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Completely asymmetric Lévy processes confined in a finite interval. (English) Zbl 0970.60055
Let $$[0,a]$$ 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.

##### 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)
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