A weak version of path-dependent functional Itô calculus

Abstract

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.MSC 2010 subject classifications: Primary 60H07; secondary 60H25

Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

“…No.x Qiu & Wei: VISCOSITY SOLUTIONS OF STOCHASTIC HJ EQUATIONS 5 the space C 2 F in [28], by Definition 2.1, we have in fact characterized the two linear operators d t and d ω which is consistent with the two differential operators w.r.t. the paths of Wiener process W in the sense of [17], defined via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales; in particular, for the operator d ω u, an earlier discussion may be found in [4,Section 5.2]. We would also note that the operators d t and d ω here are different from the path derivatives (∂ t , ∂ ω ) via the functional Itô formulas (see [2] and [10, Section 2.3]).…”

“…The Doob-Meyer decomposition theorem implies the uniqueness of the pair (d t u, d ω u) and thus makes sense of the linear operators d t and d ω which actually coincide with the two differential operators introduced by Leão, Ohashi and Simas in [17,Theorem 4.3]. In fact, an earlier discussion on operator d ω u may be found in [4,Section 5.2].…”

Weak differentiability of Wiener functionals and occupation times

“…Inspired by the discretization scheme introduced by Leão and Ohashi [32], Leão, Ohashi and Simas [33] have developed a non-anticipative differential calculus over the space of square integrable Wiener functionals which proves to be a weaker notion of the functional Itô calculus studied by previous works [17,8,9,10,43,13,36,37]. The building block of the theory is composed by what we call a stable imbedded discrete structure Y = (X k ) k≥1 , D (see Definition 2.6) equipped with a continuoustime random walk approximation D = {T , A k ; k ≥ 1} driven by a suitable class of waiting times T = {T k n ; n, k ≥ 1}.…”

“…The class T describes the evolution of the Brownian motion at small scales and (X k ) k≥1 is a suitable type of approximation (adapted to A k ) for a given target process X. In [33], the authors show that a large class of Wiener functionals admits a stable imbedded discrete structure which has to be interpreted as a rather weak concept of continuity w.r.t Brownian motion driving noise. One major advantage of this theory is the possibility of computing the sensitivities of X w.r.t B via simple and explicit differential-type operators based on D. For a concrete application of this theory, we refer the reader to [34,35,5].…”

“…where dB is the Itô integral w.r.t a Brownian motion B and V X is a continuous process which carries all possible orthogonal variations (in a sense made precise in [33]) of X w.r.t B. This class of Wiener functionals are called weakly differentiable (see Definition 2.7) which turns out to encompass semimartingales and other classes of processes with unbounded variation drifts.…”