Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

We consider a system of fully coupled forward-backward stochasticdifferential equations. First we generalize the results ofPardoux-Tang [7] concerning the regularity of the solutions withrespect to initial conditions. Then, we prove that in some particularcases this system leads to aprobabilistic representation of solutions of a second-order PDE whosesecond order coefficients depend on the gradient of the solution. Wethen give some examples in dimension 1 and dimension 2 for which theassumptions are easy to check.

In this paper we study the problem of pricing contingent claims for a large investor (i.e. the coefficients of the price equation can also depend on the wealth process of the hedger) in an incomplete market where the portfolios are constrained. We formulate this problem so as to find the minimal solution of forward-backward stochastic differential equations (FBSDEs) with constraints. We use the penalization method to construct a sequence of FBSDEs without constraints, and we show that the solutions of these equations converge to the minimal solution we are interested in.