Exercises in stochastic analysis
Franco Flandoli, Mario Maurelli, Dario Trevisan
The exercises with a P are those which have been done (totally or partially) in the previous lectures; the exercises with an N could be done (totally or partially) in the next classes. Of course, this is a pre-selection, one more selection and changes are possible.
1 Laws and marginals
Exercise 1 (P). Let τ be a random variable (r.v.) with values in [0, +∞[, define the (continuous-time) waiting process X as Xt= 1τ <t. Compute the m-dimensional distri- butions of X, for any m integer.
Exercise 2 (P). Exhibit two processes, which are not equivalent but have the same 1- and 2-dimensional distributions. Exhibit a family of compatible 2-dimensional distribions, which does not admit a process having those distributions as 2-dimensional marginals.
Exercise 3. Let (F, F ) be a measurable space, G a subset (not necessarily measurable) of F , define the restriction of F to G as F|G := {A ∩ G|A ∈ F }; it is easy to see that F|G is a σ-algebra on G.
Now consider (E, E ) a measurable space (morally, the state space for some continuous process), call C = C([0, T ]; E). Show that B(C) = (E[0,T ])|C. This allows to define the law of a continuous process on C, directly from the definition of law for a generic process.
2 Trajectories and filtrations
Exercise 4. Let X be a continuous process. Prove that its natural filtration (Ft)t (Ft= σ(Xs|s ≤ t)) is left-continuous, that is Ft=W
s<tFs.
Exercise 5. Let X, Y be two processes, a.e. continuous and modifications one of each other. Show that they are indistinguishable.
Let X be a process, such that a.e. trajectory is continuous. Show that there exists a process U , indistinguishable of X, which is everywhere continuous, i.e. every trajectory is continuous.
Exercise 6. Let X be a progressively measurable bounded process. Show that the process (Yt=Rt
0Xsds)t is progressively measurable (with respect to the same filtration).
Exercise 7. Let X, Y be two processes, modifications one of each other. Suppose that X is adapted to a filtration F = (Ft)t and that F0 is completed, that is, contains all the P -negligible sets. Prove that also Y is adapted to F .
Exercise 8 (P). Let X be a process, adapted to a filtration F . Fix t ≥ 0 and (tn)n a sequence of non-negative times converging to t. Show that the r.v.’s lim supnXtn and lim infnXtn are measurable with respect to Ft+ = ∩s>tFs; in particular, the set {∃ limnXtn = Xn} is in Ft+.
Is it true that the r.v. lim supr→tXr is Ft+-measurable? What if X is right-continuous (or left-continuous)?
3 Gaussian r.v.’s
Exercise 9 (P). Let (Xn)n be a family of d-dimensional Gaussian r.v.’s, converging in L2 to a r.v. X. Show that X is Gaussian and find its mean and convariance matrix. A stronger result holds: the thesis remains true if the convergence is only in law (see Baldi 0.16). [Recall that Y is Gaussian with mean m and cocariance matrix A if and only if its characteric function is ˆY (ξ) = exp[im · ξ − 12ξ · Aξ].]
Exercise 10. Let X be a Rd-valued non-degenerate Gaussian r.v., with mean m and covariance matrix A. Find an expression for its moments.
4 Conditional expectation and conditional law
Exercise 11 (P). Prove the conditional versions of the following theorems: Fatou, Lebesgue, Jensen.
Exercise 12. Let X1, . . . Xn a family of i.i.d. r.v.’s, call S = Pn
k=1Xk. Prove that E[Xk|S] is independent of k and so is n1S.
Exercise 13. Let X be a r.v., G, H be two filtrations such that X is independent of H.
Is it true that E[X|GW H] = E[X|G]?
Exercise 14 (P). Let X, Y be two r.v.’s with values in Rd. We know, from a mea- surability theorem by Doob (see the notes and Baldi, Chapter 0), that, for every Borel bounded function f on Rd, there exists a measurable function g = gf on Rd, such that E[f (X)|Y ] = g(Y ) a.s.. Notice that the statement remains true if we change g outside a negligible set with respect to the law of Y .
Prove that, for every f , gf depends only on the law of (X, Y ). Conversely, prove that the law of (X, Y ) depends only on the law of Y and such a family (gf)f.
Exercise 15 (P). Here is an example where the g of the previous exercise (and so the conditional expectiation) can be computed. Let X, Y be two r.v.’s with values in Rd, suppose that (X, Y ) has an absolutely continuous law (with respect to the Lebesgue measure), call ρY, ρX,Y the densities of the laws resp. of Y , (X, Y ) (which is the relation
between them?). Prove that, for every Borel bounded function f on Rd, a version of E[f (X)|Y ] = g(Y ) is given by
g(y) = Z
Rd
f (x)h(x, y)dx, (1)
h(x, y) = ρX,Y(x, y)
ρY(y) 1ρY>0(y). (2)
Notice that the statement remains true if we change h outside the set {ρY > 0}.
Exercise 16. With the notation of the previous exercise, compute h when (X, Y ) is a non-degenerate Gaussian r.v. with mean m and covariance matrix A.
Notice that, even if h is defined up to Y#P -negligible sets (and so up to Lebesgue-negligible sets, since Y is Gaussian non-degenerate), the expression (2) gives the unique continuous representative of h (we will call it h as well). Notice also that, for fixed y, h(·, y) is a Gaussian density. For such h, we will denote the corresponding gf as gf(y) = E[f (X)|Y = y].
5 Markov processes
Exercise 17. Show that the Markow property, with respect to the natural filtration, depends only on the law of a process: if X is a Markov process (with respect to Ft = σ(Xs|s ≤ t)) and Y has the same law of X, then Y is also Markov (with respect to Gt = σ(Ys|s ≤ t)). Show also that, if p is a transition function for X, then p is a transition function also for Y .
Exercise 18. Show that, if X is a Markov process with respect to a filtration F , then it is Markov also with respect to a subfiltration G (such that X is adapted to G).
Exercise 19 (P). Let Sn be a symmetric random walk, call Mn = max0≤j≤nSj. Show that M is not a Markov process.
Exercise 20. Find a Markov process X, such that |X| is not Markov (with respect to the same filtration of X or even the natural filtration of |X|). Can you guess a sufficient condition, for example on the joint law of (Xs, |Xt|) (for s < t), which makes
|X| Markov?
Exercise 21 (P). Let X be a process with independent increments. Show that X is a Markov process.
Exercise 22. Let X be a process with independent increments; call µ0 the law of X0, µs,t the law of Xt− Xs, for s < t. Show that these laws determines uniquely the law of X and that, for every s < r < t, it holds µs,t = µs,r∗ µr,t, that is
Z
Rd
Z
Rd
f (x + y)µs,r(dx)µs,r(dy) = Z
Rd
f (z)µs,t(dz). (3)
Conversely, let µ0, µs,t, 0 ≤ s < t, be a family of probability measures on Rd such that, for every s < r < t, it holds µs,t = µs,r∗ µr,t. Show that there exists a process X such that the law of X0 is µ0 and, for every s < t, the law of Xt− Xs is µs,t. Is there a filtration which makes X a process with independent increments?
Exercise 23. Let X be a process with independent increments, with values in Rd. Sup- pose that, for some s ≥ 0, Xshas a law which is absolutely continuous with respect to the Lebesgue measure (resp. non atomic). Prove that, for every t ≥ s, Xt has an absolutely continuous (resp. non atomic) law.
6 Brownian motion
Exercise 24. Show that a Brownian motion is unique in law.
Exercise 25 (P). Verify the scaling invariance of the Brownian motion: if W is a real Brownian motion, with respect to a filtration F , then
1. for every r > 0, (Ws+r− Wr)t is a Brownian motion with respect to (Fs+r)s; 2. −W is a Brownian motion with respect to (Fs)s;
3. for every c > 0, (√1cWcs)s is a Brownian motion with respect to (Fcs)s;
4. the process Z, defined as Z0 = 0, Zs = sW1/s for s > 0, is a Brownian motion with respect to its natural completed filtration.
A remark on property 3: if we put s = t/c, 3 implies that ˜Ws := √1cWt is a Brownian motion; in other words, we can say that the space scales as the square root of the time, in law. Think about the links between this and H¨older and no BV properties of the trajectories.
Exercise 26 (P). Let W be a real Brownian motion. Show that, for a.e. trajectory, limt→0
Wt
t = 0. (4)
Hint: use Exercise 25.
Exercise 27 (P). Let W be a real Brownian motion. For T > 0, ω in Ω, define the random set ST(ω) = {t ∈ [0, T ]|Wt(ω) > 0}. Show that the law of the r.v.
L1(ST)
T (5)
is independent of T (L1 is the 1-dimensional Lebesgue measure). This law is in fact the arcsine law: P (L1(ST) ≤ αT ) = π2 arcsin(√
α).
Exercise 28. Let W be a real Brownian motion. Prove the following facts:
• For a.e. ω in Ω, the trajectory W (ω) is not monotone on any interval [a, b], with a < b; in particular, there exists a dense set S in [0, +∞[ of local maximum points for W (ω).
• For a.e. ω in Ω, the trajectory W (ω) assumes different maxima on every couple of (non-trivial) compact disjoint intervals with rational extrema; in particular, every local maximum for W (ω) is strict.
Exercise 29. Let W be a real Brownian motion. Prove that, if A is a negligible set of Rd (with respect to L1), then, for a.e. ω, the random set C(ω) = {t ∈ [0, T ]|Wt(ω) ∈ A}
is negligible (again with respect to L1).
Exercise 30. Let W be a real Brownian motion. Show that, on every interval [0, δ], δ > 0, W passes through 0 infinite times, with probability 1.
Exercise 31. A d-dimensional Brownian motion, with respect to a filtration F = (Ft)t, is a process B, adapted to F , with values in Rd, such that:
• B0 = 0 a.s.;
• B has independent increments (with respect to the filtration F );
• for every s < t, the law of Bt− Bs is centered Gaussian with covariance matrix (t − s)Id (Id is the d × d identity matrix);
• B has continuous trajectories.
In case F is the natural completed filtration of B, we call B a standard Brownian motion.
Show that a d-dimensional process X is a standard Brownian motion if and only if its components Xj’s are real independent standard Brownian motions.
Exercise 32. Let W a real Brownian motion. Prove that, for every T > 0, α < 1/2, there exists c > 0 such that the event {Wt ≤ ctα, ∀t ∈ [0, T ]} occurs with positive probability.
Hint: prove first than, for any fixed c, there exists a (small) T > 0, depending only on c, α and the law of W , such that the above event is not negligible.
7 Stopping times
Exercise 33 (P). Let X be a continuous process, with values in a metric space E, adapted to a right-continuous completed filtration F ; let G, F be an open, resp. closed set in E. Define τG= inf{t ≥ 0|Xt∈ G}, ρ/ F = inf{t ≥ 0|Xt∈ F }; notice that ρF = τFc. Prove that τG, ρF are stopping times with respect to F .
Show that the laws of τG and of ρF depend only on the law of X. Prove also that the laws of (τG, X) and (ρF, X), as r.v.’s with values in [0, +∞] × C([0, T ]; E), depend only on the law of X.
Convention: unless otherwise stated, if the set {t ≥ 0|Xt ∈ G} is empty, we define/ τG= +∞; an analogous convention will be used in the sequel for similar cases.
Exercise 34. Let X be a real process, adapted to F , and let τ be a random time, such that {τ < t} is in Ft for every t. Show that τ is a stopping time with respect to the augmented filtration F+= (Ft+)t, where Ft+ = ∩s>tFs.
Exercise 35. Prove the stopping theorem for Brownian motion: If W is a real Brow- nian motion motion with respect to a filtration (Ft)t and τ is a finite stopping time (with respect to the same filtration), then (Wt+τ− Wτ)t is a standard Brownian motion, independent of Fτ.
Exercise 36. Let X be a continuous process with values in R2, such that X0 = 0 and the law of X is invariant under rotation (i.e. the processes (RXt)t and (Xt)tare equivalent, for every orthognonal matrix R). Let τ = inf{t ≥ 0||Xt| = 1} and suppose τ < +∞ a.s.
(see the convention in Exercise 33). Show that τ and Xτ are independent r.v.’s and that the law of Xτ is λS1, the renormalized Lebesgue measure on the sphere S1.
We recall that λ is the unique probability measure on S1 with is invariant under rotation and satisfies λ([cα, cβ]) = cλ([α, β]) for every c > 0, α, β angles.
Find examples of R2-valued continuous processes X (with X0= 0) such that:
• τ , Xτ are independent, but the law of Xτ is not λS1;
• the law of Xτ is λS1, but τ and Xτ are not independent;
• for every t > 0, P (Xt= 0) = 0 and the law of |XXt
t| is λS1, but the law of Xτ is not λS1.
8 Martingales: examples, stopping theorem, Doob and maximal inequalities
Exercise 37. Let M be a (continuous- or discrete-time) martingale, with respect to a filtration F ; let G be a subfiltration of F (i.e. Gt⊆ Ft for every t), such that M is still adapted to G. Verify that M is a martingale with respect to G.
Exercise 38 (P). Let (ξj)j be a sequence of i.i.d. r.v.’s. Let f be a measurable function such that E[|f (ξ1)|] < +∞. Prove that the following processes are martingales:
• Xn=Pn
j=1f (ξj) − nE[f (ξ1)];
• Yn= E[f (ξ1)]−nQn
j=1f (ξj), if E[f (ξ1)] 6= 0.
Deduce that the following are martingales: ....
Exercise 39. Find an example of a martingale which is not a Markov process.
Exercise 40 (P). Let X be a process with independent increments (with respect to a filtration F ). Prove that Xt− E[Xt] is a martingale (with respect to F ).
Let Y be a process with centered independent increments. Prove that Xt2− E[Xt2] is a martingale.
Exercise 41 (N). Let W be a Brownian motion. Prove that Wt, Wt2−t, exp[λWt−12λ2t], λ in C, are martingales.
Let X = (Xt) be a continuous process, adapted to F , with X0 = 0. Suppose that exp[λWt−12λ2t] is a martingale, for every real λ. Prove that X is a Brownian motion with respect to F .
Exercise 42 (P). Let M be a discrete-time martingale, let τ be a finite stopping time.
Prove that the process Mτ, defined by Mnτ = Mτ ∧n, is a martingale (with respect to the same filtration of M ).
Let X be a continuous-time martingale, let τ be a stopping time, suppose that the the hypotheses of the stopping theorem are satisfied. Prove that Mτ, defined as Mtτ = Mτ ∧t, is a martingale (again with respect to the same filtration).
Exercise 43 (N). Let W be a Brownian motion; for a, b > 0, call τa,b= inf{t ≥ 0|Wt∈/ ] − a, b[}. Compute E[τa,b] and P {Bτa,b = b}.
Exercise 44. Let W be a real Brownian motion; for γ > 0, a > 0, let ργ,a = inf{t ≥ 0|Wt= a + γt}. Prove that P (ργ,a< +∞) = exp[−2γa].
Hint: consider the martingale Mt = exp[2γWt− 2γ2t] and remember the behaviour of the Brownian motion at infinity.
Exercise 45 (N). Let X be a continuous sub-martingale, let h : R → R be a positive convex nondecreasing function, θ > 0. Prove that
P (sup
[0,T ]
Xt≤ λ) ≤ e−h(θλ)E[exp[h(θXT)]]. (6)
If X is a Brownian motion, taking h(x) = ex and making the infimum over θ > 0, we get the exponential bound for the Brownian motion.
9 Martingales: convergence and Doob-Meyer decomposi- tion
Exercise 46 (N). Let M be a continuous martingale in L2, with M0 = 0. We remind that, since the process A = hM i is nonnegative nondecreasing, it converges to some finite or infinite limit, as t → +∞. Prove the following dichotomy:
• on {limt→+∞At< +∞}, M converges a.s. as t → +∞;
• on {limt→+∞At= +∞}, it holds for a.e. ω: for every a ≥ 0, the trajectory |M (ω)|
reaches a at least one time; in particular, a.e. trajectory does not converge.
Exercise 47. Let (Mt)t≥0be a positive (i.e. Mt≥ 0) continuous supermartingale and let T = inf {t ≥ 0 : Mt= 0}. Prove that, if T (ω) < ∞, then for any t ≥ T (ω), Mt(ω) = 0.
10 Stochastic integrals
Through all this section, let (Bt)t≥0be a real continuous Brownian motion, starting from 0, defined on a filtered probability space (Ω, A, P, F = (Ft)t≥0), where F satisfies the usual assumptions.
Exercise 48. Prove that, for 0 ≤ α ≤ β < ∞ the space M2[α, β] of (P ⊗ L -equivalence classes of ) progressively measurable real processes (Xr)α≤r≤β such that
E
Z β α
|Xr|2dr
< ∞,
with the scalar product hX, Y i = Eh Rβ
α XrYrdri
, is a Hilbert space. (Hint: it is a closed subspace of L2(Ω, A ⊗ B ([α, β]) , P ⊗ L ).)
Exercise 49. (see Baldi, 6.10) Prove that, for α ≤ β, M2[α, β] 3 H 7→Rβ
α HrdBr is ho- mogeneus with respect to the product of bounded Fα-measurable r.v.’s, i.e.Rβ
α Y HrdBr= Y Rβ
α HrdBr if Y ∈ L∞(P, Fα).
Exercise 50. (see Baldi, 6.9) If X ∈ M2[α, β] is a continuous process such that supα≤r≤βXr ∈ L2(P) then, for every sequence (πn)n of partitions α = tn,0 < . . . <
tn,mn = β, with |πn| → 0, it holds
n→∞lim
mn−1
X
k=0
Xtn,k Btn,k+1− Btn,k = Z β
α
XrdBr
where the limit is taken in L2(P).
Exercise 51. Assuming that H has adapted C1 trajectories on [α, β], prove that a.s.
Z β α
HsdBs= HβBβ− HαBα− Z β
α
Hs0Bsds.
Exercise 52. On a probability space (Ω, A, P), let X be a real random variable, inde- pendent of a σ-algebra F ⊆ A. Assuming that the law of X is N (0, σ2), and that Y is a F -measurable real r.v. with |Y | = 1 a.s., prove that XY has the same law of X and it is independent of F .
Exercise 53. Assume that (Hs)s≥0 is adapted and |Hs| = 1 for any s. Prove that the process Zs=Rs
0 HrdBr is a Brownian motion.
Exercise 54. (Wiener integrals) Let f1, . . . , fn∈ L2((0, T ),L ). Prove that the random variables
Fi = Z T
0
fi(s) dBs∈ L2(Ω, P) (i = 1 . . . , n)
have joint Gaussian law: compute its mean vector and its covariance matrix.
Let f be a locally bounded Borel function defined on [0, ∞). Prove that the process Z·=
Z · 0
f (s) dBs is Gaussian with independent increments.
Exercise 55. (Multiple Wiener integrals) Fix T > 0, let
∆2 =(s, t) ∈ R2|0 ≤ s ≤ t ≤ T
and call rectangle any set R =]a, α]×]b, β] ⊆ ∆2. Prove the following statements.
1. For any rectangle R =]a, α]×]b, β], t 7→
Z t 0
IR(s, t) dBs =
Bα− Ba if t ∈]b, β]
0 otherwise
defines a process IR· B ∈ M2[0, T ].
2. For R, R0 rectangles, hIR· B, IR0 · BiM2 =L2(R ∩ R0).
3. Given a function f =Pn
i=1αiIRi where αi ∈ R and Ri are rectangles, f · B =
n
X
i=1
αi(IRi· B) ∈ M2[0, T ] does not depend on the representation of f .
4. The following isometry holds:
hf · B, g · BiM2 = Z
∆2
f (s, t) g (s, t) dsdt.
5. The map f 7→ f · B extends to an isometry of L2 ∆2,L2 into M2[0, T ], and in particular f · B is well-defined for any f ∈ L2 ∆2,L2 as follows:
given fn→ f in L2 ∆2,L2, put f · B = lim
n (fn· B).
6. Define the multiple Wiener integral I2(f ) =
Z T 0
dBt Z t
0
f (s, t) dBs= Z T
0
(f · B)tdBt∈ L2(P) and prove that
E
"
Z T 0
dBt
Z t 0
f (s, t) dBs
2#
= Z
∆2
f (s, t) g (s, t) dsdt.
Exercise 56. With the notation of the above exercise, is the process (0 ≤ r ≤ T ) r 7→
Z r 0
dBt
Z t 0
f (s, t) dBr∈ L2(P ) adapted? is it a martingale?
Exercise 57. Repeat the construction of the exercise above for any n > 2 and provide a precise definition of
In(f ) = Z T
0
dBtn
Z tn
0
dBtn−1
Z tn−2
0
· · · Z t2
0
dBt1f (t1, t2, . . . , tn) ∈ L2(P) , for any f ∈ L2(∆n,Ln), where
∆n= {(t1, . . . , tn) ∈ Rn|0 ≤ t1 ≤ . . . ≤ tn≤ T } .
Exercise 58. With the notation of the above exercises, prove the following orthogonality relations, for multiple Wiener integrals: given 1 ≤ n, m ≤ m, f ∈ L2 ∆n,L2, g ∈ L2 ∆m,L2, it holds
E [In(f ) Im(g)] =
hf, giL2(∆n) if n=m
0 otherwise.
Moreover, E [In(f )] = 0.
11 Itˆo’s Formula
As in the previous section, we assume that we are given (Bt)t≥0, a real continuous Brownian motion, starting from 0, defined on a filtered probability space (Ω, A, P, F = (Ft)t≥0), where F satisfies the usual assumptions.
Exercise 59. Using Itˆo’s formula, find alternative expressions for 1. Rt
0BsdBs, Rt
0Bsds;
2. Bt2, Rt
0Bs2dBs; 3. cos (Bt), Rt
0sin (Bs) dBs; 4. Rt
0exp {Bs} dBs.
Exercise 60 (Stochastic exponential). (see Baldi, sec. 7.2) For H ∈ Λ2([0, ∞[), put Xt=Rt
0 HsdBs and prove that
t 7→ E (X)t= exp
Xt−1
2 Z t
0
|Hs|2ds
is a positive local martingale.
Exercise 61. Prove that for any t > 0, E [E (B)t] = 1, and limt→∞E (B)t= 0 a.s.
Exercise 62. Define for t ∈ [0, 1[
Zt= 1
√1 − texp
− Bt2 2 (1 − t)
.
Show that t 7→ Ztis a positive martingale for t ∈ [0, 1[, with E [Zt] = 1 and limt→1Zt= 0.
Exercise 63 (Exponential inequality). (see Baldi, prop. 7.5) Let H ∈ Λ2([0, T ]) and fix k > 0. Then
P sup
0≤t≤T
Z t 0
HsdBs
> c, Z T
0
Hs2ds ≤ kT
!
≤ 2 exp
− c2 2kT
.
Exercise 64. Prove that, given p ≥ 2, Xt=Rt
0HsdBs, for H ∈ Λ2([0, T ]), it holds
|Xt|p = Z t
0
p |Xs|p−1sgn (Xs) HsdBs+1 2
Z t 0
p (p − 1) |Xs|p−2Hs2ds.
Exercise 65. (A first step towards local times) Given > 0, set f(x) = x2+ 21/2
, use Itˆo’s formula to show that for H ∈ M2([0, T ]), it holds, for Xt=Rt
0HsdBs, E
h
XT + 21/2i
= E
"
Z T 0
2
2 Xt2+ 23/2Ht2dt
# .
Deduce that, for some absolute constant C > 0, it holds lim sup
→0 E
Z T 0
1
2I{|Xt|≤}Ht2dt
≤ CE [|XT|] .
Exercise 66 (Burkholder-Davis-Gundy inequalities). (see Baldi, prop. 7.9) Fix p ≥ 2 and show that there exists some constant Cp > 0 such that, for any T > 0 and any H ∈ M2([0, T ]), it holds
E sup
0≤t≤T
Z t 0
HsdBs
p!
≤ CpE
Z T 0
Hs2ds
p/2! . In particular, this entails that
E sup
0≤t≤T
Z t 0
HsdBs
p!
≤ CpTp−22 E
Z T 0
|Hs|pds
.
(Actually, in the first inequality above, the opposite inequality holds too, with a different constant.)
Exercise 67. Given H ∈ Λ2[0, T ], define recursively In∈ Λ2[0, T ] as follows:
I0(t) = 1 In(t) =Rt
0In−1(s) HsdBs for n ≥ 1.
Prove that for n ≥ 2, it holds
nIn(t) = In−1(t) Z t
0
HsdBs− In−2(t) Z t
0
Hs2ds.
11.1 Multidimensional Itˆo’s Formula
In what follows, we assume that n ≥ 1 is fixed we are given (Bt)t≥0, an n-dimensional Brownian motion, starting from the origin, defined on (Ω, A, P, F = (Ft)t≥0), where F satisfies the usual assumptions.
Exercise 68. Let f ∈ Cb2(Rn), i.e. |∇f | and D2f
uniformly bounded. Prove that 1. if ∆f = 0, then t 7→ f (Bt) is a martingale;
2. if t 7→ f (Bt) is a martingale, then ∆f = 0.
Exercise 69. Let n = 2 and define
(Xt, Yt) = exp Bt1 cos Bt2 , exp Bt1 sin Bt2 . Compute the stochastic differential of the process (Xt− 1)2+ Yt2.
Exercise 70. Let R > 0, x0 ∈ Rn, with |x0| < R and let τR= inf {t ≥ 0 : |x0+ Bt| = R}.
Prove that
E [τR] =
R2− |x0|2 /n.
Exercise 71 (Exit time from a spherical shell). Fix 0 < r < R, T > 0 and x0 ∈ Rn, with r < |x0| < R. Let
τr = inf {t ≥ 0 : |x0+ Bt| = r} τR= inf {t ≥ 0 : |x0+ Bt| = R} τ = τr∧ τR. Assume n ≥ 3, write down and fully justify Itˆo formula for f (x0+ Btτ), where
f (x) = |x|2−n. Taking expectations and letting t → ∞, deduce that
P (τr< τR) =
|x0|2−n− R2−n /
|r|2−n− R2−n .
Letting R → ∞, prove that P (τr < ∞) = (r/ |x0|)n−2 < 1. Deduce the following asser- tions.
1. Points are polar for Bt, i.e.
P (∃t ≥ 0 : Bt= −x0) = 0.
2. The process leaves every compact set with strictly positive probability, but infact it holds limt→∞|Bt(ω)| = ∞ a.e. ω. (Hint: strong-Markov and Borel-Cantelli).
Adapt the steps above for the case n = 2, using f (x) = − log |x| and discuss the validity of the corresponent conclusions.
Exercise 72. Let X1, . . . , Xd be Itˆo processes (defined on the same space, fixed above).
Using the differential notation dXi, d [Xi, Xj], write as Itˆo processes:
1. Πni=1Xi;
2. X1/X2 (assume X2
> for some > 0);
3.
X1
k, where k ≥ 2,
4. log X1, (assume X1 > for some > 0);
5. X1X2
, (assume X1 > for some > 0);
6. sin X1 cos X2 + cos X1 sin X2;
11.2 L´evy theorem
Exercise 73. Given an n-dimensional Wiener process B, let Hs ∈ Λ2B([0, ∞]) with HsHs∗ = Id for a.e. ω, for all s ∈ [0, ∞[. Prove that
t 7→
Z t 0
HsdBs
is a Brownian motion.
Exercise 74. Given a real valued Wiener process B, let Hs ∈ Λ2B([0, ∞]) and assume thatRt
0Hs2ds = f (t) is a deterministic process. Prove that t 7→
Z t
0
Hsds is a Gaussian process.
Exercise 75 (Reflection principle). Let B be a real valued Wiener process, and let a > 0.
Write Ta= inf {t ≥ 0 : Bt≥ a}. Define Wt=
Z t 0
I{s<Ta}− I{s≥Ta} dBs
and prove that W is a Wiener process. Use this fact and the identity {Ta< t , Bt< a} = {Wt> a}
to conclude that
P (Ta< t) = P (Bt> a) + P (Wt> a) = P (|Bt| > a) .
12 SDEs
Exercise 76. Consider the matrix A =
0 1
−1 0
and the two-dimensional SDE, with additive noise, dXt= AXtdt + σdBt,
with X0 = (p0, q0), σ = (0, σ0) and dBt is the stochastic differential of a real (1- dimensional) Brownian motion. Recall the existence and uniqueness results for such an equation and prove that
E h
|Xt|2i
= |X0|2+ σ02t.
Exercise 77 (Ornstein-Uhlenbeck process). Consider a real-valued Wiener process B and two real numbers θ, σ. Consider the following SDE
dXt= −θXtdt + σdBt with X0 = x0 and prove the following assertions.
1. It holds
Xt= e−θtx0+ e−θt Z t
0
eθsσdBs. 2. The process Xt is Gaussian.
3. If θ > 0, then limt→∞Xt exists in in law but not in L2(Ω, P). Compute its mean and covariance.
Exercise 78 (Variation of constants formula). Consider a real-valued Wiener process B and two real numbers θ, σ and a continuous function f : [0, ∞[×R → R. Consider the following SDE,
dXt= (−θXt+ f (t, Xt)) dt + σdBt, with X0 = x0. Prove the following assertions.
1. Xt admits the representation Xt= e−θtx0+ e−θt
Z t 0
eθsf (Xs) ds + e−θt Z t
0
eθsσdBs.
2. Assume that f (t, x) = g (t) does not depend on x and prove that Xt is Gaussian.
3. Assuming f (t, x) = g (t) and that limt→∞e−θtRt
0eθsg (s) ds = µ, prove that limt→∞Xt
exists in law. Compute its mean and covariance.
Exercise 79. Let µ, σ be real numbers, with σ 6= 0 and let B be a real-valued Wiener process, starting from 0. Consider the following SDE:
dSt= St(µdt + σdBt) , S0= s0 ∈]0, ∞[.
Prove the following assertions.
1. Strong existence and uniqueness hold for St, which admits the following expression St= s0exp
µ − σ2/2 t + σBt > 0.
2. The law of log St is Gaussian, i.e. the law of St is log-normal. Compute its mean and variance.
3. Discuss the behaviour of St as t → ∞.
Exercise 80. Fix T, K > 0 and let µ, σ B and S as in the exercise above. Write Φ (x) =Rx
−∞exp −t2/2 /√
2π for the repartition function of a normal distribution.
1. Show the following Black-Scholes formula for the price at time 0 of a European call option with expiration time T and strike price K:
C = e−µTE(ST − K)+ = s0Φ
−c + σ√ T
− Ke−µTΦ (−c) , where
c = log (K/c0) − µ − σ2/2 T σ√
T .
2. Show Black-Scholes formula for the correspondent European put option:
P = e−µTE(K − ST)+ = s0Φ
−p + σ√ T
− Ke−µTΦ (−p) , where
p =log (K/c0) − µ + σ2/2 T σ√
T .
3. Deduce the Put-Call parity for the European option:
C − P = S0− K.
Exercise 81. Let µ, µ0, σ, σ0 be real numbers, and let B be a real-valued Wiener process, starting from 0. Consider the following SDE’s:
dSt= St(µdt + σdBt) , S0= s0 ∈]0, ∞[, dSt0 = St0 µ0dt + σ0dBt , S0= s00 ∈]0, ∞[
where s0, s00 are real numbers, with s0 6= 0. Prove that if, for some 0 ≤ t1 < t2, it holds St1 = S0t1 St2 = St02
then s0 = s00, µ = µ0 and σ = σ0.
13 Itˆo processes and stopping times
Exercise 82. On a filtered probability space
Ω, A, P, F = (Ft)t≥0
, with the usual as- sumptions, let τ be a stopping time, i.e. a [0, ∞]-valued r.v. such that, for every t ≥ 0, {τ ≤ t} ∈ Ft. Prove that there exists a decreasing sequence of stopping times τn, con- verging uniformly to τ on {τ < ∞}, such that τn(ω) ∈ N/2n∪ {∞}.
Exercise 83. Let H ∈ MB2[0, T ] and let τ be a stopping time (with respect to the natural filtration of a Brownian motion B). Prove that
s 7→ HsI[0,τ [(s) is a process in MB2[0, T ] and that, if we write Xt=Rt
0HsdBs, then Z T
0
HsI[0,τ [(s) dBs= XT ∧τ.
(Hint: assume first that the range of τ is discrete and use the locality of the stochastic integral, then use the exercise above)
