"Frontmatter". In: Analysis of Financial Time Series

STOCHASTIC PROCESSES 2253. stationary increments: if s ≤ t, w t − w s and w t−s − w 0 have the same probabilitydistribution.It can be shown that the probability distribution of the increment w t − w s is normalwith mean µ(t − s) and variance σ 2 (t − s). Furthermore, for any given time indexes0 ≤ t 1 < t 2 < ···< t k , the random vector (w t1 ,w t2 ,...,w tk ) follows a multivariatenormal distribution. Finally, a Brownian motion is standard if w 0 = 0 almost surely,µ = 0, and σ 2 = 1.Remark: An important property of Brownian motions is that their paths are notdifferentiable almost surely. In other words, for a standard Brownian motion w t ,itcan be shown that dw t /dt does not exist for all elements of except for elementsin a subset 1 ⊂ such that P( 1 ) = 0. As a result, we cannot use the usualintergation in calculus to handle integrals involving a standard Brownian motionwhen we consider the value of an asset over time. Another approach must be sought.This is the purpose of discussing Ito’s calculus in the next section.6.2.2 Generalized Wiener ProcessesThe Wiener process is a special stochastic process with zero drift and variance proportionalto the length of time interval. This means that the rate of change in expectationis zero and the rate of change in variance is 1. In practice, the mean and varianceof a stochastic process can evolve over time in a more complicated manner. Hence,further generalization of stochastic process is needed. To this end, we consider thegeneralized Wiener process in which the expectation has a drift rate µ and the rateof variance change is σ 2 . Denote such a process by x t and use the notation dy for asmall change in the variable y. Then the model for x t isdx t = µ dt + σ dw t , (6.1)where w t is a Wiener process. If we consider a discretized version of Eq. (6.1), thenfor increment from 0 to t. Consequently,x t − x 0 = µt + σɛ √ tE(x t − x 0 ) = µt, Var(x t − x 0 ) = σ 2 t.The results say that the increment in x t has a growth rate of µ for the expectationand a growth rate of σ 2 for the variance. In the literature, µ and σ of Eq. (6.1) arereferred to as the drift and volatility parameters of the generalized Wiener process x t .6.2.3 Ito’s ProcessesThe drift and volatility parameters of a generalized Wiener process are timeinvariant.If one further extends the model by allowing µ and σ to be functions