Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet

2003 ◽  
Vol 03 (02) ◽  
pp. 121-139 ◽  
Author(s):  
Mohamed Erraoui ◽  
Youssef Ouknine ◽  
David Nualart
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Strong Solution ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Approximation Scheme ◽  
Borel Function ◽  
Brownian Sheet ◽  
Fractional Brownian Sheet ◽  
Partial Differential

Let [Formula: see text] be a fractional Brownian sheet with Hurst parameters H, H′ ≤ 1/2. We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equations with additive fractional Brownian sheet of the form [Formula: see text], where b(ζ, x) is a Borel function satisfying some growth and monotonicity assumptions. We also prove the convergence of Euler's approximation scheme for this equation.

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

Reflected stochastic partial differential equations with jumps

2021 ◽  
pp. 2250002
Author(s):  
Hongchao Qian ◽  
Jun Peng
Keyword(s):  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Uniqueness Of Solutions ◽  
Partial Differential ◽  
Penalization Method

In this paper, we establish the existence and uniqueness of solutions of reflected stochastic partial differential equations (SPDEs) driven both by Brownian motion and by Poisson random measure in a convex domain. Penalization method plays a crucial role.

scholarly journals Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Mean Field ◽  
Partial Differential ◽  
Existence And Uniqueness Results ◽  
Doubly Stochastic ◽  
Uniqueness Results

Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

scholarly journals Stationary in Distributions of Numerical Solutions for Stochastic Partial Differential Equations with Markovian Switching

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yi Shen ◽  
Yan Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Markovian Switching ◽  
Partial Differential ◽  
Stationary Distributions

We investigate a class of stochastic partial differential equations with Markovian switching. By using the Euler-Maruyama scheme both in time and in space of mild solutions, we derive sufficient conditions for the existence and uniqueness of the stationary distributions of numerical solutions. Finally, one example is given to illustrate the theory.

scholarly journals Elliptic stochastic partial differential equations with two reflecting walls

2014 ◽  
Vol 17 (04) ◽  
pp. 1450025
Author(s):  
Wen Yue ◽  
Tusheng Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Multiplicative Noise ◽  
Partial Differential

In this paper, we study elliptic stochastic partial differential equations with two reflecting walls h1 and h2, driven by multiplicative noise. The existence and uniqueness of the solutions are established.

scholarly journals Foundations of the Theory of Semilinear Stochastic Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Stefan Tappe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Uniqueness Result ◽  
Review Article ◽  
Banach Fixed Point Theorem ◽  
Extended Version ◽  
Partial Differential

The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In particular, we provide a detailed study of the concepts of strong, weak, and mild solutions, establish their connections, and review a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem.

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