Abstract
In this paper, we develop a Monte Carlo based algorithm for estimating the FPT (first passage time) density of the solution of a one-dimensional time-homogeneous SDE (stochastic differential equation) through a time-dependent frontier. We consider Brownian bridges as well as local Daniels curve approximations to obtain tractable estimations of the FPT probability between successive points of a simulated path of the process. Under mild assumptions, a (unique) Daniels curve local approximation caneasily be obtained by explicitly solving a non-linear system of equations.
Highlights
Let X be a time homogeneous one-dimensional diffusion process which is the unique solution of the following stochastic differential equation: dX (t) = μ (X (t)) dt + σ (X (t)) dW (t), X (0) = x0 (1)that is the functions μ and σ satisfy regularity conditions as described for example in Karatzas and Shreve (1991)
We propose to focus on local Daniels curve approximations of the time-varying boundary
We proposed an innovative Monte Carlo based algorithm for estimating the first-passage time density of a one-dimensional diffusion process through a deterministic time-dependent boundary
Summary
If S is a time dependent boundary, we are interested in estimating either the pdf or cdf of the first-passage time of the diffusion process through this boundary that is we will study the following random variable: τS = inf {t > 0|X (t) = S (t)}. There is no explicit expression for the first-passage time density of a diffusion process through a time-varying boundary. To this date, few specific cases provide closed-form formulas for some classes of time-varying boundaries for example when the process is Gaussian (Durbin (1985), Durbin and Williams (1992), DiNardo et al (2001)) or a Bessel process (Deaconu and Herrmann (2013)).
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