Completely asymmetric Lévy processes confined in a finite interval.

*(English)*Zbl 0970.60055Let \([0,a]\) be a finite interval and consider a Lévy process \(X_t\) which has only negative jumps (i.e., it is completely asymmetric) and admits transition densities. Write \(\psi\) for the characteristic (Laplace) exponent of \(X_t\) and denote by \(T\) the first exit time from \([0,a]\). The two-sided exit problem for such a process was studied by J. Bertoin [Bull. Lond. Math. Soc. 28, No. 5, 514-520 (1996; Zbl 0863.60068) and Ann. Appl. Probab. 7, No. 1, 156-169 (1997; Zbl 0880.60077)]. Building on these earlier results, the author considers the problem to find the distribution \(\mathbb{P}_x^\updownarrow\), \(x\in[0,a]\), under which the process \(X_t\) stays in \([0,a]\). This is achieved by showing that the limit \(\lim_{t\to \infty}\mathbb{P}_x (\Lambda\mid T>t= \mathbb{P}_x^\updownarrow (\Lambda)\) exists and defines a new probability measure under which \((X_t,\mathbb{P}_x^\updownarrow)\) is a Feller process with values in \([0,a]\). It turns out that \(\mathbb{P}_x^\updownarrow\) can be realized as a Doob \(h\)-transform of the original measure \(\mathbb{P}_x\) with respect to the \(\mathbb{P}_x\)-martingale (under the original filtration)
\[
D_t=e^{\rho t}1_{\{t< T\}}{W^{(-\rho)} (X_t)\over W^{(-\rho)} (x)}.
\]
In this formula, \(W^{(q)}(x)\) is the scale function of the process \(X_t\) killed at the constant rate \(q\),
\[
\int^\infty_0 e^{-\lambda x}W^{(q)}(x)dx= {1\over\psi (\lambda)-q},
\]
and \(\rho= \rho(a)\) is the first zero of \(q\mapsto W^{(-q)}(a)\). Several properties of \(\mathbb{P}^\updownarrow\) and of \(X_t\) under \(\mathbb{P}^\updownarrow\) are established, e.g., an explicit representation of the resolvent kernels in terms of the scale function is given and the two-sided exit problem within \([0,a]\) is solved.

Moreover, the author studies some elements of fluctuation theory of \((X_t, \mathbb{P}^\updownarrow)\), in particular the excursion measure (away from a point \(x \in[0,a])\) of the confined process which is expressed in terms of the excursion measure of the original process. This is then used to study local times \(L^x_t\) (and their inverses) of the confined process, and the value of the almost sure limit \(\lim_{t\to\infty} (L^x_t/t)\) is explicitly found. If \(X_t\) is of unbounded variation, \(t(a-\sup_{s\leq t}X_s)\) converges as \(t\to\infty\) in distribution to an exponential random variable with parameter \(|\rho' (a) |\); a simple integral criterion is provided for convergence or divergence of the upper and lower limits of \((a-\sup_{s\leq t}X_s)/f(t)\), \(t\to\infty\). The proofs use variants of techniques for unconfined processes [cf. the monograph “Lévy processes” by J. Bertoin (1996; Zbl 0861.60003)], but they rely crucially on the explicit form of the law and excursion measure of the confined process.

Moreover, the author studies some elements of fluctuation theory of \((X_t, \mathbb{P}^\updownarrow)\), in particular the excursion measure (away from a point \(x \in[0,a])\) of the confined process which is expressed in terms of the excursion measure of the original process. This is then used to study local times \(L^x_t\) (and their inverses) of the confined process, and the value of the almost sure limit \(\lim_{t\to\infty} (L^x_t/t)\) is explicitly found. If \(X_t\) is of unbounded variation, \(t(a-\sup_{s\leq t}X_s)\) converges as \(t\to\infty\) in distribution to an exponential random variable with parameter \(|\rho' (a) |\); a simple integral criterion is provided for convergence or divergence of the upper and lower limits of \((a-\sup_{s\leq t}X_s)/f(t)\), \(t\to\infty\). The proofs use variants of techniques for unconfined processes [cf. the monograph “Lévy processes” by J. Bertoin (1996; Zbl 0861.60003)], but they rely crucially on the explicit form of the law and excursion measure of the confined process.

Reviewer: René L.Schilling (Brighton)

##### MSC:

60G51 | Processes with independent increments; Lévy processes |

60G17 | Sample path properties |

60J45 | Probabilistic potential theory |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |