Let (,F,P) be a probability space and let {W, t>0} be a standard Wiener process. Suppose...

 

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Let (,F,P) be a probability space and let {W, t>0} be a standard Wiener process. Suppose X, follows the geometric mean-reverting process with SDE dx=k(0- log X+)Xdt+X+dW, Xo> 0, where . 0, and are constants. (i) (10p) By applying Taylor's formula to Y = log X, show that the diffusion process can be reduced to an Ornstein-Uhlenbeck process of the form dY = Y) dt + odW. (ii) (10p) Show also that for t Let (,F,P) be a probability space and let {W, t>0} be a standard Wiener process. Suppose X, follows the geometric mean-reverting process with SDE dx=k(0- log X+)Xdt+X+dW, Xo> 0, where . 0, and are constants. (i) (10p) By applying Taylor's formula to Y = log X, show that the diffusion process can be reduced to an Ornstein-Uhlenbeck process of the form dY = Y) dt + odW. (ii) (10p) Show also that for t