(B(t), t > 0) denotes the standard Brownian motion, with the natural filtra- tion FB  = (Ft(B), t > 0).

Q1. [25 marks]

(a) [3] Evaluate

P(B(3) > B(1)).

(b) [3] Evaluate

E[B(3)B(4)].

(c) [5] Determine distribution of the random variable ξ = 2B(3) - 3B(4) + 1.

(d) [5] Let (W (t), t > 0) be another Brownian motion. Show that         the joint distribution of (B(3), B(4)) conditional on B(1) = 0.5

is the same as

the joint distribution of (W (2) + 0.5, W (3) + 0.5).

Explain which properties of the Brownian motion or result from the course you use.

(e) [5] For the random variable ξ in part (c), determine the conditional dis- tribution given that B(1) = 0.5.

(f) [4] Evaluate

E[B(3)B(4) | B(1) = 0.5, B(0.7) = -0.1].

Q2. [25 marks] Consider process (X(t), t > 0) defined by the formula

X(t) = B3 (t) - 2B2 (t) - 3tB(t) + 2t

(where Bk (t) has the same meaning as (B(t))k ).

(a) [5] Determine E[X(t)].

(b) [5] Calculate the stochastic differential dX(t).

(c) [4] Represent X(t) as a stochastic integral.

(d) [4] For t > 2 determine

E[X(t) | F2(B)].

(e) [4] Write a formula for the quadratic variation (X)(t) for t > 0.

(f) [3] Are the paths of the process (X(t), t > 0) differentiable? Justify your answer.

Q3. [25 marks] Consider the simple process

．

．

X(t) =

．

．(0,   otherwise.

(a) [5] Determine Var(X(3.5)).

(b) [3] Is the squared process (X2 (t), t > 0) also a simple process?

For the rest of the question, let

t

Y (t) =      X(s)dB(s),   t > 0.

0

(c) [3] Is the process (Y (t), t > 0) a martingale? Explain your answer.

(d) [5] Evaluate Y (6) explicitly in terms of the values of the Brownian mo- tion.

(e) [5] Find Var(Y (6)). [Hint: you may use It isometry or a direct calcu- lation.]

(f) [4] Are the paths of (Y (t), t > 0) continuous? Explain your answer.