We analyze an optimal stopping problem sup(gamma is an element of T) (xi) over bar 0[y(gamma Lambda tau 0)] with random maturity to under a nonlinear expectation (xi) over bar0[.] := sup(P is an element of P) Eg[.], w...
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We analyze an optimal stopping problem sup(gamma is an element of T) (xi) over bar 0[y(gamma Lambda tau 0)] with random maturity to under a nonlinear expectation (xi) over bar0[.] := sup(P is an element of P) Eg[.], where P is a weakly compact set of mutually singular probabilities. The maturity tau(0) is specified as the hitting time to level 0 of some continuous index process X at which the payoff process g 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-*, y(*)) for sup((P, gamma)is an element of P X T) Ep[y(gamma Lambda tau 0)], in which y(*) is the first time y meets the limit. L of its approximating (xi) over bar -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 tau(0) by an increasing sequence of Lipschitz continuous stopping times and approximate y by a sequence of uniformly continuous processes. (C) 2016 Elsevier B.V. All rights reserved.
We analyze a robust version of the Dynkin game over a set P of mutually singular probabilities. We first prove that conservative player's lower and upper value coincide (let us denote the value by V). Such a resul...
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We analyze a robust version of the Dynkin game over a set P of mutually singular probabilities. We first prove that conservative player's lower and upper value coincide (let us denote the value by V). Such a result connects' the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process V is a submartingale under an appropriately defined nonlinear expectation E up to the first time tau* when V meets the lower payoff process L. If the probability set P is weakly compact, one can even find an optimal triplet (P*, tau*, gamma*,) for the value V-0. The mutual singularity of probabilities in P causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.
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