Multidimensional Stochastic Processes as Rough Paths: Free nilpotent groups

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.

A weak version of path-dependent functional It\^o calculus

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t. Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite p-variation ( p ≥ 2 ) regularity is also discussed. Under stronger regularity conditions in the sense of finite ( p , q ) -variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.

Weak differentiability of Wiener functionals and occupation times

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth, and satisfy some other conditions, then there exists a solution-giving C0-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e., as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth, and satisfy some other conditions, then there exists a solution-giving C0-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e., as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.

Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the $C_0$-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth and satisfy some other conditions then there exists a solution-giving $C_0$-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e. as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.

In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales. Under finite (p, q)variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition Ft(Xt) = F0(X0) + ∫ t

N.N. Shamarov

Papers

Path-dependent Ito formulas under (p,q)-variations