Chapter 6 Simultaneous Stochastic Differential Equations 6.1 Correlated Brownian Motion Let w1 (t), w2 (t) be two independent Brownian Motions, that is, for each value of t, w1 , w2 are independent random variables. Then w1 (0) = w2 (0) = 0, E(w1 ) = E(w2 ) = 0 and 2 2 E(w1 ) = E(w2 ) = t E(w1 w2 ) = E(w2 w1 ) = 0 Let W = Then WWT = 2 w1 w1 w2 2 w2 w1 w2 2 E(w1 ) E(w1 w2 ) 2 E(w2 w1 ) E(w2 ) w1 w2 E(W W T ) = = 1 0 t = It 0 1 Now let z1 (t), z2 (t) be two (possibly) correlated Brownian…
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