First exit time di processi stocastici e connessione con le equazioni differenziali alle derivate parziali

Markov processes can be construted starting from a Markov semigroup $(P_t)_{t\geq0}$. If the semigroup is also strongly continuous on the space of continuous functions vanishing at infinity, we call it a Feller semigroup and we can prove that the ensuing process has the strong Markov property. Feller semigroups admit a generator A and satisfy $\tfrac{d}{dt}P_tu=AP_tu=P_tAu \; \forall\,u\in \C_\infty(\R^d)$. In particular, if $(X_t)_{t\geq0}$ is a diffusion with transition density $p(t,x,y)$, $t>0$, $x,y\in\R^d$, we can deduce the Kolmogorov's backward equation from $\tfrac{d}{dt}P_t=AP_t$, by considering $A$ as a second order differential operator acting upon $x\in\R^d$. For the transition semigroup $P_t u(x):=\E^x u(B_t)$ in particular, we will see that the generator coincides with the Laplace operator ($A=\tfrac{1}{2}\Delta$). Another focus of the first chapter is Itô calculus and its relationship with diffusion processes. In the following, we formulate a Dirichlet problem, by equipping the backward Kolmogorov's equation with boundary and initial conditions, and we show that in the one-dimentional setting the cumulative distribution function of the first exit time from the boundary solves the problem. Such result is easily obtained by the use of Itô formula. To handle the multidimensional case, we prove the existence of a weak solution of the equation under some additional hypotheses on both the process and the domain. Once this is done, we make use of some classical regularity results to prove the smoothness of such a solution, together with some standard stochastic representation arguments to identify it as the survival function of the first exit time of the process from the domain of the equation. In the last chapter, we will introduce semi-Markov processes, whose governing equations are not generally known. Such processes can be constructed as time-changed Markov processes in such a way to permit intervals of constancy in their paths. The time-change is done with the inverse process $(L_t)_{t\geq0}$ of a subordinator $(\sigma_t)_{t\geq0}$, i.e., a non-decreasing Lévy process started at zero. In particular, we will focus on the stable subordinator, which is obtained by choosing the Laplace exponent $\phi(\lambda)=\lambda^\alpha=\tfrac{\alpha}{\Gamma(1-\alpha)}\int_0^\infty(1-e^{-\lambda x})x^{-1-\alpha}dx$. The Lévy-Khintchine formula (Theorem 3.3) reveals that the stable subordinator has no drift ($d=0$) and infinite Lèvy mass, therefore it is strictly increasing and has countably many jumps in an arbitrary interval. We are going to study the case of the semi-Markov process obtained by changing the time of a Brownian motion with the inverse of the stable subordinator. Through clever use of the Laplace transform, we will give a proof of the fact that the first exit time from a boundary of such process solves a modification of the Dirichlet problem discussed in the second chapter, where we introduced a fractional derivative.