scholarly journals Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1953
Author(s):  
Ning Ma ◽  
Zhen Wu
Keyword(s):  
Differential Equations ◽  
Markov Chains ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Comparison Theorems ◽  
Yosida Approximation ◽  
Uniqueness Of Solutions ◽  
Doubly Stochastic

In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.

scholarly journals Anticipated Generalized Backward Doubly Stochastic Differential Equations

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 114
Author(s):  
Tie Wang ◽  
Jiaxin Yu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Uniqueness Theorem ◽  
Comparison Theorem ◽  
Existence And Uniqueness ◽  
Comparison Theorems ◽  
Existence And Uniqueness Theorem ◽  
New Class ◽  
Doubly Stochastic ◽  
Increasing Process

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

scholarly journals Comparison Theorems for the Multidimensional BDSDEs and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Bo Zhu ◽  
Baoyan Han
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Existence Of Solutions ◽  
Minimal Solution ◽  
Comparison Theorems ◽  
Doubly Stochastic

A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multidimensional BDSDEs. As its applications, we derive the existence of solutions for this multidimensional BDSDEs with continuous coefficients. We can also prove that this solution is the minimal solution of the BDSDE.

Comparison Theorems for the Multi-Dimensional Backward Doubly Stochastic Differential Equations

2012 ◽  
Vol 166-169 ◽  
pp. 3210-3213 ◽  
Author(s):  
Bao Yan Han
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Comparison Theorems ◽  
Doubly Stochastic

A class of backward doubly stochastic differential equations are studied. We obtain a comparison theorem of these multi-dimensional backward doubly stochastic differential equations.

scholarly journals Ulam–Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Xue Wang ◽  
Danfeng Luo ◽  
Zhiguo Luo ◽  
Akbar Zada
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lipschitz Condition ◽  
Existence And Uniqueness ◽  
Time Delays ◽  
Jensen’S Inequality ◽  
Uniqueness Of Solutions ◽  
Gronwall’S Inequality ◽  
Fractional Stochastic Differential Equations ◽  
New Criteria

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath e ´ odory approximation. Furthermore, with the help of H o ¨ lder’s inequality, Jensen’s inequality, It o ^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.

Anticipated backward doubly stochastic differential equations driven by teugels martingales with or without reflecting barrier

2020 ◽  
pp. 1-21
Author(s):  
Mostapha Saouli ◽  
B. Mansouri
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Time Interval ◽  
Present Value ◽  
Uniqueness Of Solutions ◽  
Doubly Stochastic ◽  
Teugels Martingales ◽  
The Fixed Point Theorem

We are interested in this paper on reflected anticipated backward doubly stochastic differential equations (RABDSDEs) driven by teugels martingales associated with Levy process. We obtain the existence and uniqueness of solutions to these equations by means of the fixed-point theorem where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right)$. We also show the comparison theorem for a special class of RABDSDEs under some slight stronger conditions. The novelty of our result lies in the fact that we allow the time interval to be infinite.

scholarly journals Backward Doubly SDEs with weak Monotonicity and General Growth Generators

2020 ◽  
pp. 59-85
Author(s):  
B. Mansouri ◽  
M. A. Saouli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Doubly Stochastic ◽  
General Growth ◽  
Weak Monotonicity ◽  
Probabilistic Solution

We deal with backward doubly stochastic differential equations (BDSDEs) with a weak monotonicity and general growth generators and a square integrable terminal datum. We show the existence and uniqueness of solutions. As application, we establish the existenceand uniqueness of Sobolev solutions to some semilinear stochastic partial differential equations (SPDEs) with a general growth and a weak monotonicity generators. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.

scholarly journals Comparison theorems for anticipated backward doubly stochastic differential equations with non-Lipschitz coefficients

2020 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
Sadibou Aidara
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorems ◽  
Doubly Stochastic

AbstractIn this work, we prove some comparison theorems of anticipated backward doubly stochastic differential equations with non-Lipschitz coefficients.

scholarly journals FILTERED STOCHASTIC CALCULUS

2001 ◽  
Vol 04 (03) ◽  
pp. 309-346 ◽  
Author(s):  
ROMUALD LENCZEWSKI
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Fock Space ◽  
Stochastic Calculus ◽  
Itô Formula ◽  
Uniqueness Of Solutions ◽  
Ito Formula ◽  
Itô Calculus ◽  
Boolean Independence

By introducing a color filtration to the multiplicity space [Formula: see text], we extend the quantum Itô calculus on multiple symmetric Fock space [Formula: see text] to the framework of filtered adapted biprocesses. In this new notion of adaptedness, "classical" time filtration makes the integrands similar to adapted processes, whereas "quantum" color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpolation between the main types of calculi, regardless of the type of independence, including freeness, Boolean independence (more generally, m-freeness) as well as tensor independence. Moreover, it shows how boson calculus is "deformed" by other noncommutative notions of independence. The corresponding filtered Itô formula is derived. Existence and uniqueness of solutions of a class of stochastic differential equations are established and unitarity conditions are derived.

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