David Neal

Determining Sample Sizes for Monte Carlo Integration

In an introductory course in probability, one discusses the use of random samples for estimating an unknown population average /jl. If one chooses a large enough random sample xv...,xn, then by the law of large numbers, the sample average x = (l/n)Ef=1Jc/ should reasonably approximate jjl. We can use this fact to esti? mate many types of averages via a Monte Carlo simulation. For example, to estimate the average value of a continuous function / over an interval [a,b],_one selects a random sample of points Xx,...,Xn from [a, b]. Then, Xn = (l/7z)??=1/(^) converges in probability to jjl = \/(b a)f*f(x)dx [2, p. 157]. The solution of the Dirichlet problem for Laplaces equation offers another interesting application of the Monte Carlo method [3, p. 35]. Although it may be interesting to make Monte Carlo approximations, it may not always be very practical or efficient. In particular, if students are limited to manual methods, such as coin tossing or dice throwing, the problem of obtaining a large enough sample will arise. For surveys, one may also encounter the problem of ensuring that samples are truly random. Fortunately, for mathematical computations such as Monte Carlo integration, a computer can provide both a pseudo-random number generator and the power to handle large samples. For example, in Mathematica, one can use the command

Characterization ofO-summable processes

For a Banach-valued martingaleX, we define anL1-valued measureJX on an algebra of stochastic intervals which generates the optional σ-algebraO. We discuss conditions for when the measure has a countably additive extension toO, that is, for whenX isO-summable. For a process of integrable variationV, we define another countably additive measureIV onO. The existence of these measures allows for the definition of stochastic integrals of optional processes with respect to these Banach-valued processesX andV.

The Optional Stochastic Integral

In this paper we shall study the optional (or compensated) stochastic integral CH·X. The two main problems connected with this integral will be considered. First, we wish to express HC·X in terms of an ordinary predictable stochastic integral H’·X, where H’ is a suitable predictable process associated with the optional process H. An attempt in this direction was first undertaken by Yor [8]; however, even for bounded, scalar H, the problem remained open. We shall show in this case that HC·X — H’·X exists as a certain limit in M2, the space of cadlag (Hilbert-valued) square integrable martingales, (cf. §3). Secondly, we shall develop HC·X for processes H and X which take their values in a separable Hilbert space. These integrals, in turn, will allow us in a later paper to develop HC·X for certain nuclear-valued processes. Full details of the proofs of the theorems presented here will appear elsewhere.

Commutation of variation and dual projection

For a raw process of integrable variation V, taking values in a Banach space E having the Radon-Nikodyn property, the variation of the predictable (optional) dual projection is the predictable (optional) dual projection of the variation. An analogous result holds for the associated stochastic measures. The result is applied to the stochastic integral of a real, optional process H with respect to V when V is adapted.