Jacques Printems

Optimal quadratic quantization for numerics: the Gaussian case

Optimal quantization has been recently revisited in multi-dimensional numerical integration, multi-asset American option pricing, control theory and nonlinear filtering theory. In this paper, we enlighten some numerical procedures in order to get some accurate optimal quadratic quantization of the Gaussian distribution in one and higher dimensions. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ≥ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed from mathematical finance are used to test the accuracy of our Gaussian optimal quantizers.

Optimal Quantization Methods and Applications to Numerical Problems in Finance

We review optimal quantization methods for numerically solving nonlinear problems in higher dimensions associated with Markov processes. Quantization of a Markov process consists in a spatial discretization on finite grids optimally fitted to the dynamics of the process. Two quantization methods are proposed: the first one, called marginal quantization, relies on an optimal approximation of the marginal distributions of the process, while the second one, called Markovian quantization, looks for an optimal approximation of transition probabilities of the Markov process at some points. Optimal grids and their associated weights can be computed by a stochastic gradient descent method based on Monte Carlo simulations. We illustrate this optimal quantization approach with four numerical applications arising in finance: European option pricing, optimal stopping problems and American option pricing, stochastic control problems and mean-variance hedging of options and filtering in stochastic volatility models.

Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of

AN OPTIMAL MARKOVIAN QUANTIZATION ALGORITHM FOR MULTI-DIMENSIONAL STOCHASTIC CONTROL PROBLEMS

X_t

DISCRETIZATION AND SIMULATION OF THE ZAKAI EQUATION

Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as

A Quantization Tree Method for Pricing and Hedging Multidimensional American Options

dX_t+AX_t \, dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T],

Functional quantization for numerics with an application to option pricing

driven by a Gaussian space time noise whose covariance operator

A quantization method for pricing and hedging multi-dimensional american style options

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