Universit´e Paris 1-Panth´eon Sorbonne– UFR 27

Stochastic Calculus 1

MMMEF

Written exam: October 19th, 2023

Duration: 3 hours

Question 1: Let {F } be a filtration on (Ω,A,p).

t t≥0

1) Define F when τ is a stopping time.

τ

2) Prove that if τ ≤σ are two stopping times then F ⊂F .

τ σ

3) If X is a continuous adapted process and F is a closed set, prove that θ,

defined as: θ(ω):=inf{t≥0:X (ω)∈F} is a stopping time.

t

4) If X is an {F } -martingale, prove that so is Xτ.

t t≥0

5) If X is a U.I. martingale, is Xτ also U.I.?

6) If Y ∈L1(Ω,A,p), if σ and σ are stopping times, prove that

1 2

E[E[Y|F ]|F ]]=E[Y|F ]

σ1 σ2 σ1∧σ2

(Hint: consider the process Z :=E[Y|F ])

t t

7) Define X is a local martingale.

8) Prove that a bounded local martingale is a martingale.

Question 2: On a filtration {F } , let X be a continuous adapted process,

t t≥0

B a B.M. and let Mα be defined as: M tα :=eiαXt+α 22 t.

1) If X is a B.M. prove that Mα is a martingale starting at 1.

2) Conversely, prove that if, ∀α∈R, Mα is a martingale starting at 1, then X

is a B.M.

3) Let τ ,τ ,τ denote respectively τ := inf{t ≥ 0 : B = a}, τ := inf{t ≥ 0 :

a b a t b

B =b} where a>0 and b<0 and τ :=τ ∧τ .

t a b

3-a) Prove that E[B ]=0 and E[B2]=E[τ].

τ τ

3-b) Compute p(τ <τ ).

a b

3-c) Compute E[τ].

Question 3:

1)LetX beacontinuous{F } -adaptedprocess. ProvethatX isamartingale

t t≥0

if and only if there exists a constant C such that for all discrete (i.e. taking

finitely many values) stopping times τ: E[X ]=C.

τ

2) Let τ be a discrete stopping time:

2-1) If a∈Esc, prove that b(ω,t):=a(ω,t)11 (t) also belongs to Esc.

[0,τ(ω)[

2-2) If a∈Esc and Y =(cid:82). a dB , prove that Yτ =(cid:82). b dB

0 s s 0 s s

2-3) Prove that the previous result also holds for a∈H2.

2

3) If a∈H2, prove that X

:=(cid:16)

(cid:82)t a dB

(cid:17)2

−(cid:82)t a2ds is a martingale.

2 t 0 s s 0 s

4)IsitpossibletofindamartingaleZ thatsatisfies∀s,t≥0: cov(Z ,Z )= 1 ?

s t s+t

5)Isitpossibletofindaprocessa∈Hloc suchthatR :=(cid:82). a dB isacentered

2 t 0 s s

gaussian process with ∀s,t≥0: cov(R ,R )=es∧t?

s t