Abstract
This paper considers a method of the calculation of probability of the exit from a band of the solution of a stochastic differential equation. The method is based on the approximation of the solution of the considered equation by a process which is received as a concatenation of Gauss processes, random partition of the interval, Girsanov transform and Wiener-Hopf factorization, and the Monte-Carlo method. The errors of approximation are estimated. The proposed method is illustrated by numerical examples.
Highlights
Introduction and Related WorkThe paper is dedicated to the calculation of probability of the exit from a band
This problem is very important in applications of the theory of stochastic processes, such as financial mathematics, communication lines, and so on
The method is based on (1) the approximation of the solution of the considered equation by a process which is received as a concatenation of Gauss processes; (2) random partition of the interval; (3) Girsanov transform and Wiener-Hopf factorization; and (4) Monte-Carlo method, which is the calculation core of the approach [1]
Summary
The paper is dedicated to the calculation of probability of the exit from a band. This problem is very important in applications of the theory of stochastic processes, such as financial mathematics (barrier options), communication lines (signals), and so on. We consider a method of the calculation of probability of the exit from a band of the solution of a stochastic differential equation dXt = f (Xt )dt + φ(Xt )dWt with an initial condition X0 , where W is a standard Wiener process. An approximation based on the Brownian bridges was used for calculation of the prices of barrier options [8,9] We think it impossible to use Brownian bridges for the calculation of the probability of the exit of a stochastic process from a band. An integral approach to the sustainable management based on the notion of homeostasis is presented in [15] In this connection notice that the probability of the exit from a band gives a stochastic characteristic of the possible violation of the homeostasis.
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