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Let W be the random measure in R2+, the positive quadrant of the plane, which assigns to each Borel set A a Gaussian random variable of mean zero and variance re(A), where m is Lebesgue measure, and which assigns independent random variables to disjoint sets (see [6], [14], [15], [19] and [20]). It is natural to construct stochastic integrals with respect to W (see [1], [4], [7], [10], [13], [15], [18] and [20]) but one can do more. Define a process W={W~, zeR2+} by Wz=W(Rz) , where Rz is the rectangle whose lower left hand corner (2) E{X((z, z'])] :~} =0/or each ~<<='. (b) X is an i-martingale(2) z{x((z, ~'])l:~} =o/or ~ach ~-<-<~'.(c) X is a strong martingale i/(1) X is adapted;(2) X vanishes on the axes;(3) ~{x((z, ~'])l:~ v y~}=0/or ~ach ~<<~'.Thanks to hypothesis (F4) we have the following proposition:PROPOSITION 1.1. A martingale is both a 1-and a 2.martingale.Proo/. Suppose X is a martingale and let A=((s, t), (s', t')], where s

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We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank-Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h 1/2 + k 1/4 , which is known to be optimal. We also consider cases where k is much greater than h 2 , and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank-Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method. (2000): 60H15, 60H35, 65N30, 35R60. Mathematics Subject Classifications