Convergence of multi-dimensional quantized $SDE$'s

Gilles Pagès (1) , Afef Sellami (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We quantize a multidimensional $SDE$ (in the Stratanovich sense) by solving the related $ODE$'s in which the Brownian motion has been replaced by the components of stationary quantizers. We make a connection with rough path theory to show that such quantizations converge toward the solution of the $SDE$. In some particular cases, we show that this procedure provide some rate optimal quantizations of the equation.

Mots clés: Functional quantization; Stochastic Differential Equations; Stratanovich stochastic integral; stationary quantizers; rough path theory; p-variation; Hölder norm

Domaines: Probabilités [math.PR]

Gilles Pagès, Afef Sellami. Convergence of multi-dimensional quantized $SDE$'s. 2008. ⟨hal-00202297v1⟩