Optimal stopping under nonlinear expectation

Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng

doi:10.1016/j.spa.2014.04.006

Cited by 71 publications

(109 citation statements)

References 18 publications

Supporting

Mentioning

106

Contrasting

Order By: Relevance

“…We next recall the Snell envelop characterization of the optimal stopping problem under nonlinear expectation, see Theorem 3.5 in Ekren, Touzi & Zhang [9].…”

Section: Preliminariesmentioning

confidence: 99%

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

Ren¹,

Touzi²,

Zhang³

2017

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang [11], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Souganidis [12] in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof in [11], but requires an L p −type of continuity (with respect to the path) for the viscosity semi-solutions and for the nonlinearity defining the equation. IntroductionThis paper provides a proof for the comparison result for viscosity solutions of the fully nonlinear path dependent partial differential equation:Here, T > 0 is a given terminal time, and ω ∈ Ω is a continuous path from [0, T ] to R d starting from the origin. The nonlinearity G is a mapping fromSuch equations arise naturally in many applications. For instance, the dynamic programming equation (also called Hamilton-Jacobi-Bellman equation) associated to a problem of stochastic control of non-Markov diffusions falls in the class of equations (1.1), see [10]. In particular hereditary control problems may be addressed in this context rather than embedding the problem into a PDE * CMAP, Ecole Polytechnique Paris, ren@cmap.polytechnique.fr. † CMAP, Ecole Polytechnique Paris, nizar.touzi@polytechnique.edu.

show abstract

“…We next recall the Snell envelop characterization of the optimal stopping problem under nonlinear expectation, see Theorem 3.5 in Ekren, Touzi & Zhang [9].…”

Section: Preliminariesmentioning

confidence: 99%

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

Ren¹,

Touzi²,

Zhang³

2017

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following lemma shows that the function u satisfies a dynamic programming principle (see, for example, Lemma 4.1 of [7] for a proof).…”

Section: Example Of a Duality Results For An American Optionmentioning

confidence: 99%

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

Henry-Labordère¹,

Litterer²,

Ren

2016

SIAM J. Finan. Math.

View full text Add to dashboard Cite

show abstract

“…Up to a subsequence, {P n } n∈N has a limit P in the weakly compact probability set P. Then as n → ∞ in (1.4), we can deduce Z 0 = E P [Z ν ] and thus (1.3) by leveraging the continuity estimates (4.2), (4.5) of Z as well as a similar argument to the one used in the proof of [19,Theorem 3.3] that replaces ν n 's with a sequence of quasi-continuous random variables decreasing sequence to ν.…”

Section: Introductionmentioning

confidence: 82%

Optimal stopping with random maturity under nonlinear expectations

Bayraktar

Yao

2017

Stochastic Processes and their Applications

View full text Add to dashboard Cite

We analyze an optimal stopping problem sup γ∈T E 0 Yγ∧τ 0 with random maturity τ0 under a nonlinear expec-where P is a weakly compact set of mutually singular probabilities. The maturity τ0 is specified as the hitting time to level 0 of some continuous index process X at which the payoff process Y is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.We utilize a martingale approach to construct an optimal pair (P * , γ * ) for sup (P,γ)∈P×T E P Yγ∧τ 0 , in which γ * is the first time Y meets the limit Z of its approximating E −Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process Y , we approximate τ0 by an increasing sequence of Lipschitz continuous stopping times and approximate Y by a sequence of uniformly continuous processes.This result together with its premises, Lemma A.5 and Lemma A.6, are among the main contributions of this paper. Given n, k ∈ N, connecting L and U near ℘ n with lines of slope 2 k yields a uniformly continuous process Y n,k t := L t + 1∧(2 k (t−℘ n )−1) + (U t −L t ), t ∈ [0, T ], see Lemma 5.1. Then one can apply Theorem 4.1 to Y n,k and 1 The authors would like to thank Jianfeng Zhang for an instructive discussion.

show abstract

Optimal stopping under nonlinear expectation

Cited by 71 publications

References 18 publications

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

A Dual Algorithm for Stochastic Control Problems: Applications to Uncertain Volatility Models and CVA

Optimal stopping with random maturity under nonlinear expectations

Contact Info

Product

Resources

About