Analytical approximations of BSDEs with nonsmooth driver

We provide and analyze analytical approximations of backward SDEs in the limit of small nonlinearity and short time, in the case of nonsmooth drivers. We identify the first and second order approximations within these asymptotics and consider two topical financial applications: the two interest rates pricing problem and the funding value adjustment. In the high dimensional diffusion setting, we show how to compute explicitly the first order formula by taking advantage of recent proxy techniques. Numerical tests up to dimension 20 illustrate the efficiency of the numerical schemes. We additionally investigate higher order expansions, which may hold under additional assumptions; we also provide a counterexample where the third order expansion fails to exist.