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Abstract.In the present paper, we first deal with the discretization of stochastic differential equations. We elaborate on the analysis of the weak error of the Euler scheme by Talay and Tubaro [31] to contruct schemes with quicker weak rate of convergence for SDEs corresponding to an infinitesimal generator with smooth coefficients. We also extend this analysis to the case of a discontinuous drift coefficient. In a second part, we present two applications of stochastic gradient algorithms in finance.

We consider rate swaps which pay a fixed rate against a floating rate in presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit strategy optimizing an expected function of the hedging error. We here propose an efficient algorithm based on the stochastic gradient method to compute an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numerical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost.