Question 1 - Compute the stochastic differential dZ where

1.1 Z = e^αt where α is a constant.

1.2 Z = e^αw(t).

1.3 Z = X²(t), where X has stochastic differential

dX = αX(t)dt + σX(t)dW(t).

Question 2 - Compute the stochastic differential dZ where Z = 1/X(t) and

dX = αX(t)dt + σX(t)dW(t).  

Question 3 - Suppose X_n is a submartingale. Show there exists a martingale M_n such that if A_n = X_n - M_n, then A₀ ≤ A₁ ≤ A₂ ≤ · · · and A_n is F_n-1 measurable for each n.

Question 4 - Suppose X_n is a submartingale and X_n = M_n + A_n = M'_n + A'_n. where both A_n and A'_n are F_n-1 measurable for each n, both M and M' are martingales, both A_n and A'_n increase in n, and A₀ = A'₀. Show M_n = M'_n for each n.

Question 5 - Suppose that S and T are stopping times. Show that max(S, T) and min(S, T) are also stopping times.

Question 6 - Suppose that S_n, is a stopping time for each n and S₁ ≤ S₂ ≤ · · ·. Show S = lim_n→∞ S_n is also a stopping time. Show that if instead S1 ≥ S2 ≥ · · · and S = lim _n→∞S_n, then S is again a stopping time.

Question 7 - Let W_t be Brownian motion. Show that e^{iuW_t+u^2t/2} can he written in the form ₀∫^tH_sdW_s and give an explicit formula for H_s.

Question 8 - Let X_t be the solution to

dX_t = σ(X_t)dW_t + b(X_t)dt,   X₀ = x,

where W_t is Brownian motion and σ and b are bounded C^∞ functions and σ is bounded below by a positive constant. Find a nonconstant function f such that f(X_t) is a martin-gale.

[Hint: Apply Ito's formula to fat) and obtain an ordinary differential equation that f needs to satisfy.]

Question 9 - Suppose X_t = W_t + F(t), where F is a twice continuously differentiable function, F(0) = 0, and W_t is a Brownian motion under P. Find a probability measure Q under which X_t is a Brownian motion and prove your statement. (You will need to use the general Girsanov theorem.)

Question 10 - Suppose X_t = W_t - ₀∫^tX_sds. Show that

X_t = ₀∫^te^s-tdW_s.

Assignment Files -

https://www.dropbox.com/s/6z9vhonc2j6biqr/Assignment%20Files.rar?dl=0