Let S(t) denote the price of a security at time t . A popular model for the process {S(t), t .transtutors.com/qimage/image09062014494.png” alt=” width=”> 0} supposes that the price remains unchanged until a shock occurs, at which time the price is multiplied by a random factor. If we let N(t) denote the number of shocks by time t , and let Xi denote the ith multiplicative factor, then this model supposes that
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Where.transtutors.com/qimage/image09062014496.png” alt=” width=”> is equal to 1 when N(t) = 0. Suppose that the Xi are independent exponential random variables with rate ?; that {N(t), t .transtutors.com/qimage/image09062014494.png” alt=” width=”> 0} is a Poisson process with rate ?; that {N(t), t .transtutors.com/qimage/image09062014494.png” alt=” width=”> 0} is independent of the Xi ; and that S(0) = s.
(a) Find E[S(t)].
(b) Find E[S2(t)].