Consider two stochastic differential equations:
a) $$ dX=\sqrt{a}dW_{1,t}+\sqrt{b}dW_{2,t} $$
b) $$ dX=\sqrt{a+b}dW_{3,t} $$
where $W_{1,t},W_{2,t},W_{3,t}$ are independent brownian motions. Do these two equations give the same dynamics?
Since the distribution of $\sqrt{a}W_{1,t}+\sqrt{b}W_{2,t}$ is equivalent to the distribution of $W_{1,at}+W_{2,bt}$, which is also equivalent to the distribution of $W_{3,(a+b)t}$, and eventually that is equivalent to the distribution of $\sqrt{a+b}W_{3,t}$, can we simply say that those two equations are also identical?