scholarly journals Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations

2017 ◽  
Vol 87 (311) ◽  
pp. 1353-1413 ◽  
Author(s):  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Xiaojie Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Nonlinear Stochastic Differential Equations ◽  
Exponential Integrability

A variational approach to nonlinear stochastic differential equations with linear multiplicative noise

2019 ◽  
Vol 25 ◽  
pp. 71
Author(s):  
Viorel Barbu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Parabolic Equations ◽  
Optimization Problem ◽  
Multiplicative Noise ◽  
Generalized Solution ◽  
Convex Optimization Problem ◽  
Nonlinear Stochastic Differential Equations ◽  
Infinite Dimensional ◽  
Wiener Processes

One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations with mild nonlinearity are given.

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

2019 ◽  
Vol 25 (1) ◽  
pp. 37-60
Author(s):  
Antoon Pelsser ◽  
Kossi Gnameho
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Mathematical Finance ◽  
Backward Stochastic Differential Equations ◽  
Terminal Condition ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
The Family ◽  
Infinite Dimensional System

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

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