10 4 Another way to define a diffusion process is as follows Let t x and t x be continuous functions of t and x, where t 0 E2 u Budu ,N Define Xt 5 X0 1 t 0 u Budu 1 t 0 u BudBu t $ 0 Then, fXt t $ 0g is a diffusion process with as its drift function and the diffusion function If X0 5 x0 t x 5 and ...

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10.4 Another way to define a diffusion process is as follows. Let t; x and t; x...

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10.4 Another way to define a diffusion process is as follows. Let μðt; xÞ and σðt; xÞ be continuous functions of t and x, where ðt 0 E½σ2 ðu; BðuÞÞdu ,N Define XðtÞ 5 Xð0Þ 1 ðt 0 μðu; BðuÞÞdu 1 ðt 0 σðu; BðuÞÞdBðuÞ t $ 0 Then, fXðtÞ; t $ 0g is a diffusion process with μ as its drift function and σ the diffusion function. If Xð0Þ 5 x0; μðt; xÞ 5 μ and σðt; xÞ 5 σ, show that fXðtÞg is a Brownian motion with drift, where the drift is the initial state x0

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