Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 130940, 17 pages
Research Article

Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

Adrian P. C. Lim

Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA

Received 10 May 2008; Accepted 9 September 2008

Academic Editor: Hong Kun Xu

Copyright © 2008 Adrian P. C. Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This article aims to give a formula for differentiating, with respect to T, an expression of the form λ(T,x):=𝔼x[f(XT)e0TV(Xs)ds(det(I+KX,T))P], where p0 and X is a diffusion process starting from x, taking values in a manifold, and the expectation is taken with respect to the law of this process. KX,T:L2([0,T)N)L2([0,T)N) is a trace class operator defined by KX,Tf(s)=0TH(st)Γ(X(t))f(t)dt, where H, Γ are locally Lipschitz, positive N×N matrices.