4. (5 marks) Ito calculus. (a) Let X(t) and Y(t) be two stochastic processes, such that dx MX(X(t), t)dt +ox(X(t),t)dZx dY = wy(Y(t), t)dt +oy(Y(t), t)dZy with Zx(t), Zy(t) being Brownian motions. Let X(ti) = X; and Y(t) = Y;. Show that (Xi+1 - X:)(Yi+1 - Y) = Xi+1Yi+1 - X,Y; - X(Yi+1 - Y) - Y(Xi+1 - Xi). Now, using the definition of the Ito integral which is the limit of a discrete sum, show that ["x(s)dy() [XY) - 6 - [*Y(wax(s) [*ax(s)dy(9) . (b) Let Z(t) be a Brownian motion. Using the result in part (a), show that (assuming Ito calculus) Z(8) dZ (8) Lz 3202 - Define the (Ito) stochastic integral as N-1 SCCXCC),s)dz(s) = C(X(s), s)dz(s) = lim j=0 26;AZ ; At 4. (5 marks) Ito calculus. (a) Let X(t) and Y(t) be two stochastic processes, such that dx MX(X(t), t)dt +ox(X(t),t)dZx dY = wy(Y(t), t)dt +oy(Y(t), t)dZy with Zx(t), Zy(t) being Brownian motions. Let X(ti) = X; and Y(t) = Y;. Show that (Xi+1 - X:)(Yi+1 - Y) = Xi+1Yi+1 - X,Y; - X(Yi+1 - Y) - Y(Xi+1 - Xi). Now, using the definition of the Ito integral which is the limit of a discrete sum, show that ["x(s)dy() [XY) - 6 - [*Y(wax(s) [*ax(s)dy(9) . (b) Let Z(t) be a Brownian motion. Using the result in part (a), show that (assuming Ito calculus) Z(8) dZ (8) Lz 3202 - Define the (Ito) stochastic integral as N-1 SCCXCC),s)dz(s) = C(X(s), s)dz(s) = lim j=0 26;AZ ; At