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

Abstract

In this paper, we present a discrete-type approximation scheme to solve continuoustime optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct ǫ-optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent SDEs driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte-Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra, Ohashi and Russo.

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

“…In this work, we present a Monte Carlo scheme which applies to quite general states including path-dependent payoff functionals of path-dependent stochastic differential equations (henceforth abbreviated by SDEs), stochastic volatility and other non-Markovian systems driven by Brownian motion. The Monte Carlo scheme designed in this work is based on the methodology developed by Leão and Ohashi [26,27], Leão, Ohashi and Simas [28] and Leão, Ohashi and Russo [30]. In Leão, Ohashi and Russo [30], the authors present a discretization method which yields a systematic way to approach fully non-Markovian optimal stopping problems based on the filtration F. The philosophy is to consider the supermartingale Snell envelope…”

“…Therefore, the underlying state space is infinite-dimensional. By using techniques developed by [28,30], we are able to reduce the dimension of the Brownian noise by means of a suitable discrete-time filtration generated by…”

“…

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.

…”

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].…”

“…However, we are still restricted to a fixed probability measure. For partial results in the non-linear case, we refer the reader to [35].…”