A class of integration by parts formulae in stochastic analysis I

TL;DR

This paper develops a class of integration by parts formulae in stochastic analysis on path spaces, establishing foundational tools like the Laplacian and Clark-Ocone formula, with proofs leveraging Markov properties and Bismut's approach.

Contribution

It introduces a new class of integration by parts formulae in stochastic path space analysis, connecting them with existing formulas and providing novel proofs.

Findings

Derived a general integration by parts formula from the Logarithmic Sobolev inequalities.

Showed the formula implies the Clark-Ocone martingale representation.

Connected the formulae with Bismut's original approach using torsion.

Abstract

An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s\rangle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark-Ocone formula (martingale representation theorem with specific integrand) and Logrithmic Sobolev inequalities. Interpreted appropriately, the latter formula is obviously a special case of the integration by parts formula. Here we show by the Markov property and by induction that the latter formula implies the integration by parts formula. WE also use Bismut's original approach to prove an integration by parts formula, using a connection with torsion and one on the free path space.