Showing papers by "Shige Peng published in 2012"
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TL;DR: In this paper, the authors studied comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of the BSDE driven by a G-Brownian motion.
Abstract: In this paper, we study comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of the BSDE driven by a G-Brownian motion.
111 citations
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TL;DR: In this paper, the authors established a complete representation theorem for $G$-martingales and provided the existence and uniqueness of the second order term, which corresponds to the second-order derivative in Markovian case.
Abstract: In this paper we establish a complete representation theorem for $G$-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second order term, which corresponds to the second order derivative in Markovian case. The main ingredient of the paper is a new norm for that second order term, which is based on an operator introduced by Song [26].
54 citations
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TL;DR: In this article, the authors studied backward stochastic differential equations driven by a G-Brownian motion and proved the existence and uniqueness of the solution (Y,Z,K) under a Lipschitz condition.
Abstract: In this paper, we study backward stochastic differential equations driven by a G-Brownian motion. The solution of such new type of BSDE is a triple (Y,Z,K) where K is a decreasing G-martingale. Under a Lipschitz condition for generator f and g in Y and Z. The existence and uniqueness of the solution (Y,Z,K) is proved. Although the methods used in the proof and the related estimates are quite different from the classical proof for BSDEs, stochastic calculus in G-framework plays a central role.
19 citations
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TL;DR: In this article, the authors formulate and solve the martingale problem in a nonlinear expectation space and obtain an existence theorem under the Holder continuity condition of the coefficients, which generalizes the classical Girsanov transformation method.
Abstract: We formulate and solve the martingale problem in a nonlinear expectation space. Unlike the classical work of Stroock and Varadhan (1969) where the linear operator in the associated PDE is naturally defined from the corresponding diffusion process, the main difficulty in the nonlinear setting is to identify an appropriate class of nonlinear operators for the associated fully nonlinear PDEs.
Based on the analysis of the martingale problem, we introduce the notion of weak solution for stochastic differential equations under nonlinear expectations and obtain an existence theorem under the Holder continuity condition of the coefficients. The approach to establish the existence of weak solutions generalizes the classical Girsanov transformation method in that it no longer requires the two (probability) measures to be absolutely continuous.
10 citations
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TL;DR: In this paper, a dynamic pricing mechanism of contingent claims is studied and the main result is that if a given pricing mechanism is $E^{g_\mu}$-dominated, i.e.
Abstract: In this paper we study dynamic pricing mechanism of contingent claims. A typical model of such pricing mechanism is the so-called g-expectation $E^g_{s,t}[X]$ defined by the solution of the backward stochastic differential equation with generator g and with the contingent claim X as terminal condition. The generating function g this BSDE. We also provide examples of determining the price generating function $g=g(y,z)$ by testing.
The main result of this paper is as follows: if a given dynamic pricing mechanism is $E^{g_\mu}$-dominated, i.e., the criteria (A5) $E_{s,t}[X]-E_{s,t}[X']\leq E^{g_\mu}_{s,t}[X-X']$ is satisfied for a large enough $\mu> 0$, where $g_\mu=g_{\mu}(|y|+|z|)$, then $E_{s,t}$ is a g-pricing mechanism. This domination condition was statistically tested using CME data documents. The result of test is significantly positive.
4 citations
Posted Content•
TL;DR: In this paper, a dynamic pricing mechanism of contingent claims is studied and the main result is that if a given pricing mechanism is $E^{g_\mu}$-dominated, i.e.
Abstract: In this paper we study dynamic pricing mechanism of contingent claims. A typical model of such pricing mechanism is the so-called g-expectation $E^g_{s,t}[X]$ defined by the solution of the backward stochastic differential equation with generator g and with the contingent claim X as terminal condition. The generating function g this BSDE. We also provide examples of determining the price generating function $g=g(y,z)$ by testing. The main result of this paper is as follows: if a given dynamic pricing mechanism is $E^{g_\mu}$-dominated, i.e., the criteria (A5) $E_{s,t}[X]-E_{s,t}[X']\leq E^{g_\mu}_{s,t}[X-X']$ is satisfied for a large enough $\mu> 0$, where $g_\mu=g_{\mu}(|y|+|z|)$, then $E_{s,t}$ is a g-pricing mechanism. This domination condition was statistically tested using CME data documents. The result of test is significantly positive.
4 citations
Posted Content•
TL;DR: In this article, the authors consider a class of discretionary stopping problems within the $G$-framework and establish the well-definedness of the stopping problem under the G$-expectation, by showing the quasi-continuity of the stopped process.
Abstract: We consider a class of discretionary stopping problems within the $G$-framework. We first establish the well-definedness of the stopping problem under the $G$-expectation, by showing the quasi-continuity of the stopped process. We then prove a verification theorem for $G$-optimal stopping problem. One corollary is a direct proof for the well-known fact that the $G$-optimal stopping problem is the same as the classical optimal stopping problem with appropriate parameters, when the payoff function is concave or convex.
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13 Nov 2012
TL;DR: In this paper, the authors formulate and solve the martingale problem in a nonlinear expectation space and obtain an existence theorem under the Holder continuity condition of the coefficients, which generalizes the Girsanov transformation method in that it no longer requires the two probability measures to be absolutely continuous.
Abstract: We formulate and solve the martingale problem in a nonlinear expectation space. Unlike the classical work of Stroock and Varadhan (1969) where the linear operator in the associated PDE is naturally defined from the corresponding diffusion process, the main difficulty in the nonlinear setting is to identify an appropriate class of nonlinear operators for the associated fully nonlinear PDEs. Based on the analysis of the martingale problem, we introduce the notion of weak solution for stochastic differential equations under nonlinear expectations and obtain an existence theorem under the Holder continuity condition of the coefficients. The approach to establish the existence of weak solutions generalizes the classical Girsanov transformation method in that it no longer requires the two (probability) measures to be absolutely continuous.