Applications of the Quadratic Covariation Differentiation Theory: Variants of the Clark-Ocone and Stroock's Formulas

Abstract

In a 2006 article [1], Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. In it, he defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic pathwise stochastic differentiation theory that comes complete with a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. In this current article, we use this differentiation theory in [1] to obtain variants of the celebrated Clark-Ocone and Stroock representation formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under L ²-type conditions on the random variable but with no L ^p conditions on the derivative. We do not use Malliavin calculus, weak distributional or Radon-Nikodym type derivatives, or the significant extra machinery of the Hida-Malliavin calculus. Moreover, unlike with Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond the standard Novikov condition. This is a consequence of the invariance under change of measure of the first author's derivative in [1]. The formulations and proofs are simple and natural applications of the differentiation theory in [1] and standard Itô integral calculus. Iterating our variants of the Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas and the theory in [1] by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator F = 𝕀_{[K, ∞)}(W _T ), which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator. The first author further extends and applies his differentiation theory in forthcoming articles and obtains a general stochastic calculus for a large class of processes with different orders and types of variations, including many that fall outside the classical Gaussian, Markovian, or semimartingale classes.

Keywords:

• Itô calculus
• Quadratic covariation stochastic derivative
• Quadratic covariation stochastic differentiation theory
• Stochastic calculus

Mathematics Subject Classification:

• 60H05
• 60H10
• 60H99
• 60G20
• 60G05

Notes

For any probability measure ℙ defined on ℱ _T  ⊂ ℱ and for any Y ∈ L ¹(Ω, ℙ), we will always assume that Y _t : = 𝔼[Y | ℱ _t ] is chosen from the equivalence class of 𝔼[Y | ℱ _t ] in such a way that the resulting martingale 𝔼[Y | ℱ _t ] = {𝔼[Y | ℱ _t ]; 0 ≤ t ≤ T} has paths that are right continuous with left limits (RCLL or cadlag) almost surely. This is of course possible by the right continuity and completeness of our filtration {ℱ _t }. Of course, this also means that if 𝔼[Y | ℱ _t ] is a modification of a continuous process X, then they are indistinguishable and 𝔼[Y | ℱ _t ] is continuous almost surely.