We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow.

We study a backward stochastic differential equation (BSDE) whose terminal condition is an integrable function of a local martingale and generator has bounded growth in z. When the local martingale is a strict local martingale, the BSDE admits at least two different solutions. Other than a solution whose.A backwards martingale is a stochastic process that satisfies the martingale property reversed in time, in a certain sense. In some ways, backward martingales are simpler than their forward counterparts, and in particular, satisfy a convergence theorem similar to the convergence theorem for ordinary martingales. The importance of backward martingales stems from their numerous applications. In.As application, we give a solution to reflected backward stochastic differential equations when the barrier is an optional positive process neither right-continuous nor left-continuous. Previous article in issue; Next article in issue; MSC. 60G20. 60H05. 60H15. Keywords. Optional Snell envelopes. Split stopping times. Reflected backward stochastic differential equations. Mertens decomposition.

Martingales Last updated by Serik Sagitov: May 23, 2013 Abstract This Stochastic Processes course is based on the book Probabilities and Random Processes by Geo rey Grimmett and David Stirzaker. Chapters 7.7-7.8, and 12. 1 De nitions and examples Example 1 Martingale: a betting strategy. Let X n be the gain of a gambler doubling the bet after each loss. The game stops after the rst win. X 0.

This is illustrated by a simple shift between two martingale measures such that the price process is uniformly integrable under one of them, but not under the other. To prepare for a more systematic study of dynamics on the space of martingale measures, we consider different versions of the optimal transport problem for measures on path space.

Lp Solutions of Re ected Backward Stochastic Di erential Equations with Jumps Song Yao Abstract Given p 2(1;2), we study Lp solutions of a re ected backward stochastic di erential.

Abstract. We prove that a continuous -supermartingale with uniformly continuous coeffcient on finite or infinite horizon, is a -supersolution of the corresponding backward stochastic differential equation.It is a new nonlinear Doob-Meyer decomposition theorem for the -supermartingale with continuous trajectory. 1. Introduction. In 1990, Pardoux-Peng () proposed the following nonlinear.

Martingale Theory with Applications 34. Unit aims. To stimulate through theory and examples, an interest and appreciation of the power of this elegant method in probability theory. And to lay foundations for further studies in probability theory. Unit description. The theory of martingales is of fundamental importance to probability theory, statistics, and mathematical finance. This unit is a.

Theorem 233 Let X n F n n 0 1 2 be a uniformly integrable submartingale Then X. Theorem 233 let x n f n n 0 1 2 be a uniformly School DeAnza College; Course Title MATH 10; Type. Notes. Uploaded By eelompanda. Pages 44 This preview shows page 36 - 38 out of 44 pages.

It is easiest to think of this in the nite setting, when the function X: !R takes only nitely many values. Then, as you might already suspect from (1.2), to check if Xis measurable its.

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Pr Backward submartingale convergence Suppose that X n n N is a. Pr backward submartingale convergence suppose that x School Peking Uni. Course Title MATHEMATIC 2012080032; Type. Essay. Uploaded By victory1832. Pages 12 This preview shows page 7 - 10 out of 12 pages.

Uniformly integrable family of functions, continuation of proof of an explicit formula for generator of an Ito diffusion, Dynkin's formula and its applications. Mon Feb 29th More examples for applications of Dynkin's formula (such as calculation of expected value of the first exist time of Brownian motion from a ball), the characteristic operator of an Ito diffusion and its coincidence with.

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On g pEvaluations with L Domains under Jump Filtration Song Yao Abstract Given p2(1;2), the unique Lp solutions of backward stochastic di erential equations with jumps (BSDEJs) al.

I.3. Uniformly integrable martingales 16 I.4. Stochastic integration 18 I.5. Selected results from stochastic calculus 26 I.6. The Skorokhod space 29 I.7. Martingale representation of square-integrable martingales 46 Chapter II. Backward Stochastic Di erential Equations with Jumps 51 II.1. Introduction 51 II.2. A simple counterexample and its.

Constrained Quadratic Risk Minimization via Forward and Backward Stochastic Di erential Equations Yusong Li and Harry Zhengy Abstract In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical nance. We model the asset dynamics with random market coe cients and portfolio strategies with convex constraints. Following the convex duality.

Backward stochastic differential equation, Infinite horizon, Reflected barriers, Stochastic optimal control, Stochastic differential game. I. Introduction. Nonlinear backward stochastic daerential equations (BSDE's in short) have been independently introduced by Pardoux and Peng (18) and DdEe and Epstein (7). It has already been discovered by Peng (20) that, coupled with a forward SDE, such.