Itô-Skorohod stochastic equations and applications to finance

The introduction of the anticipating (or Skorohod) integral in [8] and of the anticipating stochastic calculus in [7] has opened the question of solving anticipating stochastic differential equations. In general, the existence and uniqueness of the solution for these equations is not known. The difficulty of solving such equations is due to the fact that the classical method of Picard’s iterations cannot be applied because the mean square formula for the Skorohod integral involves the Malliavin derivation in a such way that we cannot find ”closed” formulas. Only in few particular cases some results exist, see for example [1], [2] or [3]. We have recently proved in [9] that the set of Skorohod integrals coincides with a set of integrals of Itô type. In the present work, using this correspondence between Skorohod integrals and Itô-Skorohod integrals, we introduce a class of anticipating equations (called Itô-Skorohod equations) that can be solved using standard techniques. As an application we introduce a market model where the price of the risky asset follows such an equation with a random initial condition (the price at the transaction time). We prove that our model is complete and has no arbitrage opportunities and we derive a Black-Scholes formula when the initial price of the risky asset is given by a standard normal random variable. We organized the paper as follows. Section 2 contains some preliminaries on the anticipating stochastic calculus. In Section 3 we define the class of Itô-Skorohod equations and we prove the existence and uniqueness of the solution. In Section 4 we introduce a market model with price dynamic following an Itô-Skorohod equation and we obtain a Black-Scholes option valuation formula and the expression of the replicant portfolio.