WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
First-exit Problems for Integrated Diffusion Processes with State-dependent Jumps
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Abstract: Let dX(t) = -Y (t)dt, where Y (t) is a one-dimensional diffusion process. First-exit problems from $$C \in \mathbb{R}_{2}$$ are studied for the degenerate two-dimensional diffusion process (X(t), Y (t)) when the process leaves C not later than after a random time having an exponential distribution. When Y (t) is a standard Brownian motion, the Laplace transform of the moment-generating function M of the first-exit time is computed explicitly, as well as the Laplace transforms of the mean exit time m and the probability p of leaving C through a given part of its boundary. When Y (t) is a geometric Brownian motion, the functions M, m and p are obtained by making use of the method of similarity solutions to solve the various partial differential equations, subject to the appropriate boundary conditions.
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Keywords: Kolmogorov backward equation, Brownian motion, geometric Brownian motion, method of similarity solutions
Pages: 864-868
DOI: 10.37394/23206.2022.21.98