scholarly journals A weak approximation with asymptotic expansion and multidimensional Malliavin weights

2016 ◽  
Vol 26 (2) ◽  
pp. 818-856 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Weak Approximation

A Weak Approximation of SDEs with Asymptotic Expansion

2013 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Weak Approximation

scholarly journals Approximation of SDE solutions using local asymptotic expansions

2021 ◽  
Author(s):  
A. M. Davie
Keyword(s):  
Asymptotic Expansion ◽  
Strong Approximation ◽  
Optimal Transport ◽  
Small Time ◽  
Degenerate System ◽  
Weak Approximation ◽  
Forward Equation ◽  
Smooth Coefficients ◽  
One Step ◽  
Edgeworth Type Expansion

AbstractWe develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

A control variate method for weak approximation of SDEs via discretization of numerical error of asymptotic expansion

2019 ◽  
Vol 25 (3) ◽  
pp. 239-252
Author(s):  
Yusuke Okano ◽  
Toshihiro Yamada
Keyword(s):  
Monte Carlo ◽  
Asymptotic Expansion ◽  
Stochastic Volatility ◽  
Diffusion Processes ◽  
Stochastic Volatility Model ◽  
Weak Approximation ◽  
Numerical Examples ◽  
Control Variate ◽  
Volatility Model ◽  
Multidimensional Diffusion

Abstract The paper shows a new weak approximation method for stochastic differential equations as a generalization and an extension of Heath–Platen’s scheme for multidimensional diffusion processes. We reformulate the Heath–Platen estimator from the viewpoint of asymptotic expansion. The proposed scheme is implemented by a Monte Carlo method and its variance is much reduced by the asymptotic expansion which works as a kind of control variate. Numerical examples for the local stochastic volatility model are shown to confirm the efficiency of the method.

A weak approximation with Malliavin weights for local stochastic volatility model

2017 ◽  
Vol 04 (01) ◽  
pp. 1750002
Author(s):  
Toshihiro Yamada
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Stochastic Differential Equation ◽  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Weak Approximation ◽  
Volatility Model ◽  
Sabr Model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

Influence of free convection on the diffusion current to a sphere in a laminar forced flow regime

1985 ◽  
Vol 50 (12) ◽  
pp. 2697-2714
Author(s):  
Arnošt Kimla ◽  
Jiří Míčka
Keyword(s):  
Asymptotic Expansion ◽  
Free Convection ◽  
Boundary Value Problem ◽  
Concentration Gradient ◽  
Empirical Formula ◽  
Convective Diffusion ◽  
Forced Flow ◽  
Diffusion Current ◽  
Wide Range ◽  
Free And Forced Convection

The formulation and solution of a boundary value problem is presented, describing the influence of the free convective diffusion on the forced one to a sphere for a wide range of Rayleigh, Ra, and Peclet, Pe, numbers. It is assumed that both the free and forced convection are oriented in the same sense. Numerical results obtained by the method of finite differences were approximated by an empirical formula based on an analytically derived asymptotic expansion for Pe → ∞. The concentration gradient at the surface and the total diffusion current calculated from the empirical formula agree with those obtained from the numerical solution within the limits of the estimated errors.

Sign in / Sign up

Export Citation Format

Share Document