Optimal Stopping of McKean--Vlasov Diffusions via Regression on Particle Systems

Belomestny, Denis; Schoenmakers, John

doi:10.1137/18m1195590

Cited by 7 publications

(10 citation statements)

References 24 publications

(36 reference statements)

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“…In fact, the latter appearance is observed in all error bounds concerning regression‐based backward dynamic programs for optimal stopping in the literature (e.g., Egloff et al., 2007, Zanger, 2013). It also appears in a later result by Zanger (2018), based on dependent samples, and in the convergence analysis by Belomestny and Schoenmakers (2018) in the context of optimal stopping of McKean–Vlasov processes. This factor seems to be unavoidable because at each backward step the projection error of the estimated continuation function needs to be bounded in relation to the projection error of the true continuation function.…”

Section: Introductionmentioning

confidence: 76%

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

Belomestny

Kaledin

Schoenmakers

2020

Mathematical Finance

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In this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete‐ and continuous‐time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete‐time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by ε−4logd+2false(1/εfalse) with d being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous‐time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example.

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Section: Introductionmentioning

confidence: 76%

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

Belomestny

Kaledin

Schoenmakers

2020

Mathematical Finance

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“…The procedure is illustrated in Figure 1. Note that the costs of this algorithm are of the order M • J • K 2 (see, e.g., [BS20] or Section 5).…”

Section: Reinforced Regression For Optimal Stoppingmentioning

confidence: 99%

“…As a more systematic approach, the authors of [BS20] proposed a reinforced regression algorithm. In this procedure the regression basis at each step of the backward induction is reinforced with the approximate value function from the previous step of the induction.…”

Section: Reinforced Regression For Optimal Stoppingmentioning

confidence: 99%

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Reinforced optimal control

Bayer¹,

Belomestny²,

Hager³

et al. 2020

Preprint

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Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay. Commun. Math. Sci., 18(1):109-121, 2020] proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method's efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

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“…We remark that in the last mean field game, the dynamics of X µ• does not depend on the stopping time τ , so it has a completely different structure than our optimal stopping problem. We would also like to mention Li [18], Briand, Elie & Hu [5], and Djehiche, Elie & Hamadene [12] for closely related works on mean field type reflected BSDEs, and Belomestny & Schoenmakers [1] for a numerical method for mean field type optimal stopping problems. However, in all these works again the dynamics of the state process does not depend on the stopping time τ .…”

Section: Introductionmentioning

confidence: 99%

Dynamic programming equation for the mean field optimal stopping problem

Talbi,

Touzi,

Zhang

2021

Preprint

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We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process.This specification satisfies a dynamic programming principle. The corresponding dynamic programming equation is an obstacle problem on the Wasserstein space, and is obtained by means of a general Itô formula for flows of marginal laws of càdlàg semimartingales. Our verification result characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. The effectiveness of our dynamic programming equation is illustrated by various examples including the mean-variance and expected shortfall criteria.

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Optimal Stopping of McKean--Vlasov Diffusions via Regression on Particle Systems

Cited by 7 publications

References 24 publications

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

Reinforced optimal control

Dynamic programming equation for the mean field optimal stopping problem

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