scholarly journals An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach

2012 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Malliavin Calculus ◽  
Backward Sdes

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 An asymptotic expansion of forward-backward SDEs with a perturbed driver

2015 ◽  
Vol 02 (02) ◽  
pp. 1550020 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Error Estimate ◽  
Arbitrary Order ◽  
Nonlinear Pricing ◽  
Small Noise ◽  
Small Diffusion ◽  
Noise Perturbation ◽  
Backward Sdes ◽  
Expansion Scheme

Motivated by nonlinear pricing in finance, this paper presents a mathematical validity of an asymptotic expansion scheme for a system of forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. In particular, we represent the coefficients of the expansion of the FBSDE up to an arbitrary order, and obtain the error estimate of the expansion with respect to the driver and the small noise perturbation.

Asymptotic expansion for the quadratic variations of the solution to the heat equation with additive white noise

2020 ◽  
Vol 21 (02) ◽  
pp. 2150010
Author(s):  
Héctor Araya ◽  
Ciprian A. Tudor
Keyword(s):  
Asymptotic Expansion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Heat Equation ◽  
White Noise ◽  
Malliavin Calculus ◽  
Order Term ◽  
Sharp Estimates ◽  
Additive White Noise ◽  
Quadratic Variations

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are based on sharp estimates of the correlation structure of the solution, which may have their own interest.

scholarly journals Asymptotic expansion for forward-backward SDEs with jumps

Stochastics ◽  
2018 ◽  
Vol 91 (2) ◽  
pp. 175-214 ◽  
Author(s):  
Masaaki Fujii ◽  
Akihiko Takahashi
Keyword(s):  
Asymptotic Expansion ◽  
Backward Sdes
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