A Primal–dual Algorithm for Bsdes

Abstract

We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least-squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time-discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5-dimensional nonlinear pricing problems where tight price bounds were previously unavailable.

Pathwise Dynamic Programming

“…The obvious advantage is that, in principle, this approach requires neither knowledge nor existence of associated primal and dual optimization problems. This paper generalizes the approach of Bender et al (2015) in various directions. For example, we merely require that the functions F j and G j are of polynomial growth while the corresponding condition in the latter paper is Lipschitz continuity.…”

“…Recently, Bender et al (2015) have proposed a posteriori criteria which are derived directly from a dynamic programming recursion like (1). The obvious advantage is that, in principle, this approach requires neither knowledge nor existence of associated primal and dual optimization problems.…”

“…Conceptually, our main departure from the approach of Bender et al (2015) is that we do not require that the recursion (1) satisfies a comparison principle. Suppose that two adapted processes Y low and Y up fulfill the analogs of (1) with "=" replaced, respectively, by "≤" and "≥".…”

Solving high-dimensional partial differential equations using deep learning

“…(see, e.g., [16][17][18][19][20]). Each of these effects results in a nonlinear contribution in the pricing model (see, e.g., [17,21,22]). In particular, the credit crisis and the ongoing European sovereign debt crisis have highlighted the most basic risk that has been neglected in the original Black-Scholes model, the default risk [21].…”

“…The link between (nonlinear) parabolic PDEs and backward stochastic differential equations (BSDEs) has been extensively investigated in the literature (see, e.g., [8,9,26,27] (17) for which we are seeking for a {F t } t∈[0,T ] -adapted solution process {(X t , Y t , Z t )} t∈[0,T ] with values in R d × R × R d . Under suitable regularity assumptions on the coefficient functions µ, σ, and f , one can prove existence and up-to-indistinguishability uniqueness of solutions (cf., e.g., [8,26]).…”