First passage times of bivariate correlated diffusion processes: Analytical and Numerical Results

Beschreibung

The first passage time (FPT) problem of univariate stochastic processes through boundaries is relevant in different fields (e.g. economics, engineering, finance, neuroscience and physics), and it has been extensively studied in the literature. On the contrary, results for the first passage time problem of two-dimensional correlated diffusion processes are still scarce and fragmentary. Consider a two-dimensional time homogeneous diffusion process X. Define the random variable Ti =inf{t>t0 :Xi(t)>Bi}, i=1,2 i.e. the FPT of Xi through the constant boundary Bi > x0i. The purpose of this talk is to determine the joint density of (T1,T2) for the process X. A discussion on the behavior of the process in presence of different types of boundaries, in particular, killing, absorbing or crossing boundaries, is proposed. In all scenarios, we show that the quantity f(T1,T2) depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First, we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then, we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for bivariate diffusion processes. We derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting previous misprints appearing in the literature [2]. Finally, we illustrate the application of the method through a set of examples.

Zeitraum	12 Juni 2018
Ereignistitel	40th Conference on Stochastic Processes and their applications
Veranstaltungstyp	Konferenz
Ort	SchwedenAuf Karte anzeigen