Hi Tutors,

Can you please help me out on the below question.

Question 3. "Solve" some widely used stochastic processes. In all of the below, let B, be a standard Brownian motion with Bo = 0. I will give you a SDE for X, which is built based on this BM. Your goal is to "solve" the SDE in the sense that you write Xy as a function of time and the BM. This function of time could be an integral that "adds up" other functions of time; and the function of the BM could be a stochastic integral. The main point is to not have any terms involving its own past values {X,}ter in the solution. (a) Ornstein-Uhlenbeck process. Let X, follow dXt = -A(Xt - x)dt + odBy, t20, Xo =To. Hint: apply Ito's formula to exp(At)X, and see what comes out. The idea is we are trying to "undo" the exponential decay in dX, by augmenting it with exp( At). This is the same step as what you would do to solve the non-stochastic ordinary differential equation X"(t) = -A(X(t) - I). (b) Geometric Brownian motion. Let Xt follow dXt = Xtudt + XtodBy, t20, Xo = To Hint: apply Ito's formula to Y = log(X,) and see what comes out. The idea is we are trying to get the stochastic calculus version of "change-of-variable" for this process. The is the same step as what you would do to solve the non- stochastic integral Jo x() X't dt. In other words, in regular calculus, you would use Y(t) = log(X(t)) so that Y'(t) = X'(t)/X(t), and therefore the integral is very simple fo Y'(t)dt = Y(T) - Y(0), by the fundamental theorem of calculus. (c) Brownian Bridge. Let Xt follow dXt = -J. -X,dt + dB, OStsT, X, =0 What is the probability distribution of Xy? Hint: apply Ito's formula to Y := dB, to start. You should get something close to what you want and then your job is to figure out how to adjust things to make it work