ON MARTINGALE REPRESENTATIONS OF NON-SMOOTH BROWNIAN FUNCTIONALS

The martingale representation theorem (along with Girsanov’s measure change theorem) plays an important role in modern stochastic financial mathematics. The first proof of the martingale representation theorem was implicitly provided by Ito himself (1951). Subsequently, many other works have been written about the existence of the martingale representation and its applications, but one of the pioneering works in this direction is certainly the work of Clark (1970). On the other hand, given the needs of modern financial mathematics, it is not enough to know only the existence of an integral representation; it is necessary to be able to find an explicit form of the integrand of the integral representation. It is known that for stochastically smooth functionals, the integrand is calculated using Ocone’s formula (1984) as an optional projection of the stochastic derivative of the considered functional F ( \(E[D_tF|\Im _t]\) $E[D_{t}F|{\Im }_t]$ ). Ocone’s formula was later generalized by Glonti and Purtukhia (2017), when only the filter \(g_t:=E[F|\Im _t]\) $g_t:=E[F|{\Im }_t]$ (conditional mathematical expectation) of the functional is stochastically smooth, according to which the integrand is calculated as the limit of the integrand corresponding to the filter ( \(\lim \limits _{s\uparrow T}E[D_tg_s|\Im _t]\) $\underset{s\uparrow T}{lim}E[D_t{g}_s|{\Im }_t]$ in \(L_2([0,T]\times \Omega )\) $L_2([0,T]\times \mathrm{\Omega })$ ). Here, we study functionals whose filter is no longer smooth and propose a method for finding the integrand.