Optimal Stopping Under Ambiguity in Continuous Time

We develop a theory of optimal stopping problems under ambiguity in continuous time. Using results from (backward) stochastic calculus, we characterize the value function as the smallest (nonlinear) supermartingale dominating the payoff process. For Markovian models, we derive an adjusted Hamilton-Jacobi-Bellman equation involving a nonlinear drift term that stems from the agent's ambiguity aversion. We show how to use these general results for search problems and American Options.Optimal Stopping, Ambiguity, Uncertainty Aversion, Robustness, Continuous-Time, Optimal Control

An optimal control problem of forward-backward stochastic Volterra integral equations with state constraints

This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequality. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.Comment: 19 page

Indifference Pricing and Hedging in a Multiple-Priors Model with Trading Constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.Comment: 28 pages in Science China Mathematics, 201