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.

A second-order discretization for degenerate systems of stochastic differential equations

2020 ◽  
Author(s):  
Yuga Iguchi ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Second Order ◽  
Weak Approximation ◽  
Hörmander Condition ◽  
Quasi Monte Carlo ◽  
Control Variate ◽  
The Monte Carlo Method

Abstract The paper proposes a new second-order weak approximation scheme for hypoelliptic diffusions or degenerate systems of stochastic differential equations satisfying a certain Hörmander condition. The scheme is constructed by a Gaussian process and a stochastic polynomial weight through a technique based on Malliavin calculus, and is implemented by a Monte Carlo method and a quasi-Monte Carlo method. A variance analysis for the Monte Carlo method is discussed, and further control variate methods are introduced to reduce the variance. The effectiveness of the proposed scheme is illustrated through numerical experiments for some hypoelliptic diffusions.

A third-order weak approximation of multidimensional Itô stochastic differential equations

2019 ◽  
Vol 25 (2) ◽  
pp. 97-120 ◽  
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Space ◽  
Weak Approximation ◽  
Third Order ◽  
Brownian Motions ◽  
Numerical Examples ◽  
Discretization Algorithm

Abstract This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler–Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply implemented by a Monte Carlo method. The method of Watanabe distributions on Wiener space is effectively applied in the computation of the polynomial weight of Brownian motions. Numerical examples are shown to confirm the accuracy of the scheme.

scholarly journals A regression-based Monte Carlo method to solve two-dimensional forward backward stochastic differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaofei Li ◽  
Yi Wu ◽  
Quanxin Zhu ◽  
Songbo Hu ◽  
Chuan Qin
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Backward Stochastic Differential Equations ◽  
Characteristic Functions ◽  
Two Dimensional ◽  
Least Squares Regression ◽  
European Option Pricing

AbstractThe purpose of this paper is to investigate the numerical solutions to two-dimensional forward backward stochastic differential equations(FBSDEs). Based on the Fourier cos-cos transform, the approximations of conditional expectations and their errors are studied with conditional characteristic functions. A new numerical scheme is proposed by using the least-squares regression-based Monte Carlo method to solve the initial value of FBSDEs. Finally, a numerical experiment in European option pricing is implemented to test the efficiency and stability of this scheme.

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