A structure-preserving method for the distribution of the first hitting time to a moving boundary for some Gaussian processes

A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

“…Theorem 1 (Macías-Díaz and Villa-Morales [6]). Let u : R × [0, ∞) → R be bounded, and suppose that it satisfies the following parabolic problem with initial-boundary conditions:…”

“…It is important to point out that, for the stochastic model considered in this work, the associated parabolic initial-boundary-value problem (5) is obtained from the Volterra integral equation of the second kind satisfied by the probability density of the first hitting time by making use of Itô's formula and some properties on continuous martingales. The details on this relationship are provided in reference [6]. On the other hand, some exact solutions of this model are known in exact form [6].…”

“…The details on this relationship are provided in reference [6]. On the other hand, some exact solutions of this model are known in exact form [6]. More precisely, let α, β ∈ R, X t = x + B t and f (t) = α + βt.…”

“…Indeed, there are reports which study the first hitting time and the last exit time for Brownian motions to and from a moving boundary, respectively [7]. Other works tackle the investigation of the first hitting time for reducible diffusion processes [8], one-dimensional diffusion and compound Poisson processes [9], Markov processes between moving retaining or absorbing barriers [10], time-dependent Ornstein-Uhlenbeck processes to a moving boundary [11], moving boundaries for asymptotically stable Lévy processes [12], moving boundaries for some Gaussian stochastic processes [6], the pricing of continuously monitored barrier options under stochastic volatility with jumps [13], just to mention some examples [14][15][16].…”

“…In this manuscript, we investigate a moving boundary problem which considers the presence of a deterministic and a stochastic components. It is well known that the first hitting time has a distribution of probability governed by a parabolic advection-diffusion differential equation, for which very few solutions are known in exact form [6]. Motivated by this fact, we will propose a spatially symmetric finite-difference scheme to approximate the solutions of this differential model.…”