On Integral Equations Arising in the First-Passage Problem for Brownian Motion
نویسنده
چکیده
be the first-passage time of B over g , and let F denote the distribution function of . The first-passage problem seeks to determine F when g is given. The inverse first-passage problem seeks to determine g when F is given. Both the process B and the boundary g in these formulations may be more general, and our choice of Brownian motion is primarily motivated by the tractability of the exposition. The facts to be presented below can be extended to more general Markov processes and boundaries (such as two-sided ones) and the time may also be discrete. The first-passage problem has a long history and a large number of applications. Yet explicit solutions to the first-passage problem (for Brownian motion) are known only in a limited number of special cases including linear or quadratic g . The law of is also known for a square-root boundary g but only in the form of a Laplace transform (which appears intractable to inversion). The inverse problem seems even harder. For example, it is not known if there exists a boundary g for which is exponentially distributed (cf. [20]).
منابع مشابه
A wavelet method for stochastic Volterra integral equations and its application to general stock model
In this article,we present a wavelet method for solving stochastic Volterra integral equations based on Haar wavelets. First, we approximate all functions involved in the problem by Haar Wavelets then, by substituting the obtained approximations in the problem, using the It^{o} integral formula and collocation points then, the main problem changes into a system of linear or nonlinear equation w...
متن کاملExistence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion
متن کامل
Integral Equations and the First Passage Time of Brownian Motions
The first passage time problem for Brownian motions hitting a barrier has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the barrier itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of n...
متن کاملThe Inverse First-passage Problem and Optimal Stopping
Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse firstpassage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stoppi...
متن کاملA computational wavelet method for numerical solution of stochastic Volterra-Fredholm integral equations
A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method wa...
متن کامل