The Malliavin calculus and its application to second order parabolic differential equations: Part II

1981 ◽  
Vol 14 (1) ◽  
pp. 141-171 ◽  
Author(s):  
Daniel W. Stroock
Keyword(s):  
Differential Equations ◽  
Malliavin Calculus ◽  
Second Order ◽  
Parabolic Differential Equations

A second-order discretization for forward-backward SDEs using local approximations with Malliavin calculus

2019 ◽  
Vol 25 (4) ◽  
pp. 341-361
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Malliavin Calculus ◽  
Second Order ◽  
Discretization Method ◽  
Local Approximations ◽  
Backward Sdes

Abstract The paper proposes a new second-order discretization method for forward-backward stochastic differential equations. The method is given by an algorithm with polynomials of Brownian motions where the local approximations using Malliavin calculus play a role. For the implementation, we introduce a new least squares Monte Carlo method for the scheme. A numerical example is illustrated to check the effectiveness.

scholarly journals Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Sixian Jin ◽  
Henry Schellhorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Malliavin Calculus ◽  
Series Representation ◽  
Parabolic Type ◽  
Second Order ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Path Dependent

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

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