On the Existence and Uniqueness of a Solution to a Stochastic Differential Equation in a Banach Space

2004 ◽  
Vol 11 (3) ◽  
pp. 515-526
Author(s):  
B. Mamporia
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Covariance Operator ◽  
Sufficient Condition ◽  
Existence Of A Solution ◽  
Arbitrary Banach Space ◽  
Uniqueness Of A Solution

Abstarct A sufficient condition is given for the existence of a solution to a stochastic differential equation in an arbitrary Banach space. The method is based on the concept of covariance operator and a special construction of the Itô stochastic integral in an arbitrary Banach space.

BSDE with rcll reflecting barrier driven by a Lévy process

2020 ◽  
Vol 28 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Mohamed El Jamali ◽  
Mohamed El Otmani
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Lévy Process ◽  
Backward Stochastic Differential Equation ◽  
American Option ◽  
Levy Process ◽  
Fair Price ◽  
Penalization Method ◽  
Uniqueness Of A Solution

AbstractIn this paper, we study the solution of a backward stochastic differential equation driven by a Lévy process with one rcll reflecting barrier. We show the existence and uniqueness of a solution by means of the penalization method when the coefficient is stochastic Lipschitz. As an application, we give a fair price of an American option.

Fractional anticipated BSDEs with stochastic Lipschitz coefficients

2018 ◽  
Vol 26 (3) ◽  
pp. 143-161
Author(s):  
Ahmadou Bamba Sow ◽  
Bassirou Kor Diouf
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Comparison Theorem ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Hurst Parameter ◽  
Uniqueness Of A Solution

Abstract In this paper, we deal with an anticipated backward stochastic differential equation driven by a fractional Brownian motion with Hurst parameter {H\in(1/2,1)} . We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and prove a comparison theorem in a specific case.

scholarly journals Lp-SOLUTIONS FOR REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 12 (02) ◽  
pp. 1150016 ◽  
Author(s):  
SAÏD HAMADÈNE ◽  
ALEXANDRE POPIER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Snell Envelope ◽  
Equation Problem ◽  
Class Of Functions ◽  
Uniqueness Of A Solution ◽  
Lp Solutions

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p ∈ ]1, 2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution.

scholarly journals BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients

2019 ◽  
Vol 4 (1) ◽  
pp. 139-150 ◽  
Author(s):  
Sadibou Aidara ◽  
Yaya Sagna
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Backward Stochastic Differential Equation ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Type Integral ◽  
Divergence Type ◽  
Mutually Independent ◽  
Uniqueness Of A Solution

AbstractThis paper deals with a class of backward stochastic differential equation driven by two mutually independent fractional Brownian motions. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.

scholarly journals Generalized BSDE driven by a Lévy process

2006 ◽  
Vol 2006 ◽  
pp. 1-25 ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Independent Brownian Motion ◽  
Teugels Martingales

We study the solution of one-dimensional generalized backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove existence and uniqueness of the solution when the coefficient verifies some conditions of Lipschitz. If the coefficient is left continuous, increasing, and bounded, we prove the existence of a solution.

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 369-388 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
A. OGROWSKY ◽  
B. SCHMALFUSS
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Random Dynamical System ◽  
Random Delay ◽  
Random Differential Equation ◽  
Autonomous Case ◽  
Random Differential Equations ◽  
Uniqueness Of A Solution

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

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