Approximating Random Variables by Stochastic Integrals

Least-squares Approximation of Random Variables by Stochastic Integrals∗

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale X. The criterion is minimization of L−distance, or “least-squares”. This problem has a straightforward and well-known solution when X is a Brownian motion or, more generally, a square-integrable martingale, with res...

Formulas for approximating pseudo-Boolean random variables

We consider {0, 1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Ham...

Random Variables and Stochastic Processes

Random Generation of Stochastic Area Integrals

We describe a method of random generation of the integrals A 1;2 (t; t + h) = Z t+h t Z s t dw 1 (r)dw 2 (s) ? Z t+h t Z s t dw 2 (r)dw 1 (s) together with the increments w 1 (t+h)?w 1 (t) and w 2 (t+h)?w 2 (t) of a two-dimensional Brownian path (w 1 (t);w 2 (t)). The method chosen is based on Marsaglia's `rectangle-wedge-tail' method, gen-eralised to higher dimensions. The motivation is the ne...

Multidimensional Stochastic Ordering and Associated Random Variables