  A new asymptotic expansion scheme for backward SDEs (BSDEs) is proposed.The
perturbation parameter is introduced just to scale the forward stochastic
variables within a BSDE. In contrast to the standard small-diffusion asymptotic
expansion method, the dynamics of variables given by the forward SDEs is
treated exactly. Although it requires a special form of the quadratic
covariation terms of the continuous part, it allows rather generic drift as
well as jump components to exist. The resultant approximation is given by a
polynomial function in terms of the unperturbed forward variables whose
coefficients are uniquely specified by the solution of the recursive system of
linear ODEs. Applications to a jump-extended Heston and lambda-SABR models for
European contingent claims, as well as the utility-optimization problem in the
presence of a terminal liability are discussed.
