scholarly journals Yamada-Watanabe Results for Stochastic Differential Equations with Jumps

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Mátyás Barczy ◽  
Zenghu Li ◽  
Gyula Pap
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Stochastic Equations ◽  
Original Method ◽  
Pathwise Uniqueness ◽  
General Version ◽  
Strong Existence ◽  
Weak And Strong Solutions

Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps.

scholarly journals McKean–Vlasov type stochastic differential equations arising from the random vortex method

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Zhongmin Qian ◽  
Yuhan Yao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Vortex Dynamics ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Vortex Method ◽  
Wasserstein Distance ◽  
New Approach ◽  
Lipschitz Continuous ◽  
Space Of Distributions

AbstractWe study a class of McKean–Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean–Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot–Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.

Strong solutions for functional SDEs with singular drift

2017 ◽  
Vol 18 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Xing Huang
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Stable Process ◽  
Strong Solutions ◽  
Singular Drift

By using Zvonkin type transforms, existence and uniqueness are proved for a class of functional stochastic differential equations with singular drifts. The main results extend corresponding ones in [5, 11] for stochastic differential equations driven by Brownian motion and symmetric [Formula: see text]-stable process respectively.

scholarly journals Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

2020 ◽  
Vol 28 (4) ◽  
pp. 253-268
Author(s):  
AbdulRahman Al-Hussein ◽  
Boulakhras Gherbal
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Continuation Method ◽  
Strong Solutions ◽  
Poisson Jumps ◽  
Doubly Stochastic ◽  
Backward Equation ◽  
Different Dimensions ◽  
Fully Coupled

AbstractThe paper addresses a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. These equations are allowed to live in Euclidean spaces of different dimensions, and the system is Markovian in the sense that the terminal value of the backward equation depends on the terminal value of the solution of the forward one. Under some monotonicity conditions we establish the existence and uniqueness of strong solutions of such equations by using a continuation method.

Stability of stochastic differential equations driven by multifractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oussama El Barrimi ◽  
Youssef Ouknine
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Process ◽  
Strong Stability ◽  
Pathwise Uniqueness ◽  
Uniqueness Property ◽  
Selection Theorem ◽  
Multifractional Brownian Motion ◽  
Stability Results

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

scholarly journals Fuzzy stochastic differential equations driven by fractional Brownian motion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Marek T. Malinowski ◽  
M. J. Ebadi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Financial Models ◽  
Stochastic Integral ◽  
Long Range Dependence ◽  
World Systems

AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

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