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

Stochastic Calculus 1

Written Exam: October 21st, 2021

Question 1: Let B be a Brownian Motion on (Ω,A,p) and let {F } be its

t t≥0

natural filtration.

1) State and prove the (0-1)-law.

2) Prove that, ∀t>0:Y

t

a =.s. +∞, where Y

t

:=sup{√Bs

s

:s∈]0,t]}.

(Hint: Prove that Y converges a.s. to a random variable Y as t (cid:38) 0. Prove

t 0+

next that: 0

t→0 t 0+ t

3)Provethat,withprobability1,thetrajectoryoftheB.M.isnotdifferentiable

at time 0.

4) Prove that, for α∈R, M

t

:=eαBt−α 22t is a martingale,

5)Letτ (ω):=inf{t≥0:B (ω)≥β+αt},withtheconventioninf∅:=+∞.

α,β t 2

Is τ a stopping time? Justify your answer.

α,β

6) Prove that ∀α,β >0: p(τ <∞)≤e−αβ. (Hint: compute E[Mτα,β]).

α,β T

Question 2: Let B be a Brownian Motion and let {F } be its natural

t t≥0

filtration.

1) Give the definition of H2 and Hloc. Is B in one of these two spaces? (Justify

2 2

you answer).

2) Define the Itˆo integral J(a) for a process a ∈ Hloc and prove that J(a) is a

2

local martingale.

3) If a ∈ Hloc satisfies ∀T : E[(cid:82)T a2ds] < ∞, prove that J(a) is a martingale.

2 0 s

Compute E[J(a) ] and var[J(a) ].

t t

4) Prove that B2 −T = (cid:82)T 2B dB (Hint: For a subdivision ∆ = {t ,...,t }

T 0 s s 0 n

of [0,T], compute B2 −T∆(B,B))

T t

5) Prove that the sum of two local martingales is a local martingale.

Question 3:

1)ProvethatXσ :=X isamartingaleifX isamartingaleandσ astopping

t σ∧t

time.

2) If, in the previous subquestion, X is a U.I. martingale, is it true that Xσ is

also U.I.? If this result is true, prove it. Otherwise, give a counterexample.

3) If U ∈L1(Ω,A,p), is it true that, for all pair of sub-σ-algebras (B,C),

E[E[U|B]|C]=E[E[U|C]|B]

(We do not assume any inclusion between B and C.)

If this result is true, prove it. Otherwise, give a counterexample.

4) If U ∈L1(Ω,A,p), is it true that, for all pair of stopping times (σ,τ):

E[E[U|F ]|F ]=E[E[U|F ]|F ]

τ σ σ τ

((F ) is a filtration, but we do not assume any inequality between σ and τ.)

t t≥0

If this result is true, prove it. Otherwise, give a counterexample.