Time-Non-Local Pearson Diffusions

Abstract

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.

A Sojourn-Based Approach to Semi-Markov Reinforcement Learning

“…Moreover, a common strategy to construct continuous-time semi-Markov processes is through time-change (see [11][12][13]15] for some first considerations and, for instance, [21] for a more recent one), while this kind of approach has been considered, up to our knowledge, only in [27] in the discrete-time setting. Such an approach, in the continuous-time case, is revealed to be fruitful when combined with the theory of non-local operators (leading, in particular, to the generalized fractional calculus, see [18,33]) and different properties of solutions of non-local equations are achieved via stochastic representation results (see, for instance, [2,3,19,22]). This connection has not been fully explored yet in the discrete-time setting, but in [27] the authors achieved a first result on the link between fractional powers of the finite difference operator and the Sibuya distribution.…”

“…In Sect. 3 we use such a construction to provide a bivariate Markov approach to semi-Markov decision processes in which one can observe the state of the process also during the experiment. Previously, semi-Markov decision processes with continuous time allowed the agent to apply an action, that could modify the distribution of the inter-state time, only when the process changes its state (i.e.…”

Generalized Fractional Calculus for Gompertz-Type Models

“…Explicit formulae for such functions are not known in the general case, while in the standard fractional one they are recognized as Mittag-Leffler functions in [40]. In [41], the existence, uniqueness and spectral decomposition of exact solutions of some timenonlocal parabolic equations are obtained by means of stochastic representation results. Here, we will use both the representation of the eigenfunctions as obtained in [36][37][38] and the regularity properties of solutions of generalized fractional linear differential equations proved in [41].…”

On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

“…Much is known about this process, although a lack of results emerges when dealing with its version with non-zero asymptotic mean, namely the Inhomogeneous Geometric Brownian Motion (IGBM). The IGBM belongs to the class of Pearson diffusions [1,2] but goes under different names according to the field of study. In the interest rates field, it is called the Brennan-Schwartz model [3,4], denoted as the GARCH model when used for stochastic volatility and for energy markets [5], as Lognormal diffusion process with exogenous factors when used for forecasting and analysis of growth [6,7], in real option literature, it goes under the names of Geometric Brownian motion with affine drift [8,9], Geometric Ornstein-Uhlenbeck [10] or meanreverting Geometric Brownian motion [11].…”