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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES CONTINUOUSTIME STOCHASTIC PROCESSES DiscreteTime Random Walk The concept of a Wiener process is an ex trapolation of that of a discretetime random walk. A standardised random walk is a process that is defined over the set of integers { t = 0 , 1 , 2 , . . . } , which represent dates separated by a unit time interval. It may be denoted by w ( t ) = { w t ; t = 0 , 1 , 2 , . . . } , and it can be represented by the equation w t +1 = w t + t +1 , (1) wherein t is an element of a sequence ( t ) = { t ; t = 0 , 1 , 2 , . . . } of inde pendently distributed standard normal random elements with a mean value of of E ( t ) = 0 and a variance of V ( t ) = 1, for all t , which is described as a whitenoise process. By a process of backsubstitution, the following expression can be derived: w t = w + t + t 1 + + 1 . (2) This depicts w t as the sum of an initial value w and of an accumulation of stochastic increments. If w has a fixed finite value, then the mean and the variance of w t , conditional of this value, are be given by E ( w t  w ) = w and V ( w t  w ) = t. (3) There is no central tendency in the randomwalk process; and, if its starting point is in the indefinite past rather than at time t = 0, then the mean and variance are undefined. To reduce the random walk to a stationary stochastic process, it is neces sary only to take its first differences. Thus w t +1 w t = t +1 . (4) We should also observe that we can take larger steps through time without fundamentally altering the nature of the process. Let h be any integral number of time periods. Then w t + h w t = h X j =1 t + j = t + h h, (5) where t + h is an element of a sequence { t + jh ; j = 0 , 1 , 2 , . . . } of indepen dently and identically distributed standard normal variates. The factor h has D.S.G. Pollock: stephen pollock@sigmapi.unet.com CONTINUOUSTIME STOCHASTIC PROCESSES entered the equation for the reason that we are now taking steps through time of length h whereas, previously, we were taking steps of unit length....
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 Spring '12
 D.S.G.Pollock

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