Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 1, week 1
Let (Ω, F, P ) be a probability space.
Exercise 1 Let X : Ω → W d be a bounded measurable map. Let µ = (X)∗ (P )
be the pushed forward measure on W d . Let πt : W d → Rd be the evaluation map,
πt (ω) = ωt . Let µt = (πt )∗ µ.
X
Ω
Wd
X
t
=π
t
πt
◦X
Rd
Define the random variable Xt : Ω → Rd by Xt = πt ◦ X. Check that
µt = (Xt )∗ (P ).
Exercise 2 Let µ be the Wiener measure on (W0d , B(W0d )). Write πt = (πt1 , . . . , πtd )
in components. Then for Ai ∈ B(Rd ), t1 < t2 < · · · < tn ,
µ(πt1 ∈ A1 , . . . , πtk ∈ Ak )
Z
=
pt1 (0, y1 ) . . . pti+1 −ti (yi , yi+1 ) . . . ptk −tk−1 (yk−1 , yk ) Πnk=1 dyi .
A1 ×···×Ak
Answer the following questions:
1. Prove that Eπt = 0, E(πti )2 = t and E(πti πtj ) = 0 if i 6= j.
2. What is the probability distribution of πt ?
3. Prove that E(πsi πtj ) = (s ∧ t)δi,j . Here s ∧ t stands for min(s, t) and δi,j = 0
if i 6= j and δi,i = 1.
4. Define Gs = σ{πr : r ∈ [0, s]}. Let E{πt |Gs } be the conditional expectation
of πt with respect to Gs . Show that E{πt |Gs } = πs .
(If you are not familiar with conditional expectations, leave this problem
until next week.)
5. Compute the probability distribution of |πt |2 .
6. Compute the distribution of πt − πs , for t > s.
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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7. Let f be a bounded measurable function. Define Pt f (x) = Ef (x + πt )).
Show that for any f smooth with compact support,
∂Pt f
1
= ∆Pt f.
∂t
2
Exercise 3 Let (Bt ) be a real valued standard Brownian motion. Fom the definition of the Brownian compute the marginal distributions of (Bt ).
Exercise 4 Let b : Rd → Rd be a Borel measurable functions. For each ω,
consider the equation
Z t
xt (ω) = x0 +
b(xs (ω))ds + Bt (ω).
0
Suppose that b is globally Lipschitz continuous, i.e. there is a number K > 0 s.t.
|b(x) − b(y)| ≤ K|x − y|.
Prove that for each initial value x0 there is a unique solution to the above equation.
n g (B ), where g are Borel
Hint to Problem 1(d). The class of functions πi=1
i
si
i
measurable and 0 ≤ s0 < s1 < . . . sn = s < t, are sufficient for determining
conditional expectation with respect to FsB .
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Fix 0 ≤ s1 < s2 < · · · < sn ≤ s. For each i = 1, . . . , n, let gi : Ω → R be a
bounded function measurable with respect to Gsi . Prove that
n
n
E[f (πt − πs )πi=1
gi (πsi )] = Ef (πt − πs )E[πi=1
gi (πsi )].
Hint for Problem 3. Let n ∈ N , let 0 ≤ t1 < t2 < · · · < tn and f : Rn → R
a bounded measurable function. Work on Ef (Bt1 , Bt2 , . . . Btn ).
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Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 2, week 2
Let (Ω, F, P ) be a probability space. Let (Bt ) be a one dimensional Brownian
motion unless otherwise stated.
Exercise 5
1. Write down the Fourier transform of a Gaussian measure N (a, C).
2. Let a ∈ Rd and C = (Ckl ) a positive definite symmetric matrix. Let X =
(X1 , . . . , Xd ) be an Rd valued random variable with probability distribution
N (a, C). Prove that EX = a and cov(Xk , Xl ) = Ckl .
3. Let (Bt ) be an Rd valued Brownian motion. Let a ∈ Rd and A a d × dmatrix. For each t > 0, compute the probability distribution of ABt + at.
Exercise 6 Let (Xt ) be a one dimensional process with finite dimensional distribution given below. For 0 = t0 < t1 < · · · < tk , Ak ∈ B(R),
P (Xt1 ∈ A1 , . . . , Xtk ∈ Ak )
Z
=
pt1 (0, y1 ) . . . pti+1 −ti (yi , yi+1 ) . . . ptk −tk−1 (yk−1 , yk ) Πkk=1 dyi .
A1 ×···×Ak
Prove that (Xt ) is a Brownian motion.
Exercise 7 (a) Let a 6= 0 be a real number. Show that
motion;
√1 Bat
a
is a Brownian
(b) Let t0 ≥ 0. Prove that Bt0 +t − Bt0 is a standard Brownian motion;
(c) Define a process Wt by W0 = 0 and Wt = tB 1 when t > 0. Show that Wt
t
is a Brownian motion.
(d) Show that limt→∞
Bt
t
= 0.
(e) Let Xt = Bt − tB1 , 0 ≤ t ≤ 1. Show that E(Xs Xt ) = s(1 − t) for s ≤ t.
Compute the probability distribution of Xt .
Exercise 8 A zero mean Gaussian process BtH is a fractional Brownian motion of
Hurst parameter H, H ∈ (0, 1), if its covariance is
1
E(BtH BsH ) = (t2H + s2H − |t − s|2H ).
2
Then E|BtH −BsH |p = C|t−s|pH . If H = 1/2 this is Brownian motion (Otherwise
this process is not even a semi-martingale). Show that BtH has Hölder continuous
paths of order α < H.
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 3, week 3
Exercise 9 A family of real-valued functions (fα , α ∈ I), where I is an index set,
is uniformly integrable (u.i.) if
Z
lim sup sup
|fα |dµ = 0.
C→∞
α
{|fα |≥C}
Let X : Ω → R be an integrable. Prove that the family of functions
{E{X|G} : G is a sub σ-algebra of F}
is uniformly integrable.
Exercise 10 Let X : Ω → R be integrable. Let (Ft , t ≥ 0) be a family of σalgebras such that Fs ⊂ Ft for s ≤ t (a filtration). Define Xt = E{X|Ft }, t ≥ 0.
Show that E{Xt |Fs } = Xs for any pair of numbers s, t with t > s ≥ 0.
Exercise 11 Let Bt be a Brownian motion and for s ≥ 0 define
Fs = σ(Br : 0 ≤ r ≤ t).
Show that Bt is a Gt -Markov process, which means that for any bounded Borel
measurable function f ,
E{f (Bt )|Gs } = E{f (Bt )|σ(Bs )}.
Exercise 12 If Bt is a standard Brownian motion show that
(a) For any s ≥ 0, Bt − Bs is independent of Fs where
Fs = σ{Br : 0 ≤ r ≤ s}.
(b) For any 0 ≤ s < t, E{(Bt − Bs )2 − (t − s)|Fs } = 0.
(c) Let Mt denote one of the processes: Bt , Bt2 − t and exp(Bt − t/2). Show
that martingale property holds: for all t ≥ s, E{Mt |Fs } = Ms .
Discrete Time Martingales
Exercise 13 Let X1 , X2 , . . . be independent random variables with EXi = 0. Let
Fn = σ{X1 , . . . , Xn }. Let Sn = X1 + · · · + Xn . Show that for j = 1, 2, . . . ,
E{Sn+j |Fn } = Sn .
Exercise 14 Let X1 , X2 , . . . be independent random variables with EXi = 1. Let
Fn = σ{X1 , . . . , Xn }. Let Mn = Πnk=1 Xk . Show that for all n, j = 1, 2, . . . ,
E{Mn+j |Fn } = Mn .
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Optional exercises
The following exercises are optional.
Exercise 15 Let Bt be a standard 1-dimensional Brownian motion. The Brownian
bridge from 0 to 0 in time 1 is the Brownian motion conditioned to be at 0 at time
1. Let A, Ai ∈ B(R). Compute the distribution of Bt conditioned on B1 = 0, ı.e.
P {Bt ∈ A|B1 }, and the conditional marginal distribution
P {Bt1 ∈ A1 , . . . , Btn ∈ An |B1 = 0}.
Exercise 16 Let X1 , X2 , . . . be independent random variables with EXi = 0.
Let Fn = σ{X1 , . . . , Xn }. Let S0 = 0 and Sn = X1 + · · · + Xn . Let Cn =
fn (X1 , . . . , Xn−1 ) for some Borel function fn : Rn → R. Define
X
I(C, X)n =
Ck (Sk − Sk−1 ).
1≤k≤n
This is called the martingale transform. Compute E{I(C, X)n −I(C, X)n−1 |Fn−1 }.
Exercise 17 Let {Xn , n = 0, 1, 2 . . . } be an Fn -adapted stochastic process with
Xn ∈ L1 and X0 = 0. Define Gn = E{Xn+1 − Xn |Fn }, n ≥ 1.
P
(a) Let A0 = A1 = 0 and An = n−1
j=1 Gj , n ≥ 2. Show that An ∈ Fn−1
(previsible).
(b) Let M0 = 0 and Mn = Xn − An show that Mn is a martingale. Then
Xn = Mn + An . This is the analogue of the Doob-Meyer decomposition for
continuous time processes.
(c) If Xn has another decomposition Xn = M̃n + Ãn , where {M̃n } is a martingale with M̃0 = 0, Ãn is process with Ãn ∈ Fn−1 and Ã0 = Ã1 = 0. Show
that Mn = M̃n and An = Ãn a.s. If Xn is a sub-martingale show that An is
an increasing process.
(d) Let (Mn ) be a martingale with EMn2 < ∞ and M0 = 0 show that there
is an increasing process An such that Mn2 = Nn + An where Nn is a
martingale and An is an increasing process. Show that An − An−1 =
E{(Mn − Mn−1 )2 |Fn−1 }.
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 4, week 4 (Martingales)
Let (Ω, F, Ft ) be a filtered probability space. All processes here are real valued.
Exercise 18 If Mt is an L2 bounded martingale (that is supt E(Mt )2 < ∞) show
that for s < t, E(Mt − Ms )2 = EMt2 − EMs2 .
Exercise 19 Let φ be a convex function s.t. φ(Xt ) ∈ L1 . Show that
(a) If Xt is a sub-martingale and φ is increasing then φ(Xt ) is a sub-martingale.
(b) If Xt is martingale then φ(Xt ) is a sub-martingale. Show that |Xt | is a submartingale.
Exercise 20 Let (Ω, F, Ft , P ) be a filtered probability space. Let Q be a probability measure such that Q << P . Let f = dQ
dP . Let Qt and Pt are restrictions
of Q, P to Ft . Let ft = E{f |Ft }. Show that ft is a L1 bounded martingale and
dQt
dPt = ft .
Exercise 21 Let (Bt ) be a 1-dimensional standard Brownian motion. Prove that
(Bt2 − t) is a martingale.
Exercise 22 Let (Bt ) be a 1-dimensional standard Brownian motion. Let 0 =
t0 < t1 < · · · < tn . Let Fs = σ{Br : 0 ≤ r ≤ s}. Suppose that ai be real valued
functions with ai measurable with respect to Fti .
1. Suppose that s ≥ ti+1 . Prove that
E{ai (Bti+1 − Bti )|Fs } = ai (Bti+1 − Bti ).
2. Suppose that s < ti , prove that E{ai (Bti+1 − Bti )|Fs } = 0.
3. Suppose that ti ≤ s < ti+1 , prove that
E{ai (Bti+1 − Bti )|Fs } = ai (Bs − Bti ).
4. Define
Mt =
n−1
X
ai (Bti+1 ∧t − Bti ∧t ).
i=0
Prove that Mt is a martingale. Here s ∧ t denotes min(s, t).
Pn−1
5. Let ft be a stochastic process given by ft (ω) =
i=0 ai (ω)1(ti ,ti+1 ] (t).
Prove Itô’s isometry:
Z t
2
E(Mt ) = E (fs )2 ds.
0
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 5, week 5 (Stopping Times and Martingales)
Let t ∈ [0, ∞) and let (Ω, F, Ft , P ) be a filtered probability space. All stochastic
processes are assumed to be continuous unless otherwise stated.
Exercise 23 Let (Xt ) be a continuous Ft adapted real valued process. Let (Ft ) be
a right continuous filtration. Show that τ1 := inf{t > 0 : Xt > 1} is a stopping
time.
Exercise 24 Let (Mt ) be an integrable and Ft adapted continuous stochastic process. Then Mt is a martingale if EMT = EMS for any Ft stopping times T and
S that takes at most two values.
Exercise 25 Let (Xt : t ∈ I) be an integrable continuous adapted process. Suppose that for all bounded stopping times S ≤ T , EXT ≤ EXS prove that
E{XT |FS } ≤ XS .
Exercise 26 Let 0 < t0 < t1 < t1 < · · · < tn+1 . Let Hi ∈ Fti and H−1 ∈ Ft0
be real valued. Let
n
X
Xt (ω) = H−1 (ω)1{0} (t) +
Hi (ω)1(ti ,ti+1 ] (t).
i=0
Prove that Xt is progressively measurable.
Exercise 27 Prove that if (Xt , t ≥ 0) is a sub-martingale with supt E|Xt | < ∞,
then limt→∞ Xt exists almost surely.
Exercise 28 If (Xt , t ∈ [0, ∞)) is a right continuous martingale, prove that the
following are equivalent.
1. limt→∞ Xt converges in L1 to a random variable X∞ .
2. There exists a L1 random variable XT s.t. Xt = E{XT |Ft }.
3. (Xt , t < ∞) is uniformly integrable.
Exercise 29 Using the Optional stopping theorem to prove the following: (1) A
positive local martingale Mt with M0 = 1 is a super-martingale. (2) A positive
local martingale is a martingale if E|M0 | < ∞ and EMt = EM0 for all t > 0.
Exercise 30 Let (Mt ) be a continuous local martingale. Prove that if Mt has finite
total variation on [0, T ] then Mt = M0 , any t ≤ T .
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 6: The Quadratic Variation Process
Exercise 31 Let (A1t ), (A2t ) be continuous processes of finite total variation on any
interval [0, s]. Suppose that A10 = A20 = 0. Let (Mt ) and (Nt ) be local martingales.
Show that if (Mt Nt − A1t ) and (Mt Nt − A2t ) are local martingales then A1t = A2t
a.s. for all t.
Exercise 32 Let (Bt , t ≥ 0) be a standard Brownian motion. For any u, the distributions of sups≤u Bs and |Bu | are equal.
1. Let T1 = inf{t > 0, Bt ≥ 1}. Prove that T1 < ∞ almost surely.
2. Prove that the family (Bt , 0 ≤ t ≤ 1) is uniformaly integrable. Is (Bt : t ≥
0) uniformly integrable?
Exercise 33 Let (Mt ) be a continuous local martingale with M0 = 0. Prove that
(Mt ) is a martingale if supt≤T Mt ∈ L1 .
Exercise 34 Let H02 be the space of L2 bounded sample continuous martingales
vanishing at 0. We define a norm: kM k = E(M∞ )2 . Prove that H02 is a Hilbert
space.
Exercise 35 Let M ∈ H02 . Let M∞ ≡ limt→∞ Mt . Prove the following statements.
(1) For all t > 0, E(Mt2 ) = EhM it .
(2) Let τ be a stopping time prove that E(Mτ )2 = EhM iτ and
(3)
lim EhM it = EhM i∞ = sup E(Mt2 ) = E(M∞ )2 .
t→∞
t<∞
Exercise 36 Let M n , M ∈ H02 . Show that the following statements are equivan − M )2 = 0;
lent: limn→∞ supt≤∞ E(Mtn − Mt )2 = 0; limn→∞ E(M∞
∞
n
n
limn→∞ EhM − M i∞ = 0; and limn→∞ E supt≤∞ (Mt − Mt )2 = 0.
Exercise 37 Let M, N ∈ H02 . Let τ be a stopping time. Prove that and E(Mτ Nτ ) =
EhM, N iτ and E(M∞ N∞ ) = EhM, N i∞ .
Exercise 38 Let M, N be bounded continuous martingales with M0 = N0 =
0. If Mt and Nt are furthermore independent prove that their quadratic variation
hM, N it vanishes.
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Hints: Exercise 34) How do you prove the limit of a sequence of continuous
functions is continuous? Exercise 35) We need to use uniform integrabiity, L2
convergence, and Doob’s inequality. Exercise 36) Recall Burkholder-Davis-Gundy
inequality. Exercise 38) Define Gs = σ{Mr , Nr : 0 ≤ r ≤ s}. Is (Mt ) a (Gt )
-martingale?
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 7: Stochastic Integrals
Exercise 39 Let Bt be 1-dimensional Brownian Motion,
Rt
Rt
a) Show that hBt , 0 Bs3 dBs i = 0 Bs3 ds.
Rt
Rt
b) Let fs , gs be continuous and adapted stochastic processes. Write h 0 fs dBs , 0 gs dBs i
as an integral with respect to the Lebesque measure.
R
Rt
1
c) Prove that ( 0 (Bs )2 dBs ) is a martingale and compute E 0 (Bs )2 dBs .
d) Write
down theR semi-martingale
decomposition of the stochastic process
Rt
s
0 Br d(Br + r) .
0 (2Bs + 1)d
Exercise 40 Let M, N ∈ H02 and si < si+1 .
1. Let H be bounded and Fsi measurable. Let
Xt = H(Msi+1 ∧t − Msi ∧t );
Yt = H(hM, N isi+1 ∧t − hM, N isi ∧t ).
(a) Prove that (Xt ) is a martingale.
(b) Verify that Yt = hM, N it by proving that (Xt Nt − Yt ) is a martingale.
Rt
(c) Let Ks = H1(si ,si+1 ] (t). Show that 0 Ks dMs = Xt by proving that
Z
hX, N it =
t
Ks dhM, N is .
0
2. Let Ks be an elementary process: Ks =
h
N
X
PN
i=1 Ksi 1(si ,si+1 ] (t),
Hi (M si+1 − M si ), N it =
Z
0
i=1
Prove that
t
Ks dhM, N is .
Exercise 41 Let {Bt , Wt1 , Wt2 } be independent Brownian motions. Let (xs ) be
an adapted continuous stochastic process.
1. Prove that (Wt ) defined below is a Brownian motion,
Z
Wt =
0
t
cos(xs )dWs1
Z
+
0
t
sin(xs )dWs2 .
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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2. Let sgn(x) = 1 if x > 0 and sgn(x) = −1 if x ≤ 0. Prove that (Wt ) is a
Brownian motion if it satisfies the following relation
Z t
Wt =
sgn(Ws )dBs .
0
3. Prove that the process (Xt , Yt ) is a Brownian motion if they satisfy the following relations,
Z t
Z t
1
1Xs ≤Ys dWs2
1Xs >Ys dWs +
Xt =
0
0
Z t
Z t
Yt =
1Xs ≤Ys dWs1 +
1Xs >Ys dWs2 .
0
0
Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
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Problem Sheet 8: Itô’s formula
Exercise 42 Let Bt = (Bt1 , . . . , Btd ) be a BM show that hB i , B j it = δij t and
n Z t
X
|Bt |2 = 2
Bsi dBsi + nt.
i=1
0
(Hint: exercise 38.)
Exercise 43 Prove that if H and K are continuous and adapted semi-martingales,
Z t
Z t
hHK, Bit =
Hr dhK, Bir +
Kr dhH, Bir .
0
Rt
Exercise 44 Interpret
0
0
s dBs by a Lebesque integral.
Exercise 45 Let f, g : R → R be C 2 functions, Bt one dimensional Brownian
motion. Write the following as an integral with respect to the Lebesque measure.
a) hf (Bt ), g(Bt )i.
b) h exp (M − 12 hM i), exp (N − 12 hN i)i where Mt and Nt are continuous local martingales.
1
Exercise 46 If Nt is a local martingale show that eNt − 2 hN,N it is a local martingale
1
and EeNt − 2 hN,N it ≤ 1.
Exercise 47 Show that any positive local martingale Nt with N0 = 1 can be written in the form of Nt = exp (Mt − 12 hM, M it ) where Mt is a local martingale.
Exercise 48 Let σ and b be smooth bounded functions from R to R.
(a) Let Bt be a one dimensional Brownian motion. Let (xt , t ≥ 0) be a real
valued adapted sample continuous process, s.t. for all t
Z t
Z t
xt = x0 +
σ(xs )dBs +
b(xs )ds
0
Show that E(xt
)2
0
< ∞.
Rt
Hint: Gronwall’s Lemma: If f (t) ≤ a(t) + 0 g(s)f (s)ds where a(t) is
increasing, then
Z t
f (t) ≤ a(t) exp( g(s)ds).
0
(c) Can you modify the proof to show that E(supt≤T (xt )2 ) < ∞?
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Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
Problem Sheet 9: Stochastic Differential Equations
Let Bt be a one dimensional Brownian motion on a given filtered probability space.
Let σ, b : R → R be locally bounded and Borel measurable.
Exercise 49 Give a solution to the one dimensional SDE: drt = dBt +
n−1
rt dt.
Exercise 50 Write down a solution to dxt = xt g(Bt )dBt where g : R → R is a
bounded Borel measurable function. Verify you claim.
Exercise 51 Let σ : R → R be BC 1 and f : R → R a solution of the ODE
f˙ = σ(f ). For g : R → R locally bounded and Borel measurable, assume that
dyt = dBt + g(f (yt ))dt
has a solution yt . Let b = σg + 21 σ̇σ. Show that xt = f (yt ) solves
dxt = σ(xt )dBt + b(xt )dt
Exercise 52 Consider the SDE: dxt = σ(xt )dBt + b(xt )dt, where σ, b are continuous. Assume that σ > 0. Define the scale function
−
ṡ(x) = e
2b(y)
0 σ 2 (y) dy
Rx
.
Then Ls = 0 and s is strictly increasing. Define
(
= σ(s−1 (y))ṡ(s−1 (y)),
if y is in the image of s
σ̃(y) =
= 0,
if y is not in the image of s .
Define yt = s(xt ). Show that yt solves:
dyt = σ̃(yt )dBt .
Prove that if b is bounded, then pathwise uniqueness holds for the SDE dxt =
dBt + b(xt )dt.
Exercise 53 Consider the SDE, with Stratonovitch integration,
dxt = −yt ◦ dBt ,
dyt = xt ◦ dBt .
Show that x2t + yt2 is independent of t.
Exercise 54 Consider dxt = σ(xt )dBt + b(xt )dt. Assume that σ and b are locally
Lipschitz continuous and are of at most linear growth:
|σ(x)| ≤ C(1 + |x|),
hx, b(x)i ≤ C(1 + |x|2 ).
Let xt be a solution. Prove that the SDE is complete.
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Stochastic Analysis. Lecturer: Xue-Mei Li, Support Classes: Sebastian Völlmer
Problem Sheet 10: SDEs
Exercise 55 Let σk = (σk1 , . . . , σkd ), b = (b1 , . . . , bd ) : Rd → Rd be smooth
functions. Let
!
d
d
m
X
1 X X i j
∂
∂2
L=
+
bl
.
σk σk
2
∂xi ∂xj
∂xl
i,j=1
l=1
k=1
Suppose that for C > 0,
|σ(x)|2 ≤ c(1 + |x|2 ),
hb(x), xi ≤ c(1 + |x|2 ).
Let f (x) = |x|2 + 1. Prove that Lf ≤ af for some a.
Exercise 56 Let B 1 and B 2 be independent Brownian motions. Compute the infinitesimal generator L and prove that the SDE does not explode:
dxt = (x2t + yt2 )dBt1
dyt = (x2t + yt2 )dBt2
Hint: The SDE lives in R2 . The harmonic function V (x) = C log |x| + K can be
used to build a Lyapunov function.
Exercise 57 Let b : R → R be a bounded smooth function. Let ut be a C 2,1
bounded solution to
1 d2
∂
∂ut
=
+b
ut
∂t
2 dx2
∂x
Prove that,
Z
ut (x) = Ef (x + Bt ) exp
0
t
1
b(x + Bs )dBs −
2
Z
t
2
(b(x + Bs )) ds .
0