Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the exact asymptotics for the distribution of the first time, tau_x, a Levy process X_t crosses a fixed negative level -x.  prove that P{tau_x >t\} \sim V(x) P\{X_t>0\}/t as t\to\infty for a certain function V(x). Using known results for the large deviations of random walks,  obtain asymptotics for P{tau_x>t} explicitly in both light- and heavy-tailed cases.
['Denis Denisov', 'Vsevolod Shneer']
I am writing a research paper on a idea Asymptotics for the first passage times of Lévy processes and random walks
[{"from": "human", "value": "I am writing a research paper on a idea Asymptotics for the first passage times of L\u00e9vy processes and random walks"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study the exact asymptotics for the distribution of the first time, tau_x, a Levy process X_t crosses a fixed negative level -x.  prove that P{tau_x >t\\} \\sim V(x) P\\{X_t>0\\}/t as t\\to\\infty for a certain function V(x). Using known results for the large deviations of random walks,  obtain asymptotics for P{tau_x>t} explicitly in both light- and heavy-tailed cases."}]