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Register a new userSupermartingale Decomposition Theorem under G-expectation
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The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales have the decomposition similar to the classical case. The main ideas are to apply the uniformly continuous property of $S_G^beta(0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.
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We develop a theory of optimal stopping problems under G-expectation framework. We first define a new kind of random times, called G-stopping times, which is suitable for this problem. For the discrete time case with finite horizon, the value function is defined backwardly and we show that it is the smallest G-supermartingale dominating the payoff process and the optimal stopping time exists. Then we extend this result both to the infinite horizon and to the continuous time case. We also establish the relation between the value function and solution of reflected BSDE driven by G-Brownian motion.
A real valued function defined on}$mathbb{R}$ {small is called}$g${small --convex if it satisfies the following textquotedblleft generalized Jensens inequalitytextquotedblright under a given}$g${small -expectation, i.e., }$h(mathbb{E}^{g}[X])leq mathbb{E}% ^{g}[h(X)]${small, for all random variables}$X$ {small such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient conditions for a }$C^{2}${small -function being}$% g ${small -convex. We also studied some more general situations. We also studied}$g${small -concave and}$g${small -affine functions.
Let $mathbb{hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $mathcal{P}$-semimartingale under $mathbb{hat{E}}$. Given an open set $Qsubsetmathbb{R}^{d}$, the exit time of $Y$ from $Q$ is defined by [ {tau}_{Q}:=inf{tgeq0:Y_{t}in Q^{c}}. ] The main objective of this paper is to study the quasi-continuity properties of ${tau}_{Q}$ under the nonlinear expectation $mathbb{hat{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that ${tau}_{Q}wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.
In this paper, we study the optimal multiple stopping problem under the filtration consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions rather than an RCLL process. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.
In this paper, we propose a monotone approximation scheme for a class of fully nonlinear partial integro-differential equations (PIDEs) which characterize the nonlinear $alpha$-stable L{e}vy processes under sublinear expectation space with $alpha in(1,2)$. Two main results are obtained: (i) the error bounds for the monotone approximation scheme of nonlinear PIDEs, and (ii) the convergence rates of a generalized central limit theorem of Bayraktar-Munk for $alpha$-stable random variables under sublinear expectation. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.
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