Convergence of Multi-Dimensional Quantized SDE’s

Abstract

We quantize a multidimensional SDE (in the Stratonovich sense) by solving the related system of ODE's in which the d-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the ODE converge toward the solution of the SDE. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for 1 q -Hölder distance, q > 2, in L p (P).

A variance reduction technique using a quantized Brownian motion as a control variate

“…On the other hand, G. Pagès and A. Sellami have proved in [26] With the previous results, we have seen that it is sufficient to consider how close to B is B i (B) . The distance between B and B i(B) is due to three factors: (i) the truncation number m; (ii) the quantization error on the coefficients ξ; (iii) the difference between the exact Karhunen-Loève decomposition and the one given by the least squares approximation, but which can be neglected in practice for good choices of orthonormal basis of L 2 ([0, T ]; R) if one chooses to quantize normal distribution with covariance matrix equal to Id.…”

“…The functional quantization has been proposed and studied by G. Pagès et al as a way to compute quickly an approximation of E[Φ(B)] with the expression E[Φ( B)] (see [18,26] This is important, especially when diffusion processes are simulated, since one can choose the most convenient way to do so and only record the vector of marginals of the underlying Brownian motion.…”

On rough differential equations

“…The Lipschitz continuity of the Itô map can be important for applications, since it allows one to deduce for example rates of convergence of approximate solutions of differential equations from the approximations of the driving signal (see for example [18] for a practical application to functional quantization).…”

Partial functional quantization and generalized bridges

“…This work provides a better justification of the functional stratification scheme of [6]. The second observation is that one of the main purposes of the (full) functional quantization of X is to perform a quantization of the solution of a SDE with respect to X, when a stochastic integration with respect to X can be defined (see [25,20,27]). As (full) functional quantizers of X will typically have bounded variations, one needs to add a correction term to the SDE.…”