Modern Stochastics: Theory and Applications

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes.

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364].

The main object of this paper is the planar wave equation with random source f. The latter is, in certain sense, a symmetric α-stable spatial white noise multiplied by some regular function σ. We define a candidate solution U to the equation via Poisson’s formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that U is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where σ does not vanish. Finally, we prove that U is a generalized solution to the equation.

We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process Z with Hurst parameter $H>1/2$ and stability index $\alpha >1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for Z, converge to the solution.