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

Le but de cet article est de discuter un probleme de controle optimal avec vibrations rapides. Le modele mathematique est le suivant: pour $\varepsilon > 0$, assez petit, on considere \[( * )_1 \qquad \frac{dx^\varepsilon }{dt} = g\left(\frac{t}{\varepsilon },x^\varepsilon (t),v(t)\right),\quad x^\varepsilon (0) = x_0 \]ou $v( \cdot ) \in L^2 ({0,T;\mathbb{R}^k })$ est la fonction de controle. $x^\varepsilon ( \cdot ) \in H^1 (0,T;\mathbb{R}^n )$ est la fonction d’etat. $g(\tau ,x,v)$ est 1-periodique par rapport a $\tau \in \mathbb{R}$. On cherche a minimiser la fonction cout \[( * )_2 \qquad \int_0^T {l\left(\frac{t} {\varepsilon },x^\varepsilon (t),v(t)\right) dt} \] ou $l(\tau ,x,v)$ est aussi 1-periodique par rapport a $\tau $.On s’interesse tout d’abord au comportement du systeme quand $\varepsilon \to 0$. On pourrait pent-etre croire que ce probleme va tendre vers le probleme “moyen” \[\frac{{dx}}{{dt}} = \bar g(x(t),v(t)),\qquad \mathop {\min }\limits_{v( \cdot )} \int_0^T {\bar l(x(t),v(t)) dt} \...

Asymptotic analysis and the homogeneous problem in optimal control with rapid vibrations

We formulate and solve the martingale problem in a nonlinear expectation space. Unlike the classical work of Stroock and Varadhan (Commun Pure Appl Math 22:345–400, 479–530, 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 Lipschitz 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.

Martingale problem under nonlinear expectations

The authors prove the existence and uniqueness of smallest g-supersolution with an equality constraint on (y, z) for one demensional stochastic differential equations whose drift coefficients are continuous and linearly growing, and whose terminal conditions are square integrable.

SMALLEST g-SUPERSOLUTION FOR BSDE WITH CONTINUOUS DRIFT COEFFICIENTS

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the solution of ergodic second order partial differential equations.

Ergodic Backward SDE and Associated PDE

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.

Shige Peng

Papers

Asymptotic analysis and the homogeneous problem in optimal control with rapid vibrations

Martingale problem under nonlinear expectations

SMALLEST g-SUPERSOLUTION FOR BSDE WITH CONTINUOUS DRIFT COEFFICIENTS

Ergodic Backward SDE and Associated PDE

The Pricing Mechanism of Contingent Claims and its Generating Function