1, Ito's Formula:

Let (W_t)t≥0, be a standard Brownian motion. Use Ito's Formula to write the following expressions in the form S⁽ⁱ⁾_T = const + ₀∫^T.....dt + ₀∫^T.....dW_t, i = 1, 2:

(a) S⁽⁰⁾_T = (W_T)².
(b) S⁽¹⁾_T = (W_T)³.
(c) S⁽²⁾_T = W_T+3/W²_T+1.
For the following problems, use the following generalized version of Ito's Formula:

df(t,W_t) = ∂/∂tf(t,W_t))dt +  ∂/∂W_tf(t,W_t)dW_t +1/2.∂²/∂W_t² f(t, W_t)dt,

so we have the first term on the right-hand side in addition to the remaining terms as in class.

2. Geometric Brownian motion:

Let (W_t)t ≥ 0 be a standard Brownian motion and σ, μ > 0. Show that

S_t = S₀.exp{(μ -1/2σ²).t + σW_t}

solves the stochastic differential equation

dS_t = S_t.μdt + S_t.σdW_t, S(0) = S₀ > 0.

Under which condition is (S_t)_t≥0 a martingale?

3. Ito's Formula 2:

The stochastic process (R_t)_t≥0 is given by

R_t = R₀e^-t + 0.05(1-e^-t) + 0.1₀∫¹e^s-t√R_sdW_s

where (W_t)_t≥0 is a standard Brownian motion.

Define X_t = R²_t.

Find dX_t,