Could someone please check my work?

Show That - X(+), t20 is a Brownian Motion process and find The drift parameter and variance if we are giver That X(t), +20 is a Brownian Motion process with drift parameter s and variance of where X(O)= 0 Let YCt ) = - X ( t ) > E [ Y CE ) ] = E [ - X C + ) ] = - = [ x(+ )] @ -> Var ( YCt)) = yor (- XC+1) = (- 1)2 Var (XCt) = Var (x(+)) @ Since XCO)= 0, and XCt), +zo is a Brownian Motion process -> x (t+ y ) - x (4 ) ~ N ( ut, 02 t ) with drift parameter me and variance 5 2 > > X ( + +0) - ON N (MT, 0't ) -> x (t ) ~ N ( MT, o t ), which means and Var (XC+ ) ) = 02t + Using O and 3 = > E [ Y Ct ) ] = - E [ x (+ ) ] = - MT using 2) and + -) Var ( YCE ) ) = Var (X(t ) ) = oat By the definition of Brownian motion of YCt) YCT ) ~ N ( - it , 02 t ) Since YCt) = - X(+) - X(t) ~ N( - mr, oft ), which means - X (t), +20 is a Brownian Motion process with drift parameter -in and variance o 2