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Unformatted text preview: ACTL2003 Stochastic Modelling for Actuaries Week 12 Outline: I Brownian motiona review I Stochastic differential equations and It os formula I Stochastic integration Readings: Ross, 9th Edition, Chapter 10 (10.110.3, 10.510.7) 1/18 Brownian motiona review I Definition A stochastic process { X t , t } is said to be a Brownian motion process, or simply Brownian motion , if: (1) X = 0; (2) { X t , t } has stationary and independent increments; (3) and for every t > 0, X t N ( , 2 t ) . I Definition: A stochastic process { X t , t } is said to be a Brownian motion process with drift coefficient and variance parameter 2 if: I X = 0; I { X t , t } has stationary; and independent increments; I For every t > 0, X t N ( t , 2 t ) . 2/18 Some of the strange behaviour of Brownian motion: I A Brownian motion { X t , t } is continuous with respect to time t everywhere, but it is (with probability one) differentiable nowhere. I Brownian motion will eventually hit any and every real value no matter how large, or how negative. No matter how far above the axis, it will (with probability one) be back down to zero at some later time. I Once Brownian motion hits a value, it immediately hits it again infinitely often (and will continue to return after arbitrarily large times). I It doesnt matter what scale you examine Brownian motion on, it looks just the same. Brownian motion is a fractal. 3/18 Stochastic Differential Equations (SDEs) I As early as 1900, Bachelier proposed Brownian motion as a model of the fluctuations of stock prices. I However, Brownian motion is clearly inadequate as a market model, not least because it would predict negative stock prices. I However, by considering functions of Brownian motion we can produce a wide class of potential models....
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 Three '11
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