다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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

This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $\alpha \in(1,2)$, the i.i.d. sequence \[ \left \{ \left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i},\frac{1}{n}\sum _{i=1}^{n}Y_{i},\frac{1}{\sqrt[\alpha]{n}}\sum_{i=1}^{n}Z_{i}\right) \right \} _{n=1}^{\infty} \] converges in distribution to $\tilde{L}_{1}$, where $\tilde{L}_{t}=(\tilde {\xi}_{t},\tilde{\eta}_{t},\tilde{\zeta}_{t})$, $t\in [0,1]$, is a multidimensional nonlinear L\'{e}vy process with an uncertainty set $\Theta$ as a set of L\'{e}vy triplets. This nonlinear L\'{e}vy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) \[ \left \{ \begin{array} [c]{l} \displaystyle \partial_{t}u(t,x,y,z)-\sup \limits_{(F_{\mu},q,Q)\in \Theta }\left \{ \int_{\mathbb{R}^{d}}\delta_{\lambda}u(t,x,y,z)F_{\mu}(d\lambda)\right. \\ \displaystyle \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. +\langle D_{y}u(t,x,y,z),q\rangle+\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\right \} =0,\\ \displaystyle u(0,x,y,z)=\phi(x,y,z),\ \ \forall(t,x,y,z)\in \lbrack 0,1]\times \mathbb{R}^{3d}, \end{array} \right. \] with $\delta_{\lambda}u(t,x,y,z):=u(t,x,y,z+\lambda)-u(t,x,y,z)-\langle D_{z}u(t,x,y,z),\lambda \rangle$. To construct the limit process $(\tilde{L}_{t})_{t\in \lbrack0,1]}$, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of L\'{e}vy-Khintchine representation formula to characterize $(\tilde{L}_{t})_{t\in [0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.

A universal robust limit theorem for nonlinear L\'evy processes under sublinear expectation

Under the framework of sublinear expectation, we introduce a new type of G-Gaussian random fields, which contains a type of spatial white noise as a special case. Based on this result, we also introduce a spatial-temporal G-white noise. Different from the case of linear expectation, in which the probability measure needs to be known, under the uncertainty of probability measures, spatial white noises are intrinsically different from temporal cases.

Spatial and temporal white noises under sublinear G -expectation

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.

Supermartingale Decomposition Theorem under G-expectation

The purpose of this paper is to study 2-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, statistically identical and have the same cost-functional. The game is studied in a weak formulation; this means in particular, we can study it as a game of the type "feedback control against feedback control". The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which driving coefficient is assumed to satisfy strict concavity-convexity with respect to the control parameters. This ensures the existence of saddle point feedback controls for the game with $N$ minor agents. We study the limit behavior of these saddle point controls and of the associated Hamiltonian, and we characterize the limit of the saddle point controls as the unique saddle point control of the limit mean-field stochastic differential game.

Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents

The controllable set of a controlled ordinary differential dynamic system to a given set is defined. Under certain reasonable conditions, the controllable set is characterised by a level set of the unique viscosity solution to some Hamilton-Jacobi-Bellman equation. The result is used to determine the asymptotic stable set of nonlinear autonomous differential equations.

/pdf/determination-of-a-controllable-set-for-a-controlled-dynamic-12ofiiirj8.pdf

Shige Peng

Papers

A universal robust limit theorem for nonlinear L\'evy processes under sublinear expectation

Spatial and temporal white noises under sublinear G -expectation

Supermartingale Decomposition Theorem under G-expectation

Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents

Determination of a controllable set for a controlled dynamic system