Natural Decomposition of Processes and Weak Dirichlet Processes

Abstract

A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C 1 -transformations and under absolutely continuous change of measure. If a weak Dirichlet process has finite energy, as defined by Graversen and Rao, its Doob-Meyer type decomposition is unique. The developed methods have been applied to a study of generalized martingale convolutions. Mathematics Subject Classification: 60G48, 60H05

Weak Dirichlet processes with jumps

“…X is denominated special weak Dirichlet process if it admits a decomposition of the same type but where A is predictable. This concept is compatible with the one introduced in [8] using the discretization language. The authors of [8] were the first to introduce a notion of weak Dirichlet process in the framework of jump processes.…”

“…This concept is compatible with the one introduced in [8] using the discretization language. The authors of [8] were the first to introduce a notion of weak Dirichlet process in the framework of jump processes. The decomposition of a special weak Dirichlet process is now unique, see Proposition 5.9, at least fixing A 0 = 0.…”

On pathwise quadratic variation for càdlàg functions

“…Föllmer's definition [14] contains the condition (L) on the Lebesgue decomposition of the limit µ π : the atoms of µ should correspond exactly to the jumps of x and their mass should be |∆x(t)| 2 or, equivalently, the discontinuity points of [x] π should coincide with those of x, with ∆[x] π (t) = |∆x(t)| 2 . This condition can not be removed: as shown by Coquet et al [9], there are counterexamples of continuous functions x such that (1) converges to a limit with atoms. Conversely, one can give examples of discontinuous functions for which (1) converges to an atomless measure.…”

Strong existence and uniqueness for stable stochastic differential equations with distributional drift

“…Now let us consider a function v λ := u λ 0,f , where λ 1; this function is well-defined by Lemma 4.1. We apply Itô's formula for Dirichlet processes [10,Theorem 3.4] (see also [3,Theorem 5.15(ii)…”