Open AccessThe ruin problem for a Wiener process with state-dependent jumps
Author(s) -
Mario Lefebvre
Publication year - 2020
Publication title -
journal of applied mathematics, statistics and informatics
Language(s) - English
Resource type - Journals
eISSN - 1339-0015
pISSN - 1336-9180
DOI - 10.2478/jamsi-2020-0002
Subject(s) - mathematics , jump , jump diffusion , moment (physics) , interval (graph theory) , wiener process , ordinary differential equation , state (computer science) , function (biology) , diffusion process , jump process , differential equation , differential (mechanical device) , mathematical analysis , diffusion , combinatorics , physics , quantum mechanics , thermodynamics , algorithm , computer science , innovation diffusion , knowledge management , evolutionary biology , biology
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 ).