In this article, we propose several quantization-based strati ed sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of strati cation lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with strati ed sampling. We rst put the emphasis on the consistency of quantization for partitioning the state space in strati ed sampling methods in both nite and in nite-dimensional cases. We show that the proposed quantization-based strata design has uniform e ciency among the class of Lipschitz continuous functionals. Then a strati ed sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multifactor di usions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein-Uhlenbeck processes. We derive in detail the case of Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction factor.

The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.

In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen-Loève coordinates of a continuous Gaussian semimartingale X.Using filtration enlargement techniques, we prove that the conditional distribution of X knowing its first Karhunen-Loève coordinates is a Gaussian semimartingale with respect to a bigger filtration. This allows us to define the partial quantization of a solution of a stochastic differential equation with respect to X by simply plugging the partial functional quantization of X in the SDE.Then we provide an upper bound of the L p -partial quantization error for the solution of SDEs involving the L p+ε -partial quantization error for X, for ε > 0. The a.s. convergence is also investigated.Incidentally, we show that the conditional distribution of a Gaussian semimartingale X, knowing that it stands in some given Voronoi cell of its functional quantization, is a (non-Gaussian) semimartingale. As a consequence, the functional stratification method developed in [6] amounted, in the case of solutions of SDEs, to using the Euler scheme of these SDEs in each Voronoi cell. (1) is an optimal quantizer of X. The corresponding quantization error is denoted by A solution toOne usually drops the p subscript in the quadratic case (p = 2). This problem, initially investigated as a signal discretization method [10], has then been introduced in numerical probability to devise cubature methods [24] or to solve multidimensional stochastic control problems [3]. Since the early 2000's, the infinite-dimensional setting has been extensively investigated from both constructive numerical and theoretical viewpoints with a special attention paid to functional quantization, especially in the quadratic case [18] but also in some other Banach spaces [29]. Stochastic processes are viewed as random variables taking values in functional spaces.

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