# The assumption a that the initial value w 0 is zero

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The assumption (a) that the initial value w (0) is zero is unrestrictive, so long as it can be assumed that the process takes a finite value at time t = 0. For that value can subtracted from the process to yield condition (a). D.S.G. Pollock: stephen [email protected]

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EC3070 FINANCIAL DERIVATIVES Arithmetic Brownian Motion The standard Wiener process is inappropri- ate to much of financial modelling. However, some quite general continuous stochastic processes can be derived that are functions of a standard Wiener process. A straightforward generalisation corresponds to a so-called random walk with drift. In discrete time, this can be represented by the equation x ( t + 1) = x ( t ) + µ + σε ( t + 1) . (7) The continuous-time analogue of this process is described by dx ( t ) = µdt + σdw ( t ) . (8) A generalisation of the latter is the Ito process, where the drift parameter µ and the variance or volatility parameter σ 2 become time-dependent functions of the level of the process: dx ( t ) = µ ( x, t ) dt + σ ( x, t ) dw ( t ) . (9) Geometric Brownian Motion The domain of a normally distributed random variable is the entire real line, which extends from −∞ to + . Many variables, such as those that represent physical quantities, are constrained to lie in the interval [0 , ). Therefore, it is inappropriate to describe them with a model that corresponds to a linear function of a normal random variable. In finance, there is an evident constraint that nominal interest rates must be nonnegative. Also, asset values cannot become negative. Such diﬃculties in modelling can sometimes be overcome by replacing the variables in question by their logarithms. The logarithmic transformation maps from the interval [0 , ) to the interval ( −∞ , ). The logarithmic version of the random walk is described by the following equation: ln x ( t + 1) = ln x ( t ) + σε ( t + 1) . (10) The corresponding continuous-time version can be written as d { ln x ( t ) } = σdw ( t ) . (11) Given that d dt ln x = 1 x dx dt or d { ln x ( t ) } = dx x , it follows that equation (11) can also be written as dx = σxdw ( t ) . (12) D.S.G. Pollock: stephen [email protected]
CONTINUOUS-TIME STOCHASTIC PROCESSES This equation might be used in describing the trajectories of financial assets.

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