Dorival Leão

Weak approximations for Wiener functionals.

In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The discretization is given in terms of discrete-jumping filtrations which allow us to approximate nonsmooth processes by means of a stochastic derivative operator on the Wiener space. As a by-product, we provide a robust semimartingale approximation for weak Dirichlet-type processes. The underlying semimartingale skeleton is intrinsically constructed in such way that all the relevant structure is amenable to a robust numerical scheme. In order to illustrate the results, we provide an easily implementable approximation scheme for the classical Clark-Ocone formula in full generality. Unlike in previous works, our methodology does not assume an underlying Markovian structure and does not require Malliavin weights. We conclude by proposing a method that enables us to compute optimal stopping times for possibly non-Markovian systems arising, for example, from the fractional Brownian motion.

A General Multidimensional Monte Carlo Approach for Dynamic Hedging under Stochastic Volatility

In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology can be applied to quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate the method with numerical examples based on generalized Follmer-Schweizer decompositions, locally-risk minimizing and mean-variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models.

Corrigendum Weak approximations for Wiener functionals [Ann. Appl. Probab. (2013) 23 1660–1691]

The proofs of Theorem 3.1 and Corollary 4.1 in Le\~ao and Ohashi (2013) are incomplete. The reason is a wrong statement in Remark 2.2. The hypotheses and statements of Theorem 3.1 and Corollary 4.1 in Le\~ao and Ohashi (2013) remain unchanged but the proofs have to be modified. In this short note, we provide the details.