Clark-Ocone type formula for non-semimartingales with finite quadratic variation

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

Infinite dimensional stochastic calculus via regularization with financial motivations

Calculus via regularization. [chi]-quadratic variation. Evaluations of [chi]-quadratic variations. Stability of [chi]-quadratic variation and of [chi]-covariation. Ito's formula. A generalized Clark-Ocone formula

On stochastic calculus related to financial assets without semimartingales

This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class $\mathcal{A}$ of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus related to $\mathcal{A}$-martingales with some examples is developed. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage conditions of Bender-Sottinen-Valkeila type

Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the study of strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution.Comment: arXiv admin note: text overlap with arXiv:1401.503

Infinite dimensional stochastic calculus via regularization

This paper develops some aspects of stochastic calculus via regularization to Banach valued processes. An original concept of $\chi$-quadratic variation is introduced, where $\chi$ is a subspace of the dual of a tensor product $B \otimes B$ where $B$ is the values space of some process $X$ process. Particular interest is devoted to the case when $B$ is the space of real continuous functions defined on $[-\tau,0]$, $\tau>0$. Itô formulae and stability of finite $\chi$-quadratic variation processes are established. Attention is deserved to a finite real quadratic variation (for instance Dirichlet, weak Dirichlet) process $X$. The $C([-\tau,0])$-valued process $X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process. Let $T >0$. If $X$ is a finite quadratic variation process such that $[X]_t = t$ and $h = H(X_T(\cdot))$ where $H:C([-T,0])\longrightarrow \R$ is $L^{2}([-T,0])$-smooth or $H$ non smooth but finitely based it is possible to represent $h$ as a sum of a real $H_{0}$ plus a forward integral of type $\int_0^T \xi d^-X$ where $H_{0}$ and $\xi$ are explicitly given. This representation result will be strictly linked with a function $u:[0,T]\times C([-T,0])\longrightarrow \R$ which in general solves an infinite dimensional partial differential equation with the property $H_{0}=u(0, X_{0}(\cdot))$, $\xi_{t}=D^{\delta_{0}}u(t, X_{t}(\cdot)):=Du(t,X_{t}(\cdot))(\{0\})$. This decomposition generalizes the Clark-Ocone formula which is true when $X$ is the standard Brownian motion $W$. The financial perspective of this work is related to hedging theory of path dependent options without semimartingales

Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes

This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of M\'etivier and Pellaumail which is quite restrictive. We make use of the notion of $\chi$-covariation which is a generalized notion of covariation for processes with values in two Banach spaces $B_{1}$ and $B_{2}$. $\chi$ refers to a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$. We investigate some $C^{1}$ type transformations for various classes of stochastic processes admitting a $\chi$-quadratic variation and related properties. If \X^1 and \X^2 admit a $\chi$-covariation, $F^i: B_i \rightarrow \R$, $i = 1, 2$ are of class $C^1$ with some supplementary assumptions then the covariation of the real processes F^1(\X^1) and F^2(\X^2) exist. A detailed analysis will be devoted to the so-called window processes. Let $X$ be a real continuous process; the $C([-\tau,0])$-valued process $X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for $B=\R$. If \X=X is a real valued Dirichlet process and $F:B \rightarrow \R$ of class $C^{1}(B)$ then F(\X) is still a Dirichlet process. If \X=X is a weak Dirichlet process with finite quadratic variation, and $F: C^{0,1}([0,T]\times B)$ is of class $C^{0,1}$, then [ F(t, \X_t) ] is a weak Dirichlet process. We specify corresponding results when $B=C([-\tau,0])$ and \X=X(\cdot). This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables

Representation of stochastic optimal control problems with delay in the control variable

In this manuscript we provide a representation in infinite dimension for stochastic optimal control problems with delay in the control variable. The main novelty consists in the fact that the representation can be applied also to dynamics where the delay in the control appears as a nonlinear term and in the diffusion coefficient. We then apply the representation to a LQ case where an explicit solution can be found

About classical solutions of the path-dependent heat equation

This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C([−T, 0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of final conditions: the first one is given by a cylindrical non necessarily smooth r.v., the second one is a smooth generic functional

Generalized covariation for Banach space valued processes, Ito formula and applications

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗̂πB2) ∗), we define the so-called χ-covariation of X and Y. If X = Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗̂πB1) ∗ then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−τ, 0])-valued process X: = X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h = H(XT (·)), H: C([−T, 0]) − → R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u: [0, T]×C([−T, 0]) − → R solving an infinite dimensional partial differential equation

GENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B_1 and B_2 and χ is a suitable subspace of the dual of the projective tensor product of B_1 and B_2 (denoted by (B_1 ⊗^^_ B_2)^), we define the so-called χ-covariation of X and Y. If X = Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier–Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B_1 ⊗^^_ B_2)^ then the -quadratic variation coincides with the quadratic variation of a B_1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B_1 = B_2 = C([-τ, 0]) for some τ > 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([-τ, 0])-valued process X = X(・) defined by X_t(y) = X_ , where y ∈ [-τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark–Ocone type for a class of path dependent random variables of type h = H(X_T(・)), H : C([-T,0])→R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u : [0, T] × C([-T, 0]) → R solving an infinite dimensional partial differential equation