Itô formula for an asymptotically $4$-stable process

Abstract

Abstract. An Itô-type formula is given for an asymptotically 4-stable process.

The first hitting time and place of a half-line by a biharmonic pseudo process

“…Further to them, Burdzy and Madrecki [5] treated BPP as an extension of the usaul stable process and defined stochastic integrals and Ito's formula.…”

High order heat-type equations and random walks on the complex plane

“…Complex valued processes, connected to PDE of the form (3) have been also proposed by other authors by means of different techniques [6,26,7]. It is worthwhile also to mention a completely different approach proposed by R. Léandre [23], which has some analogies with the mathematical realization of Feynman path integrals by means of white noise calculus [13].…”

Explicit solutions of some fractional partial differential equations via stable subordinators

“…This process is called Krylov motion or pseudoprocess and it has p 2n (t,x) as transition density [27]. After that, the question how to generalize the stochastic calculus to pseudoprocesses gave rise to several papers [4,13,22,[25][26][27], and so forth, where particular questions such as stochastic integral, Itô formula, first hitting time and first hitting place, Girsanov formula, stochastic differential equations, have been studied. Another formal way is to use the subordination analogy theory as in [9,23].…”

“…In [4], Burdzy and Madrecki gave the process connected to the fourth-order equation as a "wide limit" of a sequence of C N -valued random processes. In this construction, they used three independent Brownian motions on a product of two probabilistic spaces, and they mentioned that they can replace two of them by stable processes.…”