Computing conditional expectations of multidimensional diffusion processes

Stochastics ◽  
2005 ◽  
Vol 77 (4) ◽  
pp. 315-326
Author(s):  
Alberto Lanconelli
Keyword(s):  
Diffusion Processes ◽  
Conditional Expectations ◽  
Multidimensional Diffusion

Jump Diffusion Inference in Complex Scenes

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Markov Process ◽  
Parameter Space ◽  
Diffusion Processes ◽  
Jump Process ◽  
Jump Diffusion ◽  
Jump Diffusion Processes ◽  
Transition Distribution ◽  
Conditional Expectations ◽  
Sampling Algorithms ◽  
Lie Manifolds

This chapter explores random sampling algorithms introduced in for generating conditional expectations in hypothesis spaces in which there is a mixture of discrete, disconnected subsets. Random samples are generated via the direct simulation of a Markov process whose state moves through the hypothesis space with the ergodic property that the transition distribution of the Markov process converges to the posterior distribution. This allows for the empirical generation of conditional expectations under the posterior. To accommodate the connected and disconnected nature of the state spaces, the Markov process is forced to satisfy jump–diffusion dynamics. Through the connected parts of the parameter space (Lie manifolds) the algorithm searches continuously, with sample paths corresponding to solutions of standard diffusion equations. Across the disconnected parts of parameter space the jump process determines the dynamics. The infinitesimal properties of these jump–diffusion processes are selected so that various sample statistics converge to their expectation under the posterior.

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.

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