Journal
STATISTICS & PROBABILITY LETTERS
Volume 205, Issue -, Pages -Publisher
ELSEVIER
DOI: 10.1016/j.spl.2023.109956
Keywords
Kolmogorov backward equation; Brownian motion; Vasicek model; Similarity variable
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This passage describes the analytical solution method for an Ornstein-Uhlenbeck process with Poissonian jumps, considering both exponentially and uniformly distributed jumps.
Let dX(t) =Y(t) dt, where Y (t) is an Ornstein-Uhlenbeck process with Poissonian jumps, and let T (x,y) be the first time that X(t) + Y(t) = 0, given that X (0) = x and Y (0) = y. The moment-generating function of T (x,y) is obtained in the case when the jumps are exponentially distributed by solving the integro-differential equation it satisfies, subject to the appropriate boundary conditions. The case when the jumps are uniformly distributed is also considered.
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