Applications of anticipating stochastic calculus to stochastic differential equations

1990 ◽  
pp. 63-105 ◽  
Author(s):  
Etienne Pardoux
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Calculus ◽  
Anticipating Stochastic Calculus

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.

Exponential formulae in quantum stochastic calculus

1996 ◽  
Vol 126 (2) ◽  
pp. 375-389 ◽  
Author(s):  
A. S. Holevo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Degrees Of Freedom ◽  
Stochastic Integral ◽  
Stochastic Calculus ◽  
Exponential Equation ◽  
Quantum Stochastic Calculus ◽  
Rigorous Definition ◽  
Definition Of

The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.

THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION

1991 ◽  
Vol 01 (02) ◽  
pp. 277-286 ◽  
Author(s):  
P. E. KLOEDEN ◽  
E. PLATEN ◽  
H. SCHURZ
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Rapid Development ◽  
Stochastic Calculus ◽  
Stochastic Dynamical Systems ◽  
The Subject

The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

scholarly journals Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

2021 ◽  
Vol 5 (4) ◽  
pp. 239
Author(s):  
Mahmoud Abouagwa ◽  
Rashad A. R. Bantan ◽  
Waleed Almutiry ◽  
Anas D. Khalaf ◽  
Mohammed Elgarhy
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Calculus ◽  
Variable Delay ◽  
Existence And Uniqueness Theorem ◽  
Poisson Jumps ◽  
New Class ◽  
Special Cases ◽  
Lipschitz Conditions ◽  
Theoretical Results

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

scholarly journals Lp-SOLUTIONS OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 12 (03) ◽  
pp. 1150025 ◽  
Author(s):  
AUGUSTE AMAN
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Stochastic Calculus ◽  
Technical Aspects ◽  
Doubly Stochastic ◽  
Priori Estimates ◽  
Lp Solutions

The goal of this paper is to solve backward doubly stochastic differential equations (BDSDEs, in short) under weak assumptions on the data. The first part is devoted to the development of some new technical aspects of stochastic calculus related to this BDSDEs. Then we derive a priori estimates and prove the existence and uniqueness of solution in Lp, p ∈ (1, 2), extending the work of Pardoux and Peng (see [12]).

Stochastic Calculus and Introduction to Stochastic Differential Equations

2018 ◽  
pp. 153-175
Author(s):  
Ettore Vitali ◽  
Mario Motta ◽  
Davide Emilio Galli
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Calculus

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law
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