{W} 3. (0) Let w, be the standard Wiener process. Consider the process x, = Derive the stochastic differential equation for X,. Consider the following stochastic differential equation dS, = us dt + oS dw, (ii) where u and o are positive constants and w, is standard Brownian motion. Check that the conditions for existence and uniqueness of the solution of the stochastic differential equation are verified. (iii) Let B, be a smooth deterministic function and x, be a Brownian motion with- out drift, such that dx, = odw, where w, is the standard Wiener process. Derive d(B,X,). {W} 3. (0) Let w, be the standard Wiener process. Consider the process x, = Derive the stochastic differential equation for X,. Consider the following stochastic differential equation dS, = us dt + oS dw, (ii) where u and o are positive constants and w, is standard Brownian motion. Check that the conditions for existence and uniqueness of the solution of the stochastic differential equation are verified. (iii) Let B, be a smooth deterministic function and x, be a Brownian motion with- out drift, such that dx, = odw, where w, is the standard Wiener process. Derive d(B,X,)