Lévy processes with negative drift conditioned to stay positive (Q5946909)

Let \(\mathbb X = \{X_t\}_{t\geq 0}\) be a one-dimensional Lévy process and denote by \(\tau = \tau_{(-\infty, 0]}\) and \(\tau_s = \tau_{(s,\infty)}\), \(s>0\), the first passage times for the intervals \((-\infty, 0]\) and \((s,\infty)\), respectively. Assume throughout that (i) \(\mathbb X\) starts at \(X_0=x >0\), that (ii) it admits a Laplace exponent \(\mathbb E^x e^{\xi X_t} = e^{t\varphi(\xi)}\) such that for some \(\alpha > 0\) the exponent \(\varphi\) is finite in a neighbourhood of \(\alpha\) and \(\varphi(\alpha) = 0\) (this is an exponential integrability condition entailing that \(\mathbb E X_1 < 0\), i.e., \(X_t \to -\infty\) as \(t\to\infty\)), and that (iii) \(\mathbb X\) is either of non-lattice type or of lattice type but without drift. The main aim of the paper is to investigate the law of \(\mathbb X\) conditional on \(\{\tau = \infty\}\), i.e., conditional on \(\mathbb X\) staying positive. Since \(\{\tau = \infty\}\) is a null set, the author considers two different approximations for this conditional law: \(\lim_{s\to\infty} \mathbb P^x(X\in\bullet \mid \tau > s)\) and \(\lim_{s\to\infty} \mathbb P^x(X\in\bullet \mid \tau > \tau_s)\). NEWLINENEWLINENEWLINESimilar approximations for Brownian motion with negative drift have been considered by \textit{S. Martinez} and \textit{J. San Martin} [J. Appl. Probab. 31, 911--920 (1994; Zbl 0818.60071)]. Using typical tools from fluctuation theory for Lévy processes, in particular the inverse local time at \(0\) and the ladder height process, the above limits can be identified to be the distributions \(\mathbb P^x(Y_t\in\bullet)\) and \(\mathbb P^x(Z_t\in\bullet)\), respectively, of two time-homogeneous Markov processes \(Y_t\) and \(Z_t\) whose transition functions \(q(t,x,dy)\) and \(r(t,x,dy)\) are given by NEWLINE\[NEWLINE q(t,x,dy) = \frac{\overline U(y)}{\overline U(x)} \widehat{\mathbb P}^x(X_t\in dy, \tau > 1), \quad r(t,x,dy) = \frac{\overline U_*(y)}{\overline U_*(x)} \widehat{\mathbb P}^x_*(X_t\in dy, \tau > 1), NEWLINE\]NEWLINE where \(\overline U\) (\(\overline U_*\)) is the usual renewal function associated to the ladder height process \(\overline H_t\) of the dual Lévy process \(\overline X_t\) under the measure \(\widehat{\mathbb P}\) (\(\widehat{\mathbb P}_*\)) which is, restricted on \(\mathcal F_t\), given by \(e^{\alpha X_t - t\varphi(\alpha)} \mathbb P\) (\(e^{\omega X_t} \mathbb P\)). Note that in the latter case one has to reinforce assumption (ii) from above by the Cramèr-type estimate (ii)': there exists \(\omega > 0\) such that \(\varphi(\omega) = 0\) and \(\mathbb E^x(X_1 e^{\omega X_1}) < \infty\).