Functional Itô calculus and stochastic integral representation of martingales

Functional Ito calculus and stochastic integral representation of martingales

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by B Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative ad...

Stochastic Calculus of Variations for Martingales

The stochastic calculus of variations for the Wiener process, initiated in Malliavin , aims to obtain conditions for the regularity of the density of Wiener functionals given by the values of diffusion processes. It also developed as an extension to anticipating processes of the Itô calculus, by means of the Skorohod integral, cf. Nualart-Pardoux , Üstünel . In the case of point processes we ca...

Integral representation of martingales and endogenous completeness of financial models

Let Q and P be equivalent probability measures and let ψ be a Jdimensional vector of random variables such that dQ dP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that any local martingale under Q is a stochastic integral with r...

Stochastic integral representations of quantum martingales on multiple Fock space

In this paper a quantum stochastic integral representation theorem is obtained for unbounded regular martingales with respect to multidimensional quantum noise. This simultaneously extends results of Parthasarathy and Sinha to unbounded martingales and those of the author to multidimensions.

Stochastic Calculus ——– an introduction to option pricing with martingales ——–