On Itô formulas for jump processes

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of $$L_p$$ L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.

White noise delta functions and infinite-dimensional Laplacians

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion

Let B = B t 1 , … , B t d t ≥ 0 be a d -dimensional bifractional Brownian motion and R t = B t 1 2 + ⋯ + B t d 2 be the bifractional Bessel process with the index 2 HK ≥ 1 . The Itô formula for the bifractional Brownian motion leads to the equation R t = ∑ i = 1 d ∫ 0 t B s i / R s d B s i + HK d − 1 ∫ 0 t s 2 HK − 1 / R s d s . In the Brownian motion case K = 1 and H = 1 / 2 , X t ≔ ∑ i = 1 d ∫ 0 t B s i / R s d B s i , d ≥ 1 is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process X t is not a bifractional Brownian motion unless K = 1 and H = 1 / 2 . We also study some other properties and their application of this stochastic process.

An Itô Formula for rough partial differential equations and some applications

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u$∂tu−Atu−f=(Ẋt(x)⋅∇+Ẏt(x))u on $[0,T]\times \mathbb {R}^{d}.$[0,T]×ℝd. To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u↦F(u), where F is C2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F(u) = |u|p with p ≥ 2, but also when Y ≠ 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of Lp-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.

Change of variable formulas for non-anticipative functionals

For non-anticipative functionals, differentiable in Chitashvili’s sense, the Itô formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.

Dynamical Behavior of a Nonautonomous Stochastic Modified Bazykin Model

In this article, a nonautonomous stochastic modified Bazykin model is introduced. The positive solution is proved to be unique and global for any initial data via stochastic comparison theorem and Itô formula. Stochastic ultimate boundedness and stochastic permanence are also considered. Finally, some simulations are performed to verify the validity of results.