction, and the PDE is not used as an input to construct the BSDE. The rewriting of the stochastic integral in terms of the ranked Brownian motions (9)-(10), the independence of the β_l, and the strong uniqueness of the rank-based SDE are imported from external published results [2,7,19,22,35]; among these, [6,7] are prior papers of Feng, but they are independent published technical lemmas and are not unverified assertions manufactured for this paper, so the self-citations do not carry the load in a circular way. The genuine weakness is non-circular: Lemma 7 states that w=u-v is a viscosity subsolution of the auxiliary problem (30) including the boundary condition sum_{l=1}^j(∂w/∂x_{k_l}-∂w/∂x_{k_l+1})=0 on each O_j, but its proof establishes only the interior inequality (31) and never verifies the O_j condition; the proof of Theorem 6 then invokes that boundary condition when the maximum point lies in O_j (around (34)). This is an omitted-proof/completeness gap in the uniqueness argument, not a case of a conclusion being assumed as an input. Theorem 13's American game-option pricing result is a standard hedging argument built on the BSDE solution, again with no fitted input. The score is set to 2 rather than 0 only to register the presence of self-citations among the external technical lemmas; it is not meant to indicate that a derivation reduces to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities and fits no parameters. The model coefficients σ_i, δ_i, r_0 are inputs. The central claim depends on standard external theorems for rank-based SDEs, two-barrier BSDE well-posedness, viscosity solution comparison, and Dynkin games. Self-citation appears via [6] and [7], sharing first author Xinwei Feng, but those are published results, not results constructed for this paper.

assumptions (5)
  • domain assumption Rank-based SDE (1) has a unique strong solution, and the associated processes β_l in (3) are independent Brownian motions.
    Invoked in Section 2 to write ranked dynamics and in Theorem 1 to reduce the BSDE to a standard driven-by-Brownian form; cited from [2], [22, Theorem 2] and [35, Theorem 1.4].
  • standard math The two-barrier reflected BSDE (10) has a unique solution under assumptions (A1)-(A3) by Hamadène-Hassani [19].
    Theorem 1 proof is solely a reduction to [19]; the hypotheses such as L<U, L_T≤g≤U_T, and Lipschitz generator are stated but the verification for the transformed equation is not carried out.
  • standard math The penalization approximation and continuity result for one-barrier reflected BSDEs with rank-based data [7, Theorem 3.1] hold; Lemma 8 in the paper is taken from [6].
    Proposition 2 uses [7, Theorem 3.1]; Theorem 6 uses Lemma 8 proved in [6]. Both are external published results, one from an earlier paper by the same first author.
  • standard math Dynkin game value representation of Cvitanic-Karatzas [9, Theorem 4.1] applies to the rank-based game (36).
    Theorem 9 is obtained directly from [9, Theorem 4.1] after identifying the reflected BSDE solution as the game value.
  • standard math Standard viscosity solution comparison machinery (Crandall-Ishii-Lions [8]), Girsanov theorem, and Itô-Tanaka formula for ranked semimartingales are applicable.
    Used throughout Sections 4 and 5 without proof: Ishii's lemma in Lemma 7, Girsanov change of measure in Theorem 5, and local time terms in the Itô formula for ranked processes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Two-barriers-reflected BSDE with Rank-based Data." pith.science (2026). https://pith.science/paper/R757ICEN

@misc{pith2026241117104,
  author       = {Pith},
  title        = {Pith review of: Two-barriers-reflected BSDE with Rank-based Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R757ICEN}},
  note         = {Machine review of arXiv:2411.17104}
}
read the original abstract

We investigate two-barriers-reflected backward stochastic differential equations with data from rank-based stochastic differential equation. More specifically, we focus on the solution of backward stochastic differential equations restricted to two prescribed upper-boundary and lower-boundary processes. We rigorously show that this solution gives a probabilistic expression to the viscosity solution of some obstacle problems for the corresponding parabolic partial differential equations. As an application, the pricing problem of an American game option is studied.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [19]

    BSDEs with two reflecting barri ers: the general result

    Hamad` ene S., Hassani M. BSDEs with two reflecting barri ers: the general result. Probab. Theory Relat. Fields, 2005 , 132: 237-264

    work page 2005

  2. [1]

    D., Fernholz R., Karatzas I

    Banner A. D., Fernholz R., Karatzas I. Atlas models of equ ity markets. Ann. Appl. Probab., 2005, 15: 2296-2330

    work page 2005

  3. [2]

    D., Ghomrasni R

    Banner A. D., Ghomrasni R. Local times of ranked continuo us semimartingales. Stoch. Process Appl., 2008, 118: 1244- 1253

    work page 2008

  4. [3]

    Bismut J. M. Conjugate convex functions in optimal stoch astic control. J. Math Anal. Appl., 1973, 44: 384-404

    work page 1973

  5. [4]

    A phase transition behavior for Brow nian motions interacting through their ranks

    Chatterjee S, Pal S. A phase transition behavior for Brow nian motions interacting through their ranks. Probab. Theo ry Relat. Fields, 2010, 147: 123-159

    work page 2010

  6. [5]

    Chen Z., Epstein L. G. Ambiguity, risk, and asset returns in continuous time. Econometrica, 2002, 70: 1403-1443

    work page 2002

  7. [6]

    Backward stochastic differential equ ations with rank-based data

    Chen Z.-Q., Feng X. Backward stochastic differential equ ations with rank-based data. Sci. China Math., 2018, 61: 27- 56

    work page 2018

  8. [7]

    Reflected backward stochastic differe ntial equation with rank-based data

    Chen Z.-Q., Feng X. Reflected backward stochastic differe ntial equation with rank-based data. J. Theor. Probab., 202 1, 34: 1213-1247

    work page

Show all 37 references
  1. [8]

    G., Ishii H., Lions P

    Crandall M. G., Ishii H., Lions P. L. User¡ /macron.ts1s guide to viscosity solutions of second order partial differ ential equations. Bull. Amer. Math. Soc., 1992, 27: 1-67

    1992

  2. [9]

    Backward stochastic different ial equations with reflection and Dynkin games

    Cvitani´ c J., Karatzas I. Backward stochastic different ial equations with reflection and Dynkin games. Ann. Probab. , 1996, 24: 2024-2056

    1996

  3. [10]

    Lp-Solutions for Doubly Reflected Backward Stochastic Differe ntial Equations

    El Asri B., Hamad` ene S., W ang H. Lp-Solutions for Doubly Reflected Backward Stochastic Differe ntial Equations. Stoch. Anal. Appl., 2011, 29: 907-932

    2011

  4. [11]

    El Karoui N., Kapoudjian C., Pardoux E., Peng S., Quenez M. C. Reflected solutions of backward SDE’s and related obsta cle problems for PDE’s. Ann. Probab., 1997, 25: 702-737

    1997

  5. [12]

    El Karoui N., Peng S., Quenez M. C. Backward stochastic d ifferential equations in finance. Math. Financ., 2014, 1: 1-7 1

    2014

  6. [13]

    El Karoui N., Quenez M. C. Non-linear pricing theory and backward stochastic differential equations. Financial mat hematics, 1997, 1656: 191-246

    1997

  7. [14]

    H., Hassani M

    Essaky E. H., Hassani M. BSDE approach for dynkin game an d American game option. Stat. Method Appl., 2013, 158: 211-2 25

    2013

  8. [15]

    L1 solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

    Fan S. L1 solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions. Ele ctron. J. Probab., 2019, 24: 1-48

    2019

  9. [16]

    R., Ichiba T., Karatzas I

    Fernholz E. R., Ichiba T., Karatzas I. Planar diffusions with rank-based characteristics and perturbed Tanaka equa tions. Probab. Theory Relat. Fields, 2013, 156: 343-374

    2013

  10. [17]

    R., Karatzas I

    Fernholz E. R., Karatzas I. Stochastic portfolio theor y: an overview. Handbook of numerical analysis, 2009, 15: 89 -167

    2009

  11. [18]

    Mixed zero-sum stochastic differental gam e and American game options

    Hamad` ene S. Mixed zero-sum stochastic differental gam e and American game options. Siam J. Control Optim., 2007, 45 : 496-518

    2007

  12. [20]

    A note on the reflected backward stochastic d ifferential equations driven by a L /dieresis.ts1¦vy process with stochastic Lipschitz condition

    Hu L., Ren Y. A note on the reflected backward stochastic d ifferential equations driven by a L /dieresis.ts1¦vy process with stochastic Lipschitz condition. Appl. Math . Comput., 2011, 218: 4325-4332

    2011

  13. [21]

    Anticipated backward stochastic diff erential equations with quadratic growth

    Hu Y., Li X., W en J. Anticipated backward stochastic diff erential equations with quadratic growth. J. Differential E quations, 2021, 270: 1298-1331

    2021

  14. [22]

    Strong solutions of stochastic equations with rank-based coefficients

    Ichiba T., Karatzas I., Shkolnikov M. Strong solutions of stochastic equations with rank-based coefficients. Proba b. Theory Relat. Fields, 2013, 156: 229-248. 12

    2013

  15. [23]

    Convergence rates for ran k-based models with applications to portfolio theory

    Ichiba T, Pal S, Shkolnikov M. Convergence rates for ran k-based models with applications to portfolio theory. Prob ab. Theory Relat. Fields, 2013, 156: 415-448

    2013

  16. [24]

    Hybrid atlas models

    Ichiba T., Papathanakos V., Banner A., Karatzas I., Fer nholz R. Hybrid atlas models. Ann. Appl. Probab., 2011, 21: 6 09-644

    2011

  17. [25]

    Capital distribution and portfo lio performance in the mean-field Atlas model

    Jourdain B, Reygner J. Capital distribution and portfo lio performance in the mean-field Atlas model. Ann. Finance, 2015, 11: 151-198

    2015

  18. [26]

    Game options

    Kifer Y. Game options. Finance Stoch., 2000, 4: 443-463

    2000

  19. [27]

    Li M., Shi Y.Solving the double barrier reflected BSDEs v ia penalization method.Statist. Probab. Lett., 2016, 110: 74-83

    2016

  20. [28]

    Reflected BSDEs with time-delayed generators and nonlinear resistance, Statist

    Luo P. Reflected BSDEs with time-delayed generators and nonlinear resistance, Statist. Probab. Lett., 2020, 163: 1 08765

    2020

  21. [29]

    Reflected forward-backward SDE and o bstacle problems with boundary conditions

    Ma J., Cvitani´ c J. Reflected forward-backward SDE and o bstacle problems with boundary conditions. J. Appl. Math. S toch. Anal., 2001, 14: 113-138

    2001

  22. [30]

    Adapted solution of a backward stoch astic differential equation

    Pardoux E., Peng S. Adapted solution of a backward stoch astic differential equation. Syst. Control Lett., 1990, 14: 55-61

    1990

  23. [31]

    Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order

    Pardoux E. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. Stoch. Anal. Relat. Topics VI., 1998, 42: 79-127

    1998

  24. [32]

    Backward stochastic dfferential equ ations and quasilinearparabolic partial dfferential equat ions

    Pardoux E., Peng S. Backward stochastic dfferential equ ations and quasilinearparabolic partial dfferential equat ions. Stochastic Partial Differential Equa- tionsand Their Applications, 1992, 176: 200-217

    1992

  25. [33]

    BSDEs, weak convergence and homogenization of semilinear PDEs

    Pardoux E. BSDEs, weak convergence and homogenization of semilinear PDEs. Nonlinear Analysis, Dfferential Equati ons and Control, Montreal, 1999: 503-549

    1999

  26. [34]

    Penalization method for reflected ba ckward stochastic differential equations with one r.c.l.l

    Lepeltier J., Xu M. Penalization method for reflected ba ckward stochastic differential equations with one r.c.l.l. barrier. Statist. Probab. Lett., 2005, 75: 58-66

    2005

  27. [35]

    Triple and simultaneous collisions of com peting Brownian particles

    Sarantsev A. Triple and simultaneous collisions of com peting Brownian particles. Electron. J. Probab., 2015, 20: 1-28

    2015

  28. [36]

    Competing particle systems evolving by i nteracting L´ evy processes

    Shkolnikov M. Competing particle systems evolving by i nteracting L´ evy processes. Ann. Appl. Probab., 2011, 21: 1 911-1932

    2011

  29. [37]

    and Zhou X

    Yong J. and Zhou X. , Stochastic Controls: Hamiltonian S ystems and HJB Equations, Springer, New York, 1999. 13

    1999

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.