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Register a new userA Complex Version of G-Expectation and its Application to Conformal Martingale
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Huilin Zhang
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This paper is concerned with the connection between G-Brownian Motion and analytic functions. We introduce the complex version of sublinear expectation, and then do the stochastic analysis in this framework. Furthermore, the conformal G-Brownian Motion is introduced together with a representation, and the corresponding conformal invariance is shown.
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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.
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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