Approximation of stochastic differential equations and application of the stochastic calculus of variations to the rate of convergence

2006 ◽  
pp. 267-287 ◽  
Author(s):  
Jean Picard
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Calculus Of Variations ◽  
Stochastic Calculus ◽  
Stochastic Calculus Of Variations

L2-regularity result for solutions of backward doubly stochastic differential equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050012
Author(s):  
Achref Bachouch ◽  
Anis Matoussi
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Scheme ◽  
Regularity Result ◽  
Time Discretization ◽  
Lipschitz Continuous ◽  
Doubly Stochastic

We prove an [Formula: see text]-regularity result for the solutions of Forward Backward doubly stochastic differential equations (F-BDSDEs) under globally Lipschitz continuous assumptions on the coefficients. As an application of our result, we derive the rate of convergence in time for the (Euler time discretization-based) numerical scheme for F-BDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.

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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