Universit´e Paris 1- Panth´eon Sorbonne-M2-MMMEF

Stochastic Calculus I- Written Exam

October 20th, 2022

Question1: LetB1,B2betwoindependentBrownianMotionsonafiltration

(F ) . Justify all your answers to the following questions:

t t≥0

1) Is the process X defined as X :=B1 −B1 a Brownian Motion with respect

t 2t t

to its own filtration σ(X ,s≤t)?

s

2) Is the process Y :=B1 −B2 a Brownian Motion with respect to its own

t t/4 3t/4

filtration σ(Y ,s≤t)?

s

3) Is Y it a Brownian Motion with respect to (F ) .

t t≥0

4) Is X :=B1B2 an (F ) -martingale?

t t t t t≥0

5) For t≤1, compute E[B1|B1]

t 1

6) Let R be the process R :=B1−tB1. Prove that σ(R ,l≤1)⊥⊥σ(B1)

t t 1 l 1

7) If s,t∈[0,1], compute cov(R ,R ).

s t

8) For t≤s≤1, compute M :=E[R |R ].

t s t

9) If t≤s≤1, prove that R −M ⊥⊥σ(R ,l≤t).

s t l

10) If t≤s≤1, compute E[R |σ(R ,l≤t)].

s l

Question 2: LetM bean(F )-martingale, B an(F )-BrownianMotionand

t t

τ be a stopping time.

1) Prove that Mτ defined as Mτ :=M is an (F )-martingale.

t τ∧t t

2) If M is uniformly integrable, prove that Mτ is also uniformly integrable.

3) If M ∈M2, prove that ∃M ∈L2(F ):M (cid:107). →(cid:107) L2 M as t→∞.

∞ ∞ t ∞

4) If a ∈ Esc, prove that both X := (cid:82)• a dB and S := X2 − (cid:82)t a2ds are

0 s s t t 0 s

martingales.

(cid:82)1

5) Compute the expectation and variance of Y := B dB

0 s s

6) (Difficult): Prove that Y is not normally distributed. (Hint: There is a link

between Y and B2!)

1

Question 3:

1)Provethat,if{G } isafamilyofσ-algebrasonaprobabilityspace(Ω,A,p)

s s∈S

and if U ∈L1(Ω,A,p), then the family {X } is U.I., where X :=E[U|G ].

s s∈S s s

2) Define M is a local martingale.

3) Prove that a bounded local martingale is a U.I. martingale.

1