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139) Recall that a martingale process, which we shall here simply denote by ft , is a stochastic process for which EtP fT = ft , t ≤ T , under a given probability measure P. Recall that this is a driftless process, in the sense that its expected value, under P, is constant over all future times. We have already encountered a simple example of such a process, namely, the standard Brownian motion, or Wiener process Wt . 90) provides a method of generating a martingale process. Based on Itˆo’s Lemma we now have the following result.

6. That is, by considering E exp Wti for nonzero parameter and applying a Taylor expansion of the exponential and matching terms in the power series in n , one obtains E Wti n for any n ≥ 0. For this problem you only need terms up to n = 4. Problem 5. 86) approaches the onedimensional Dirac delta function x − x0 in the limit t → 0. Problem 6. (i) Obtain the joint marginal pdf of the random variables Ws and Wt , s = t. Evaluate E Wt − Ws 2 for all s t ≥ 0. (ii) Compute Et Ws3 for s > t. Problem 7.

The foregoing derivation of Itˆo’s lemma for one underlying random variable can be xn t depending on n random variables extended to the general case of a function f x1 xn and time t. 5 Stochastic Differential Equations and Itˆo’s Formula 35 Here the coefficients ai = ai x1 xn t and bi = bi x1 xn t are any smooth functions of the arguments. g. 131) k=1 When i = j this gives E dxi 2 = bi2 dt. 133) This procedure can be straightforwardly applied or extended to stochastic differentials of various processes that are dependent on groups of underlying random variables.