Optimal stopping with random maturity under nonlinear expectations

Abstract

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.

Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

“…To overcome this difficulty, we shall modify the hitting time, inspired by [1]. While we will study the new hitting time in details in next section, we present a special case here to see how it helps overcome the difficulty in above example.…”

“…As explained in Subsection 2.4 above, we shall introduce a new type of hitting times, strongly motivated by the recent work [1].…”

“…shares these properties in uniformly non-degenerate case, but does not in degenerate case. The work [1] introduced a different hitting time which has stronger regulairty:…”

“…However, in the degenerate case, the h n used in [9,10] has very bad regularity, and consequently the PDE (1.3) in Q n typically does not have a continuous viscosity solution. Strongly motivated by the recent work Bayraktar & Yao [1], we shall use some slightly modified stopping times h n which enjoy all the desired properties.…”

“…In Section 2 we review the basic materials concerning PPDEs. In Section 3 we introduce pseudo Markovian viscosity solutions, and in particular the new hitting times inspired by [1]. The comparison principle is proved in Section 4, and Section 5 is devoted to existence.…”