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Convex Analysisの二,三の進展について

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

Backward Stochastic Differential Equations in Finance

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations

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.

The Pricing Mechanism of Contingent Claims and its Generating Function

The Numerical Algorithms and simulations for BSDEs

In this paper, we build the equivalence between rough differential equations driven by the lifted $G$-Brownian motion and the corresponding Stratonovich type SDE through the Wong-Zakai approximation. The quasi-surely convergence rate of Wong-Zakai approximation to $G-$SDEs with mesh-size $\frac{1}{n}$ in the $\alpha$-Holder norm is estimated as $(\frac{1}{n})^{\frac12-}.$ As corollary, we obtain the quasi-surely continuity of the above RDEs with respect to uniform norm.

Wong-Zakai Approximation for SDEs Driven by $G-$Brownian Motion.

The G-normal distribution was introduced by Peng [2007] as the limiting distribution in the central limit theorem for sublinear expectation spaces. Equivalently, it can be interpreted as the solution to a stochastic control problem where we have a sequence of random variables, whose variances can be chosen based on all past information. In this note we study the tail behavior of the G-normal distribution through analyzing a nonlinear heat equation. Asymptotic results are provided so that the tail "probabilities" can be easily evaluated with high accuracy. This study also has a significant impact on the hypothesis testing theory for heteroscedastic data; we show that even if the data are generated under the null hypothesis, it is possible to cheat and attain statistical significance by sequentially manipulating the error variances of the observations.

Shige Peng

Papers

Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations

The Pricing Mechanism of Contingent Claims and its Generating Function

The Numerical Algorithms and simulations for BSDEs

Wong-Zakai Approximation for SDEs Driven by $G-$Brownian Motion.

A hypothesis-testing perspective on the G-normal distribution theory.