A representation theorem for smooth Brownian martingales

Abstract

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a "frozen path". The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.

Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

“…In the first part, we give a rigorous proof of the result (7). Indeed, we complete the proof of Theorem 2.3 in our article [5]; although the statement was correct in that paper, one step of the proof was not obvious to finish. In the second part we characterize the generator of the semilinear PPDE.…”

“…The following theorem is a generalization of Theorem 2.2. in [5] to functionals that are not discrete.…”

“…For a detailed introduction, we refer to [8] and our paper [5]. Let Ω = ([0, ], R) and (Ω, F, {F } ≥0 , P) be the complete filtered probability space, where the filtration {F } ≥0 is the usual augmentation of the filtration generated by Brownian motion on R. The canonical Brownian motion can be also denoted by ( ) = ( , ) = ( ), ∈ [0, ], ∈ Ω, by emphasizing its sample path.…”

“…By Itô's lemma, it is obvious that V( ( ), ) is a martingale, say , and that the value of this martingale is the conditional expectation at time of Ψ( ( )). Consider now a general path-dependent terminal condition Ψ( ), in [5], Jin et al gave a new representation of Brownian martingales (with ≤ ) as an exponential of a timedependent generator, applied to the terminal value ≡ Ψ( ):…”

“…An important difference is that is now viewed as a random variable, and the (first-order) Malliavin derivative is a stochastic process in 2 International Journal of Stochastic Analysis the canonical probability space for Brownian motion. The stopping path operator was introduced in [5]. Informally, the action of the stopping path operator (which we define rigorously later) is to freeze the path after time :…”