Question: Consider the process x t = 1 w t1 + 2 w t2 , where 1 , 2 are numbers. Let
Consider the process xt = σ1wt1 +σ2wt2, where σ1, σ2 are numbers. Let σ2 = σ21 +σ22. Show that the process xt/σis a standard Brownian motion, and hence xt may be represented as σwt, where wt is a standard Brownian motion. Next, consider the case σ1 = −1, σ2 = 0. Explain why the fact that −wt is Brownian motion is almost obvious. (Advice: First, show that xt is the process with independent increments. Secondly, consider the distribution of xΔ.)
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From a heuristic point of view the fact that the process x is a process with independent increments should be clear Since a formal proof requires some technique maybe it makes sense to suggest the student to provide just a heuristic proof Formally one may proceed as follows Note that x W12WA2 where w is the increment ... View full answer
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