<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>X</italic>(<italic>t</italic>) be a jump-diffusion process whose continuous part is a Wiener process, and let <italic>T</italic> (<italic>x</italic>) be the first time it leaves the interval (0,<italic>b</italic>), where <italic>x</italic> = <italic>X</italic>(0). The jumps are negative and their sizes depend on the value of <italic>X</italic>(<italic>t</italic>). Moreover there can be a jump from <italic>X</italic>(<italic>t</italic>) to 0. We transform the integro-differential equation satisfied by the probability <italic>p</italic>(<italic>x</italic>) := <italic>P</italic>[<italic>X</italic>(<italic>T</italic> (<italic>x</italic>)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of <italic>T</italic> (<italic>x</italic>).</p></abstract>

Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

The ruin problem for a Wiener process with state-dependent jumps

Department of Mathematics and Industrial Engineering

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"The ruin problem for a Wiener process with state-dependent jumps"}

Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0)....

The ruin problem for a Wiener process with state-dependent jumps

Published Online: Jul 09, 2020

Page range: 13 - 23

DOI: https://doi.org/10.2478/jamsi-2020-0002

Keywords
First exit time, Brownian motion, Poisson process, jump size, integro-differential equation

© 2020 M. Lefebvre, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The ruin problem for a Wiener process with state-dependent jumps

Published Online: Jul 09, 2020

Page range: 13 - 23

DOI: https://doi.org/10.2478/jamsi-2020-0002

KeywordsFirst exit time, Brownian motion, Poisson process, jump size, integro-differential equation

© 2020 M. Lefebvre, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
First exit time, Brownian motion, Poisson process, jump size, integro-differential equation