Consider the process x t 1 w t1 2 w t2 , where 1 , 2 are numbers Let 2 21 22 Show that the process x t is a standard Brownian motion, and hence x t may be represented as w t , where w t is a standard Brownian motion Next, consider the case 1 1, 2 0 Explain why the fact that w t is Brownian motion i...

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 ...

Consider the process x t = 1 w t1 + 2 w t2 , where

Question:

Consider the process x_t = σ₁w_t1 +σ₂w_t2, where σ₁, σ₂ are numbers. Let σ₂ = σ₂₁ +σ₂₂. Show that the process x_t/σis a standard Brownian motion, and hence x_t may be represented as σw_t, where w_t is a standard Brownian motion. Next, consider the case σ₁ = −1, σ₂ = 0. Explain why the fact that −w_t is Brownian motion is almost obvious. (Advice: First, show that xt is the process with independent increments. Secondly, consider the distribution of x_Δ.)