Functional quantization of the Brownian motion and the Brownian bridge

If is a bi-measurable stochastic process on , it can be considered as a random variable valued in the Hilbert space .

In [1], it is shown that in the Gaussian case, if the covariance function is continuous, linear subspaces of spanned by -stationary codebooks correspond to principal components of , in other words, are spanned by eigenvectors of the covariance operator of .

Thus, the quantization consists first in exploiting the Karhunen-Loève decomposition. The discretization consists in truncating the decomposition at a fixed order and to quantize the -value Gaussian vector constituted of the first coordinates of the process on its Karhunen-Loève decomposition.

To reach optimal quantization, one has both to determine the optimal rank of truncation (the quantization dimension) and to determine the optimal -dimensional Gaussian quantizer corresponding to the first coordinates.

Formally, if is a bi-measurable Gaussian process, with a continuous covariance function, its Karhunen-Loève expansion writes:

where is a sequence of independent Gaussian random variables.

The terms of the Karhunen-Loève decomposition are explicit for classical Gaussian processes (the standard Brownian motion, the Brownian bridge and the Ornstein-Uhlenbeck process).

• The Karhunen-Loève decomposition of the standard Brownian motion on is:
• The Karhunen-Loève decomposition of the standard Brownian bridge on is:

In this case, the -dimensional random vector to be quantized is a Gaussian vector with diagonal variance-covariance matrix with

Optimal quantization of the Brownian motion

brownian_optimal_grids.zip

The compressed folder brownian_optimal_grids.zip contains optimal quantization grids of the standard Brownian motion.

To get optimal quantization, the point now is to quantize the finite-dimensional Gaussian vector optimally.

Hence, the method is the same as for the standard distribution except that the simulated Gaussian vector is not standard. For a given size , all possible dimensions are tested, and the one that yields the smaller quadratic distortion () is kept.

For a given size , the text files are organized as follows. It presents in the form of a matrix with rows and columns.

• On row : Element of the grid and its companion parameters. Consider
• On last row :

In particular we can verify that

and

For further details and further reading, let us refer to [1].

Product quantization of the Brownian motion and the Brownian bridge

An other way to get a good quantizer of a Gaussian process is Product Quantization. In practice, being settled, one determines the truncation threshold of the decomposition and then, is approximated by where is a quantizer of the -valued random vector .

The product quantization consists in choosing the quantizer of as a Cartesian product of one dimensional quantization grids.

Thus, one replaces by where , are optimized quantizers of the -dimensional Gaussian distribution, of size , and where the values are such that . A database of optimal quadratic quantizers of the standard Gaussian distribution is available here.

After all, one has for a settled integer to determine among all its possible product decomposition the one that minimizes the distortion error.

In article [2], the optimal product decompositions are used to compute Asian option prices in a stochastic volatility model.

 RECORD_QF.TXT RECORD_QF_BB.TXT

The text file RECORD_QF.TXT contains optimal product decompositions for the standard Brownian motion of size to .

The text file RECORD_QF_BB.TXT contains optimal product decompositions for the standard Brownian bridge of size to .

The both cases two first columns give, for a number , the value of the distortion of the optimal product quantization.

The following columns give the size of the best product quantizer for a maximum number of points of , and the corresponding distortion. At least, the corresponding optimal product decomposition is given.

References

1. Harald Luschgy, and Gilles Pagès, Journal of Functional Analysis, vol. 196, no. 2: Academic Press, pp. 486–531, December, 2002.
2. Gilles Pagès, and Jacques Printems, Monte Carlo Methods and Appl., vol. 11, no. 11, pp. 407-446, 2005.