Semimartingale properties of the lower Snell envelope in optimal stopping under model uncertainty

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

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Addendum to “Optimal stopping under model uncertainty: Randomized stopping times approach”

The Annals of Applied Probability ◽

10.1214/16-aap1226 ◽

2017 ◽

Vol 27 (2) ◽

pp. 1289-1293 ◽

Cited By ~ 1

Author(s):

Denis Belomestny ◽

Volker Krätschmer

Keyword(s):

Optimal Stopping ◽

Model Uncertainty ◽

Stopping Times

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Optimal Stopping Theory and American Options

Arbitrage Theory in Continuous Time ◽

10.1093/oso/9780198851615.003.0028 ◽

2019 ◽

pp. 376-396

Author(s):

Tomas Björk

Keyword(s):

Dynamic Programming ◽

Optimal Stopping ◽

Continuous Time ◽

American Options ◽

Programming Approach ◽

Free Boundary Value Problem ◽

Snell Envelope ◽

Dynamic Programming Approach ◽

Optimal Stopping Theory ◽

Time Theory

In this chapter we present the dynamic programming approach to optimal stopping problems. We start by presenting the discrete time theory, deriving the relevant Bellman equation. We present the Snell envelope and prove the Snell Envelope Theorem. For Markovian models we explore the connection to alpha-excessive functions. The continuous time theory is presented by deriving the free boundary value problem connected to the stopping problem, and we also derive the associated system of variational inequalities. American options are discussed in some detail.

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𝕃p solutions of reflected backward stochastic differential equations with jumps

ESAIM Probability and Statistics ◽

10.1051/ps/2020026 ◽

2020 ◽

Vol 24 ◽

pp. 935-962

Author(s):

Song Yao

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Optimal Stopping ◽

Backward Stochastic Differential Equation ◽

Stopping Time ◽

Snell Envelope ◽

Lipschitz Continuous ◽

Optimal Stopping Theory ◽

First Time ◽

Point Argument

Given p ∈ (1, 2), we study 𝕃p-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator g is Lipschitz continuous in (y, z, u). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with p-integrable parameters admits a unique 𝕃p solution via a fixed-point argument. The Y -component of the unique 𝕃p solution can be viewed as the Snell envelope of the reflecting obstacle 𝔏 under g-evaluations, and the first time Y meets 𝔏 is an optimal stopping time for maximizing the g-evaluation of reward 𝔏.

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Approximations of the optimal stopping problem in partial observation

Journal of Applied Probability ◽

10.1017/s002190020002965x ◽

1986 ◽

Vol 23 (02) ◽

pp. 341-354 ◽

Cited By ~ 2

Author(s):

G. Mazziotto

Keyword(s):

Optimal Stopping ◽

Measure Space ◽

Stopping Time ◽

Optimal Stopping Problem ◽

Snell Envelope ◽

Infinite Dimensional ◽

Markov State ◽

State Process ◽

Partially Observed ◽

True State

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

Download Full-text