Exercise 1 is perhaps more technical than the other two. You can start by exercise 2 and 3.

Introduction to mathematical finance

Sheet 12 HS 2014

Hand-in your solutions until 17.12.2014, in Martina Dal Borgo’s mailbox on K floor.

Exercise 1 is perhaps more technical than the other two. You can start by exercise 2 and 3.

Exercise 1

In what follows W is a real Brownian motion on the filtered probability space (Ω, F , (F _t ) _t∈[0,T] , P ) where the usual hypotheses hold. Recall that a stochastic process Y belongs to L ^p if

- Y is (F _t )-adapted,

- Y ∈ L ^p ([0, T ] × Ω) that is E

Z T

0

|Y s | ^p ds

= Z T

0

E [|Y s | ^p ] ds < ∞.

Consider an Itˆ o process, that is a stochastic process X of the form X _t = X ₀ +

Z t

0

ν _s ds + Z t

0

Y _s dW _s , t ∈ [0, T ],

where X ₀ is a F ₀ -measurable random variable, ν ∈ L ¹ , Y ∈ L ² . We want to prove that:

X _t is a martingale ⇔ ν = 0 a.s

The implication from right to left is obvious from the construction of the stochastic integral seen in class.

Indeed, if ν = 0, X _t is only a stochastic integral against Brownian motion, which is a martingale by construction. The implication from left to right requires a little more work.

• Prove that if X is a martingale, then for all times t and s:

E

Z t+s

t

ν _u du|F _t

= 0

• Deduce that:

ν _t = E 1

s Z t+s

t

(ν _t − ν _s )du|F _t

= 1 s

Z t+s

t

E (ν _t − ν _s |F _t ) du

• Assuming for example that ν is a.s continuous and bounded, show that ν = 0 a.s. If possible, you can replace the path-wise continuity and boundedness hypotheses by s 7→ E (|ν _s |) continuous on [0, T ].

Exercise 2[Polarisation of Ito isometry]. We know from class that if Y ∈ L ² , the stochastic integral t 7→ R t

0 Y _s dW _s is an L ² -martingale satisfying the Ito isometry:

E

"

Z t

0

Y _s dW _s 2 #

= E Z t

0

Y _s ² ds

,

Now consider two processes Y, Z ∈ L ² . Prove the polarised version:

E Z t

0

Y _s dW _s × Z t

0

Z _s dW _s

= E Z t

0

Y _s Z _s ds

, (1)

1

Hint: Apply the Ito isometry to R t

0 (Y s +Z s ) dW s and R t

0 (Y s − Z s ) dW s , make the difference then develop.

Exercise 3. Let W be a real Brownian motion on a filtered (natural filtration) probability space (Ω, F , (F t ) t∈[0,T ] , P ) where the usual hypotheses hold.

• Apply the Itˆ o formula to

i) X _t = W _t , f (x) = x ⁿ , n ∈ N ; ii) X _t = W _t , f (x) = e ^σx , σ ∈ R ; ii) X _t = W _t , f (t, x) = xt.

to give the explicit decomposition of f (t, X t ) as an Ito process.

• Recall from exercise 1 and the class, that the martingale part has zero expectation. Using the Itˆ o formula and recurrence, compute E [W _t ⁴ ], E [W _t ⁶ ]. Prove that E [W _t ⁿ ] = 0 if n is odd and

E [W _t ⁿ ] = t

2

n!

(n/2)! .

2