Estimating Parametric Derivatives of First Exit Times of Diffusions by Approximation of Wiener Processes Sergey A. Gusev * , Institute of Computational Mathematics and Mathematical Geophysics SB RAS, prospect Akademika Lavrentjeva, 6, Novosibirsk, 630090, Russia, Email: sag@osmf.sscc.ru Abstract—The problem of obtaining estimates of derivatives with respect to parameters of mathematical expectations of functionals of diffusion processes with absorbing boundary is considered in the paper. The problem demands to obtain the parametric derivatives of first exit times for the random pro- cesses. These derivatives can be obtained from the differentiation of the equation which is the result of applying the Ito’s formula to some function that vanishes on the boundary. The problem of differentiating the Ito integral, that arises here, is solved by approximating the Wiener process by a Gaussian one with exponential correlation function, consistent with the step length of the Euler method. Index Terms—diffusion process, first exit time, parametric derivatives, exponential correlation function, Euler method. I. I NTRODUCTION U Sually a study of a mathematical model is associated with the investigation of the nature of dependence of the model equations on the parameters. If the problem containing a diffusion process with absorbing boundary are in the consider- ation, then estimating the mean values of certain characteristics of the model is always connected with the first exit time τ of the process from the domain. This is due to the fact that theoretically the process can exit from a bounded domain at any time. Therefore, the first exit time of the diffusion process is a random value which depends on the parameters, and it is necessary to study this dependence. In particular, it is of interest if it is possible evaluating derivatives of τ with respect to the parameters. Evaluation of these derivatives is also important in solution of problems of stochastic optimization by gradient methods. In [1] it was proposed an idea of finding estimates of the parametric derivatives of τ from the equation that is result of applying the Ito formula to a function g which vanishes on the boundary. The main problem here was differentiation with respect to upper limit of the stochastic integral over the Wiener process. To overcome this difficulty there was suggested in this paper to approximate the Wiener process in the stochastic integral by a Gaussian process with an exponential correlation function and use the Euler method for the numerical solution of the stochastic differential equation (SDE). Wherein this * This work was supported by Russian Leading Science Schools Pro- gramme grant 5111.2014.1, Russian Foundation for Basic Research grant 14- 01-00340-a, correlation function matched with the length of the step in the Euler method. In this paper, the method [1] has been further promoted and more fully justified. In [2], [3] a variant of the method is considered when the function g vanishes on the boundary together with its first derivatives, but this option requires more stringent constraints on the smoothness of the input data. II. STATEMENT OF THE PROBLEM Let G ⊂ R d be a bounded domain with regular boundary ∂G. We suppose that we are given a complete filtered probabil- ity space (Ω, F , F t ,P ), t ≥ 0. Let W · be given d-dimensional Wiener process W · which is measurable at any t ≥ 0 with respect to F t and W s − W t for s>t is independent on σ- algebra F t . Let us denote E t,x a mathematical expectation with respect to the probability measure that corresponds to the random process initiating from the point x at the time item t. Let us introduce for a given time segment [0,T ] a space of random functions h(t) which are measurable at each t ∈ [0,T ] with respect to F t and which have with probability one the finite integral T  0 g 2 (t)dt. We also enter into consideration Q T = (0,T ) ×G a cylinder in R + ×R d and an open set U ⊂ R m . For x ∈ G and t ∈ [0,T ) we define a d-dimensional diffusion process X · which depends on vector parameter θ ∈ U , and this process is described by the following SDE X s (θ)= x+ s  t a(v,X v (θ),θ)dv + s  t σ(v,X v (θ),θ)dW v (1) where a : [0, ∞) × R d × U → R d and σ : [0, ∞) × R d × U → R d×d are measurable functions. We assume that that the following condition holds for the coefficients of the equation (1). (A) the functions a, σ are bounded and there exists a constant K such that for all θ ∈ U , v ≥ 0, x,y ∈ R d , i,j ∈{1,...,d} the following inequality is valid |a i (v,x,θ)−a i (v,y,θ)|+|σ ij (v,x,θ)−σ ij (v,y,θ)| ≤ K|x−y|. It is supposed in the paper that the absorbing boundary condition holds, i.e. each path of X · ends when the given INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 2, 2015 ISSN: 2313-0571 55