11 - Copyright c 2010 by Karl Sigman 1 IEOR 4106 Notes on...

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Unformatted text preview: Copyright c 2010 by Karl Sigman 1 IEOR 4106: Notes on Brownian Motion We present an introduction to Brownian motion , an important continuous-time stochastic pro- cess that serves as a continuous-time analog to the simple symmetric random walk on the one hand, and shares fundamental properties with the Poisson counting process on the other hand. Throughout, we use the following notation for the real numbers, the non-negative real numbers, the integers, and the non-negative integers respectively: IR def = (-∞ , ∞ ) (1) IR + def = [0 , ∞ ) (2) ZZ def = {··· ,- 2 ,- 1 , , 1 , 2 , ···} (3) IN def = { , 1 , 2 , ···} . (4) 1.1 Normal distribution Of particular importance in our study is the normal distribution, N ( μ,σ 2 ), with mean-∞ < μ < ∞ and variance 0 < σ 2 < ∞ ; the density and cdf are given by f ( x ) = 1 σ √ 2 π e- ( x- μ ) 2 2 σ 2 , x ∈ IR , (5) F ( x ) = 1 σ √ 2 π Z x-∞ e- ( y- μ ) 2 2 σ 2 dy, x ∈ IR . (6) The normal distribution is also called the Gaussian distribution after the famous German mathematician and physicist Carl Friedrich Gauss (1777 - 1855). When μ = 0 and σ 2 = 1 we obtain the standard (or unit ) normal distribution, N (0 , 1), and the density and cdf reduce to φ ( x ) def = 1 √ 2 π e- x 2 2 , (7) Φ( x ) def = 1 √ 2 π Z x-∞ e- y 2 2 dy. (8) As we shall see over and over again in our study of Brownian motion, one of its nice features is that many computations involving it are based on evaluating Φ( x ), and hence are computa- tionally elementary. If Z ∼ N (0 , 1), then X = σZ + μ has the N ( μ,σ 2 ) distribution. Conversely, if X ∼ N ( μ,σ 2 ), then Z = ( X- μ ) /σ has the standard normal distribution. It thus follows that that if X ∼ N ( μ,σ 2 ), then F ( x ) = P ( X ≤ x ) = Φ(( x- μ ) /σ ) . 1 Moment generating function of a normal distribution Letting X ∼ N ( μ,σ 2 ), the moment generating function of the normal distribution is given by M X ( s ) = E ( e sX ) = Z ∞-∞ e sx f ( x ) dx = e sμ + s 2 σ 2 / 2 ,-∞ < s < ∞ . (9) Deriving (9): First we derive M Z ( s ) = e s 2 / 2 , that is, the case when X = Z is the unit normal. M Z ( s ) = E ( e sZ ) = 1 √ 2 π Z ∞-∞ e sx e- x 2 2 dx = 1 √ 2 π Z ∞-∞ e- ( x 2- 2 sx ) 2 dx = 1 √ 2 π Z ∞-∞ e- ( x- s ) 2 2 + e s 2 / 2 dx = e s 2 / 2 1 √ 2 π Z ∞-∞ e- ( x- s ) 2 2 dx = e s 2 / 2 1 √ 2 π Z ∞-∞ e- u 2 2 du = e s 2 / 2 × 1 = e s 2 / 2 . To obtain the general form in (9): If X ∼ N ( μ,σ 2 ), then it can be expressed as X = σZ + μ , and thus M X ( s ) = E ( e sX ) = e sμ E ( e σsZ ) = e sμ M Z ( σs ) = e sμ e ( σs ) 2 / 2 = e sμ + s 2 σ 2 / 2 ; we have derived (9)....
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11 - Copyright c 2010 by Karl Sigman 1 IEOR 4106 Notes on...

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