An estimate of the rate of convergence of a sequence of additive functionals of difference approximations for a multidimensional diffusion process

2015 ◽  
Vol 90 ◽  
pp. 23-41
Author(s):  
Iu. V. Ganychenko
Keyword(s):  
Diffusion Process ◽  
Rate Of Convergence ◽  
Additive Functionals ◽  
Difference Approximations ◽  
Multidimensional Diffusion

scholarly journals Simulation of elliptic and hypo-elliptic conditional diffusions

2020 ◽  
Vol 52 (1) ◽  
pp. 173-212
Author(s):  
Joris Bierkens ◽  
Frank van der Meulen ◽  
Moritz Schauer
Keyword(s):  
Diffusion Process ◽  
The State ◽  
Identity Matrix ◽  
Excellent Performance ◽  
Linear Combinations ◽  
Time Zero ◽  
Partially Observed ◽  
Multidimensional Diffusion

AbstractSuppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

DRIFT TRANSFORMATIONS OF SYMMETRIC DIFFUSIONS, AND DUALITY

2007 ◽  
Vol 10 (04) ◽  
pp. 613-631 ◽  
Author(s):  
P. J. FITZSIMMONS
Keyword(s):  
Diffusion Process ◽  
Dirichlet Form ◽  
Infinitesimal Generator ◽  
Probabilistic Approach ◽  
Additive Functionals ◽  
Excessive Measure ◽  
Girsanov’S Theorem ◽  
Markov Diffusion ◽  
Symmetry Measure

Starting with a symmetric Markov diffusion process X (with symmetry measure m and L2 (m) infinitesimal generator A) and a suitable core [Formula: see text] for the Dirichlet form of X, we describe a class of derivations defined on [Formula: see text]. Associated with each such derivation B is a drift transformation of X, obtained through Girsanov's theorem. The transformed process XB is typically non-symmetric, but we are able to show that if the "divergence" of B is positive, then m is an excessive measure for XB, and the L2 (m) infinitesimal generator of XB is an extension of f ↦ Af + B (f). The methods used are mainly probabilistic, and involve the notions of even and odd continuous additive functionals, and Nakao's stochastic divergence. These methods yield a probabilistic approach to the adjoint of the semigroup of XB, and in particular lead to a solution of a problem of W. Stannat.

Sign in / Sign up

Export Citation Format

Share Document