1.  Let Ω = {1, 2, 3, 4} and we consider the σ-algebra given by the power set 2Ω . Let P defined on 2Ω  be given by P(1) = P(2) = P(3) = P(4) =  .

i)  Write down the σ-algebra generated by A = {1, 2} given by σ(A).

ii)  Consider the random variables X and Y : Ω → R given by X(ω) = ω and Y = 4X .

a)  Compute E(X) and E(Y).

b)  Compute E[X|Y].

iii)  Consider another random variable Z given by

Z = { 1

Compute E(X|Z).

X ≥ 3

X < 3.

2.  Let W be a one-dimensional Wiener process on the space (Ω , F, P) and let Xt  ='0(t) e −2udWu  for t ∈ [0,T].

i)  Show that X is a true martingale (not just a local martingale) and com- pute it’s quadratic variation ⟨X⟩t .

ii)  Show that X admits the following representation   Xt  = e−2tWt + 2 \0t e −2uWu du

3.  Given a Brownian motion W, the Brownian bridge is given by

Xt  := Wt − tW1 ,    t ∈ [0, 1]

where we note that X1  = 0.

i)  Compute the expected value E(Xt ) and the covariance function of Cov(Xt ,Xs ).

ii)  Another representation of the Brownian bridge is

b − Xt

where B is a standard Brownian motion.

a)  Show that Xt  = a(1 − t) + bt + (1 − t) '0(t) dBs  for t ∈ [0, 1].

b)  Hence or otherwise state the distribution of Xt  for t ∈ [0, 1).

c)  Hence show that limt→ 1 Var(Xt ) = 0.

4.  Consider the stochastic differential equation

dXt  =  tanh ( ) dt + dBt ,    t ∈ [0, ∞)                      (1)

X0  = x

where B  and a Brownian motion and we recall that tanh(x) =    and ∂x tanh(x) = sech2 (x) and ∂x sech(x) = −sech(x)tanh(x)

i)  Show that these exists a unique strong solution to the SDE (1).

ii)  Let f be the standard logit function given by f(x) =  =  . Show that Yt  = f(Xt ) satisfies the SDE

dYt  = Yt (1 − Yt )dBt

Y0  = f(x)

iii)  Argue that the process Y = (Yt )t≥0  only takes values in (0, 1).

5.  On the standard filtered probability space (Ω , F, P), for t ∈ [0,T] consider the process St  = e(µ− )t+σWt   and the Radon-Nikod´ym density process given by

 'Ft   = ηt  = exp ( Wt −  t) .

where W is a Brownian motion under P.

i)  Show that the dynamic of S under P∗  is given by

dSt  = rSt dt + σSt dWt∗

where W ∗  is a Brownian motion under P∗ .

ii)  By using i) show that (e−rtSt )t≤T  is a (local) martingale under P∗ . Here you are not required to show that the integrand is in L2 (W).

iii)  By using the fact that (e−rtSt )t≤T  is a martingale under P∗  show that e −r(T−t)EP ∗   [  \0 T Su du F't] = v(t,St ,Yt ),

where Yt  =l0(t) Su du and

v(t,x,y) =  [e −r(T−t)y + x(1 − e−r(T−t))] .