Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II

Abstract

In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leão, Ohashi and Russo [30] and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes such as functionals of pathdependent stochastic differential equations and fractional Brownian motions. Based on statistical learning theory techniques, we provide overall error estimates in terms of concrete approximation architecture spaces with finite Vapnik-Chervonenkis dimension. Analytical properties of continuation values for path-dependent SDEs and concrete linear architecture approximating spaces are also discussed.

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

“…, e(k, T ) − 1, so that one can postulate max Z k 0 (0); U k 0 (0) as a possible approximation for (3.8). Having said that, we are now able to produce a Longstaff-Schwartz-type Monte-Carlo scheme as demonstrated by [7] which we briefly outline here. For further details, we refer the reader to [7].…”

“…In order to compute a precise number of steps in our scheme, we just need to apply Propositions 5.1 and 5.3 combined with Corollary 4.1 in [7]. In this case, similar to the classical Markovian case, smoothness of the continuation values U k j ; j = 0, .…”

“…Next, we investigate the global numerical error e = e 1 + e 2 one may occur in a concrete fully non-Markovian example. The error e can be decomposed as the sum of two terms: the first one (e 1 ), which we study in this paper, is the discrete-type filtration approximation error; the second one (e 2 ), which we study in [7], it is related to the numerical approximation of the conditional expectations associated with the continuation values. By using the fact that α and F only depend on the present and performing a similar computation as described in the proof of Th 5.1 in [7], b k n → U k n (b k n ) is Lipschitz from S n k to R, where S n k := ((0, +∞) × B r (0)) × .…”

“…The underlying state A k * e(k * ,T ) lives in a e(k * , T )(d + 1)-dimensional space. A concrete Monte-Carlo scheme is developed in the companion paper by Bezerra, Ohashi and Russo [7]. Lastly, we mention that there is no conceptual obstruction to compute sensitivities of S w.r.t B (hedging strategies) by projecting the optimization problem onto the filtration (F k t ) 0≤t≤T and working with the differential operators introduced in [28] which describe Doob-Meyer decompositions for test processes.…”

Weak differentiability of Wiener functionals and occupation times

“…In [33], the authors show that a large class of Wiener functionals admits a stable imbedded discrete structure which has to be interpreted as a rather weak concept of continuity w.r.t Brownian motion driving noise. One major advantage of this theory is the possibility of computing the sensitivities of X w.r.t B via simple and explicit differential-type operators based on D. For a concrete application of this theory, we refer the reader to [34,35,5].…”