Abstract:
A white noise proof of the classical Clark-Ocone formula is first provided. This formula is proven for functions in a Sobolev space which is a subset of the space of square-integrable functions over a white noise space. Later, the formula is generalized to a larger class of operators.

Abstract:
In this paper we explore the fundamentals of the Martingale Representation Theorem (MRT) and a closely related result, the Clark-Ocone formula. We also investigate how far these theorems can be taken, notably beyond the regular Sobolev spaces, through changes of measures and enlargement of filtrations. We look at Brownian motion (B.M.) driven continuous martingales as well as Jump and Levy process-driven martingales.

Abstract:
Hedging strategies in bond markets are computed by martingale representation and the Clark-Ocone formula under the choice of a suitable of numeraire, in a model driven by the dynamics of bond prices. Applications are given to the hedging of swaptions and other interest rate derivatives, and our approach is compared to delta hedging when the underlying swap rate is modeled by a diffusion process.

Abstract:
In this paper we first prove a Clark--Ocone formula for any bounded measurable functional on Poisson space. Then using this formula, under some conditions on the intensity measure of Poisson random measure, we prove a variational representation formula for the Laplace transform of bounded Poisson functionals, which has been conjectured by Dupuis and Ellis [A Weak Convergence Approach to the Theory of Large Deviations (1997) Wiley], p. 122.

Abstract:
In this paper, we will establish a discrete-time version of Clark(-Ocone-Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense. The formula is applied to the estimation of the error caused by the martingale representation. In the way, we use another distribution theory with respect to Gaussian rather than Lebesgue measure, which can be seen as a discrete Malliavin calculus.

Abstract:
In this paper, we construct a Malliavin derivative for functionals of square-integrable L\'evy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a L\'evy process.

Abstract:
We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. If $X$ is a real finite quadratic variation process, an appropriated It\^o formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.

Abstract:
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for It\^o's integral calculus. 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 differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to It\^o's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under $L^{2}$-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike 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 Novikov's condition. This is due to the invariance under change of measure of the first author's derivative in \cite{A1}. The formulations and proofs are natural applications of the differentiation theory in \cite{A1} and standard It\^o integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator $F=\mathbb{I}_{[K,\infty)}(W_{T})$, 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.

Abstract:
We generalise the Clark-Ocone formula for functions to give analogous representations for differential forms on the classical Wiener space. Such formulae provide explicit expressions for closed and co-closed differential forms and, as a by-product, a new proof of the triviality of the L^2 de Rham cohomology groups on the Wiener space, alternative to Shigekawa's approach [16] and the chaos-theoretic version [18]. This new approach has the potential of carrying over to curved path spaces, as indicated by the vanishing result for harmonic one-forms in [6]. For the flat path group, the generalised Clark-Ocone formulae can be proved directly using the It\^o map.

Abstract:
We establish a martingale representation formula for functionals of diffusion processes with Lipschitz coefficients, as stochastic integrals with respect to the Brownian motion.