On the two-boundary first-crossing-time problem for diffusion processes

Abstract

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new systems of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such funcitons a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single one

On the densities of certain bounded diffusion processes

“…Proof The proof follows from Proposition 1 by recalling (9) and by taking the inverse LT of (26), (27), (28) and (29).…”

“…(See, for instance, [3], [39]). Unfortunately, analytical solutions are not available except for very few and fragmentary cases (cf., for instance, [4,9,15,34,35]). …”

“…In general closed form results are not available, whereas the related Laplace transforms have occasionally been obtained. Also, numerical and algorithmic approaches have at times proved to be particularly effective (see, for instance, [8,9,11,16,24,28]). A very general approach focussing on Brownian motion and its functionals can finally be found in [30].…”

Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

“…[22]) or particular time-dependent boundaries, corresponding to special symmetries, for specific diffusions (cf. [7] and [10]). The existing results generally focus on the first exit time from the strip, while our interest lies in the joint distribution of the times when the process first attains the upper and lower boundaries.…”

Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model

“…For instance, the idea of conditioning to characterize the law of the first passage time in the class of time-(in)homogeneous one-dimensional diffusions via integral equations is well known in the applied probability literature, e.g. Park and Schuurmann (1976), Nobile et al (1985), Giorno et al (1989), Buonocore et al (1990), and Gutiérrez et al (1997). Based on these insights, Kuan and Webber (2003) price single and double barrier European-style options by recovering the first passage time density of the underlying asset price to the barrier level(s) through the numerical solution of an integral equation that only involves the transition density function of the model's state variable.…”