BMSC Theory

Question 1

Define one-dim Brownian motion.

Define Brownian bridge.

Answer 1

Def 1.1.:
We say that the real-valued process (B_t){t>=0} is a one-dim Brownian motion started from 0 if:
* B(0)=0 almost surely, and for all t>=0, the law of B_t is N(0,t) (centered Gaussian w. variance t).
* For every positive integer k and all strictly incr. {t_1,…,t_k} the k increments B{t_1}-B_0,B_{t_2}-B_{t_1},…,B_{t_k}-B_{t_{k-1}} are independent random variables.
* For each t>=0 and h>0, the law of B_{t+h}-B_t is the same as the law of B_h-B_0
* There exists a m-able set A with probability 1, s.t. for all w in A, the map t->B_t is continuous on R_+.

page 22, definition 3.7

Question 2

Prove two “std. prob. theory results” used in construction of BM:

• X,Y indep Gaussian r.v.s with var.s sig_X^2,sig_Y^2 then X+Y is a Gaussian r.v. with var sig_X^2+sig_Y^2
• X and Y have the same variance sig^2, then X+Y and X-Y are two independent centered Gaussian r.v.s with variance 2*sig^2

Answer 2

• Hint: char function E[exp(ilamX)]=exp(-sig^2lam^2/2) and by indep: E[exp(i(lam(X+Y)+mu(X-Y))]=E[exp(ilam(X+Y))]E[exp(imu(X-Y))]=exp(-(sig_X^2+sig_Y^2)lam^2/2)
• E[exp(i(lam[X+Y]+mu[X-Y]))]=E[exp(i[lam+mu]X)exp(i[lam-mu]Y)]
  =E[exp(i[lam+mu]X)]E[exp(i[lam-mu]Y)]
  =exp(-sig^2([lam+mu]^2+[lam-mu]^2)/2)
  =exp(-sig^2lam^2)exp(-sig^2mu^2)
  =E[exp(ilam(X+Y)]E[exp(imu(X-Y))]

Question 3

STATE LAST PART OF PROOF OF KOLMOGOROV, i.e. starting from |f_n-f_{n-1}| leq 2^{-n*gamma}

What is a continuous modification?

State and prove Kolmogorov’s continuity criterion.

Answer 3

Y is a cont. mod. of X if for all t: [ X_t=Y_t a.s.]

pg. 17, Proposition 2.1

Hint: Cantelli, Markov ( P(|X|>=a)<=E[phi(|X|)]/phi(a) where phi is mon. increasing., non-neg

Question 4

Define centered Gaussian vector and centered Gaussian process.
Show that the law of a centered Gaussian vector X is completely characterized by its covariance matrix.

Answer 4

• A random vector (X1,…,Xn) in R^n is a centered Gaussian vector if any linear combination of the Xi is a centered Gaussian random variable.
• A stochastic process (X_t){t in I} is a centered Gaussian process if any finite-dimensional vector (X{t_1},…,X_{t_p}) is a centered Gaussian vector.
• For any lam_1,…,lam_n in R, the char func E[exp(iSum(lam_iX_i)] is equal to E[exp(iY)] where Y=Sum(lam_iX_i) is a centered Gaussian var. and it is thus equal to exp(-sig_Y^2/2) but sig_Y^2=E[Y^2]=Sum(lam_ilam_jE[X_iX_j]) is determined by the covar matrix.

Question 5

Show that BM is a Gaussian process.

Answer 5

Indeed, for any t_1,…,t_p the vector (B_{t_1},…,B_{t_p}) is a linear combination of indep centered Gaussian vars (B_{t_1},B_{t_2}-B_{t_1},…,B_{t_p}B_{t_{p-1
}) and is thus a centered Gaussian vector.. Since for all t>=0 and h>=0, E[B_tB_{t+h}]=E[B_t^2]+E[B_t(B_{t+h}-B_t)]=t+E[B_t]E[B_{t+h}-B_t]=t and thus the covar matrix has entries: Sig_B(i,j)=min(i,j).
BTW: This characterises BM: BM is a real-valued centered Gaussian process (B_t)_{t>=0} with covar function Sig_B(i,j)=min(i,j), s.t. the map t->B_t is almost surely continuous.

Question 6

State and prove invar. unter time reversal, scale invariance, inversion invariance

Answer 6

page 21, Proposition 3.4

Question 7

State and prove Blumental’s 0-1 law.

Answer 7

page 28, Proposition 4.3

Question 8

Show that almost surely, limsup B_t/sqrt(t)=inf and liminf B_t/sqrt(t)=-inf (as t–>0).

Answer 8

page 29, Proposition 4.5

Question 9

State the two versions of the Law of the iterated logarithm and the convergence results of B_t/sqrt(t) for t –>0 or inf respecitvely.
[Prove the Laws of iterated logirthm? Update!]

1) limsup B_t/sqrt(t) for t –> 0
2) liminf B_t/sqrt(t) for t –> 0
3) limsup B_t/sqrt(t) for t –> inf
4) liminf B_t/sqrt(t) for t –> inf

Answer 9

page 30, Remark 4.9
Sheet 3, ex 4

1) inf
2) -inf
3) inf
4) -inf

Question 10

State and prove the Reflection Principle and its corollaries

Answer 10

Page 32, Proposition 4.13 and Corollary 4.14, Corollary 4.15, Corollary 4.18

Question 11

Define regular boundary points.

Answer 11

U open in R^d.
We say that x in Boundary(D) is a regular boundary point of D if when one considers a BM B started from x, then almost surely inf{t>0: B_t not in D}=0. We say that D has a regular boundary for BM (or in short a regular boundary) if every boundary point is regular.

Question 12

Define harmonic function and solutions of the Dirichlet problem.

Answer 12

page 36, definition 5.1, definition 5.2

Question 13

Let D be a bounded open subset of R^d with compact boundary and f a continuous function defined on the boundary of D.
Prove that if the solution to the Dirichlet problem in D with boundary values f exists, then it is necessarily equal to the function U(x):=E_x[f(B_T)]
(i.e. B BM started from x)

Answer 13

page 36, proposition 5.3

Question 14

Granted that (last prop 5.3) if the Dirichlet problem in bounded open D equal to f on D's boundary has a solution U it is U(x):=E_x[f(B_T]], then:
If all boundary points of the bounded domain D are regular, then there exists (for any continuous function f on D's boundary) a unique solution to the Dirichlet problem, and this solution is equal to the function U.

Answer 14

page 37, Theorem 5.5 (maybe clearer: lecture 7, part 1),
note that this came in Jonas’ exam and after doing the whole thing… the examiner said he only wanted the part after the bullet points…

Hints:
i,ii,iii
i uses same idea as iii and ii uses MP

Question 15

Calculate P_x(T_r < inf) for x in D:=B(0,R) \ [ B(0,r) + dB(0,r)] in R^3.

Answer 15

P_x(T_r < inf)=r/||x||

page 40, section 5.3

Question 16

Prove that in R^3 that ||B_t||–>inf as t–>inf

Answer 16

page 41, Proposition 5.11

Question 17

Prove that for R^d with d>3 that ||B_t||–>inf as t–>inf

Answer 17

page 41, proposition 5.12

Question 18

Let B be a 2-dim BM(x).

Prove B a.s. never hits 0, i.e. T_0.

Answer 18

page 39, proposition 5.13

P_x(T_0 < inf) = 0

Question 19

Prove that {B_t: t>=0} is a.s. dense in R^2 for planar BM.

Lebesgue measure?

Answer 19

page 41, proposition 5.15

Lebesgue( {B_t: t>=0} )=0, Corollary 5.14

Question 20

Define polar sets (for planar BM).

List some props/examples

Answer 20

Definition 5.20 : We say that the compact set K is polar for BM if almost surely {B_t: t>=0} n K = { }

• K in R^2 non polar then a.s. {t>0:Bt in K} is not empty (ex. sheets, check proof length. UPDATE!!)
• K singleton, then polar (because planar BM a.s. doesn’t visit specified points). Consequently, K countable, then polar (union of countably many events with P=0 has P=0)
• K positive Lebesgue measure, then non polar (because P of B1 in K is positive)
• If K is a segment [a,b] between distinct points, then it is non polar (because two indep BMs correspond to two coordinates of B parallel and orthogonal to b-a). (example of a non-polar Lebesgue=0 set).
• Cantor set non polar (proposition 5.21)

page 44

Question 21

State the “cousins” of the Dirichlet problem and their solutions.

Answer 21

page 46, Proposition 5.22, remark 5.23, proposition 5.24
Check exercise sheet UPDATE (add excises of examples!!)

Note: LapH= - k^2H is called the Helmholtz equation.
Note: Lap*H=f is called the Poisson equation

Question 22

State and prove Donsker’s theorem.

Answer 22

Refer to document BMpart1l.pdf
page 19, theorem 2.4,
page 48, theorem 6.1,
proof via coupling page 50, 6.2 Proof via coupling
(might ask proofs individually ie refers to them, he prefers to ask the first one ie proof via coupling)

Question 23

Just Levy
What follows from Donsker’s theorem?
What is Donsker’s theorem useful for?

Answer 23

Useful for deriving limiting distribution of continuous functionals of simple random walks in terms of continous functionals of BM, e.g.
page 51 Proposition 6.5 (Paul Lévy): ( max_{t>=s} B_s -B_t){t>=0} has the same law as (|B_t|){t>=0}.
Exercise 3, sheet 5, especially part 1

https://math.stackexchange.com/questions/392042/an-application-of-donskers-theorem

Question 24

What do we always implicitly assume about our filtration? (ie “usual conditions”)

Answer 24

• F_0 contains all sets A’ contained in measureable sets A (A in F) with probability 0
• The filtration is right-continuous, i.e. t>=0 F_t=Intersection( F_{t+eps} : eps>0 )

Question 25

Define continuous martingales.

Name an example.

Answer 25

page 56, Definition 1.1:
A process (M_t)_{t>=0} defined in this filtered probability space is said to be a continuous martingale with respect to the filtration (F_t)_{t>=0} if it is an L^1 process (i.e. each M_t in L^1(F_t,P) ) that is adapted to this filtration, such that:
* For all t>=s: M_s=E[M_t|F_s] almost surely
* There exists an event of probability 1 such that on this event, t-->M_t is a continuous function of R_+

Example par excellence: BM page 56, Definition 1.1.

Question 26

State Doob’s L^2 inequality for continuous martingales and briefly explain why it follows from the discrete case.

L^p inequality
Maximal inequality

Answer 26

page 57, proposition 1.2:
If (M_t){t>=0} is a continuous martingale and M_t in L^2 for some given t, then:
4*E[M_t ^2] >= E[sup[0,t] M_s ^2].

Pf: discrete for (M_{tj2^-n})t implies 4E[M_t ^2] >= E[sup{ M^2_{jt2^{-n} : 2^n>=j} ].
A.s. continuity implies max_{2^n>=j} M_{jt*2^{-n}} ^2 –> max[0,t] M_s ^2, so that by mon. conv. follows.

L^p: (p/(p-1))^p * E[|M_t|^p] >= E[sup_[0,t] |M_s|^p]
((E[sup_[0,t] |M_s|^p]=lim E[max_{2^n>=j} |M_{jt2^{-n}|^p]))

Maximal inequality: E[|M_t|] >= lam* P(sup_[0,t] |M_s|>lam)
((P[sup_[0,t] |M_s| > lam ] =lim P[max_{2^n>=j} |M_{jt2^{-n}| >lam] leq E[ |M_t| 1{sup_[0,t] |M_s| > lam} ))

Question 27

Def. (X_i)_i uniformly integrable.

State implications between L^1-convergence, L^p boundedness, uniformly integrable

Answer 27

*lim (sup_i E[ |X_i| * 1{|X_i|>A} ] = 0 ( lim of A–>inf)

• (X_i)_i converges in L^1 iff converges in Prob and uniformly integrable.
• (X_i)_i bounded in L^p for some p>1 implies uniformly integrable
• (X_i)_i uniformly integrable implies bounded in L^1

Recall that bounded in L^p when sup_i E[|X_i|^p]< inf.

Question 28

State convergence criteria for continuous martingale (M_t)_t.

Answer 28

Prop 1.2

• bounded in L^1 implies converges almost surely to some r.v. M_{inf} as t–>inf
• uniformly integrable implies converges almost surely and in L^1 as t–>inf to some r.v. M_{inf} and M_t=E[M_{inf} | F_t] almost surely (in part. E[M_{inf}]=E[M_0]
• bounded in L^p for some p>1 implies converges almost surely and in L^p to some r.v. M_{inf}

Question 29

Define stopping time.

Answer 29

The r.v. T with values in [0,inf] is said to be a stopping time for the filtration (F_t)_{t>=0} if for any positive t, the event {t>=T} is in F_t.

Question 30

Only i) implies ii)!!

State the stopping time lemma (equivalence) and its corollary.
Prove? UPDATE

Answer 30

Page 57, lemma 1.10

in both (Optional stopping, bounded stopping times)

Question 31

State the optimal stopping theorem (simple version).

Answer 31

part 2: proposition 1.12, pg 58

2019: page 60, proposition 1.9

Question 32

State and prove the full optimal stopping theorem.

Answer 32

prop 1.13

page 61, proposition 1.10

Question 33

Define the quadratic and exponential martingale and prove that they are martingales.

Answer 33

lemma 1.14

page 61, lemma 1.11

Question 34

Define a bounded martingale.

Answer 34

(M_t)_t is a bounded martingale if

• it is a martingale
• |M| is almost surely bounded by some deterministic constant C (i.e. almost surely for all t>=0, C>=|M_t|)
• For some deterministic integer K, M is almost surely constant on [K,inf) (i.e. almost surely for all t>=K, M_t=M_K)

BTW: It follows that then M_t is in L^2, thus variance of increment is sum of variances of disjoint increments (in the increment). Same after conditioning. Thus (M_t^2)_t is a submartingale E[diff | F_t]>=0.

Question 35

How do we define quadratic variation?

Answer 35

• M bounded martingale, then there exists a unique non-decreasing process A (called quadratic variation) s.t. A_0=0 and ((M_t)^2-A_t)_{t>=0} is an L^2-martingale. (page 63, Proposition 2.2)
• Some equivalences: eg exists unique L2 mart. X_t s.t. (A_t:=(M_t)^2-2*X_t)_t is non-decr. and A_0=0, M mart both L2 adapted
• Exists unique decomp of M^2_t=2*X_t+A_t, A non-decr A_0=0, X
• We simultaneously show that V_{delta_n}:=Sum[ (M_{t^n_{i+1}}-M_{t^n_t})^2 ; i=1,…,m_n ] converges in L^2 to A_t (page 66, proposition 2.5, still for bounded mart.s)
• We then prove in page 68, theorem 2.8 (quadratic variation for L^2-martingales) : M L^2 mart.
  There exists a unique adapted contin. non-decr. pr. A with A_0=0 s.t. (M^2-A) is a mart.. Furthermore, we have V converges in probability to A.
• We then prove in page 70, Theorem 2.13 (Quadratic variation of local martingales) the props with convergence in prob for local martingales. i.e.
  There exists a unique adapted continuous non-decreasing process (A_t){t>=0} with A_0=0 s.t. ((M_t)^2-A_t){t>=0} is a local martingale. Furthermore, for all t>=0, and for all choices of nested sequences delta_n of subdivisions of [0,t] with |delta_n|–>0, V_{delta_n} does converge in probability to A_t as n–>inf.

Question 36

Define local martingale.

Answer 36

A continuous adapted process (M_t){t>=0} is said to be a local martingale started from 0 (in some filtered prob space) if M_0=0 almost surely and if there exists a sequence tau_k of stopping times such that tau_k—>inf almost surely and such that for all k>=1, M^{tau_k} is a continuous martingale started from 0.
A continuous adapted process (M_t){t>=0} is said to be a local martingale if the process (M_t-M_0)_{t>=0} is a local martingale.

Question 37

Characterize local martingales.

Answer 37

A continuous adapted process (M_t){t>=0} is a local martingale started from 0 if and only if for each k>=1, the process M^{T_k} is a continuous martingale started from 0, where T_k:=inf{t>=0; |M_t|=k}.
A continuous adapted process (M_t){t>=0} is a local martingale if and only if for each k>=1, the process (M^{T_k}-M_0)_{t>=0} is a continuous martingale started from 0, where T_k:=inf{t>=0; |M_t-M_0|=k}.

Question 38

What follows if a local martingale started from 0 has a bounded variation?

Answer 38

Bounded variation means it can written as the difference between two adapted continuous non-decreasing processes and (by page 70, lemma 2.12)
What follows is that it is almost surely equal to 0 for all times.

Potentially UPDATE with proof

Question 39

State and prove the theorem on quadratic variation for local martingales.

Answer 39

page 70, theorem 2.13

Question 40

Is a martingale a local maringale?

Answer 40

Yes

Question 41

Define cross-variation between two local martingales, say what it is the limit of and use it to define a scalar product on the space of martingales bounded in L^2.
What can you say about this space?
Prove it.

Answer 41

page 71-,

bottom of page 77 lemma 3.2 (M^2 is a Hilbert space)

Question 42

What is the Doob-Meyer decomposition?

Prove the exercise.

Answer 42

solution 7, exercise 3

page 72, 2.6.2. Doob-Meyer decomposition.
lemma 2.14
theorem 2.15
results without proof

Question 43

Define càdlàg processes.

Answer 43

A random process (X_t){t>=0} is called a cadlag process if there exists an event of prob 1, s.t. on this event, for all t>=0, the function s–>X_s is right-continuous at t (i.e. lim X{t+h}=X_t as h–>0 from above) and has left-limit denoted by X_{t-} (equal to lim X_{t+h} as h–>0 from below).

Question 44

When is a local martingale bounded in L^p?

Answer 44

page 76, proposition 2.17

Question 45

State and prove the Kunita-Watanabe inequality (i.e. theorem with assumptions).

Answer 45

page 74, proposition 2.18

exercise 5, sheet 8

Question 46

Define elementary process.

In what spaces do they live in?

Answer 46

A process (K_t){t>=0} with K_t:=Sum[ Y{a_j}*1{ t in (a_j,a_j+1] } ; j =0,…,p-1 }, where p pos. int., where a_0,…,a_p are str. mon. non-neg times and each Y_{a_j} is an F_{a_j} m-able r.v..

We denote by E the set of all elementary processes. We denote E_B the set of al elementary processes s.t. each K_t is in L^2 (i.e. the previous r.v. Y_{a_j} are in L^2). These are vector spaces.

Question 47

How are stochastic integrals defined?

Answer 47

• We first define stochastic integrals w.r.t. BM, where the integrand is an elementary process (K_t)t by I(K)t := Sum[ Y{a_j}*(B{min(t,a_{j+1})-B_{min(t,a_j}}) ; j=0,…,p-1 ]=:Int( K_s dB_s ; [0,t] )
• One proves that E_B is dense in P_B (K in E_B iff K in L2 iff I(K) bounded in L2) (P=progr. measurable.; P_B:= in P and E[Int(K^2 ds)] finite)
• One defines stochastic integral w.r.t. BM with integrands from P_B (page 82). K–>I(K) is an isometry between E_B and M^2 (:=cont. L2-mart.s on a filtered space), which we extend to an isometry on P_B=bar(E)_B:
  -Consider sequence K^n in E_B, which converges in P_B
  -Sequence (K^n)_n is Cauchy in norm, i.e. E[Int( (K^n_s-K^l_s)^2ds)]–>0, note that this implies (I(K^n))_n is Cauchy in M^2 because norm of I(K^n)-I(K^l) in M^2 is equal to the distance between K^n and K^l in P_B. (Jonas notes: ->!)
  -The space M^2 is complete, so that there exists a continuous martingale I(K) in M^2 s.t. I(K^n)–>I(K) in this space
  -We note that the continuous mart. I(K) does not depend on our choice of sequence (K^n)_n that approximates K in P_B (indeed if (K~^n) is another such sequence, then I(K^n-K~^n) does converge to 0 in M^2)
• We generalize stochastic integrals to bounded mart.s: I(K)t:=Sum( K{a_j}(M_{min(t,a_{j+1})-M_{min(t,a_j)}) ; j=0,…,p-1).
  One defines E_M as elementary processes s.t. E[Y^2(A_{a_{j+1}}-A_{a_j})] is finite and P_M s.t. E[Int(H dA)] is fintie.
  Aagain notes that I(K) is an isometry, from E_M to M^2 that can be extended on P_M (since E_M is dense)
• We then further generalize this to stochastic integrals w.r.t. local martingales
• We further generalize this to stochastic integrals w.r.t. semimartingales: Int(K_s dZ_s)=Int(K_s dM_s)+Int(K_s dV_s)

Question 48

Define progressively measureable processes.

Define the space they live in.

What are sufficient conditions?

Answer 48

• The process (K_t){t>=0} is said to be progressively measurable with respect to the filtration (F_t){t>=0} if for all t>=0, the map (s,w)–>K_s(w) defined on [0,t]xOmega is mable with respect to the product sig-field B_[0,t] \otimes F_t.
• Set of prog. mable processes denoted P. Subset P_B of P sonsists of prog mable processes s.t. E[Int( (K_s)^2 ds : [0,inf) )] finite. Endowing P_B with the scalar product:
  (K,K’)_B := E[ Int( K_s * K’_s ds; [0,inf] ) ], makes it a Hilbert space.
• Adapted process that a.s. has right or left cont. paths (page 81, lemma 3.5)

Question 49

Prove that the set E_B is dense in the Hilbert space P_B. In other words, for any process K in P_B, one can find a sequence of elementary processes K^n in E_B s.t. lim E[ Int( (K_s - K^n_s)^2 ds ; [0,inf] ]=0 as n–>inf.

Answer 49

page 79, lemma 3.6

Question 50

Let K be in P_B. Show that the quadratic variation of the stochastic integral I(K) is Int( (K_s)^2 ds; [0,t] ).

Answer 50

page 80

Question 51

Show that the cross-variation of I(K) and I(K’) for K, K’ in P_B is
Int( K_s*K’_s ds ; [0,t] ).

Answer 51

page 80

Question 52

τ_{a,b} = inf{t ≥ 0 : B t ∈ {a, b}}
Show: E(τ_{-1,1})=1
Show: a.s. finite
Show: Deduce using optional stopping:
E(exp(−λ^2 τ_{a,b} /2))=(1 + exp(−λ(a+b))/(exp( −λa) + exp(−λb))

Answer 52

part ii, pg. 59

Sheet 6, exercise 1

Question 53

Show that when K in P_B and N in M^2 that almost surely for all t>=0 that CV{I(K),N}_t=Int( K_s dCV{B,N} ; [0,t] ).

Answer 53

Ex 6, sheet 8

page 81, proposition 3.8

Question 54

Characterize I(K).

Answer 54

page 81

Question 55

Define the sets E_M and P_M.

Hilbertness?

Answer 55

Let A be quadr. var. of M.

• E_M is the set of elementary processes (i.e. subset of E) s.t. K_t=Sum( K_{a_j}1{a_j,a_{j+1}] ; j=0,…,p-1 ), where a0,…,a_p are str. incr. non-neg times and K_{a_j} are F_{a_j} mable r.v.s s.t. E[ (K_{a_j})^2(A_{a_{j+1}}-A_{a_j}) ] finite for all j.
• K prog. mable s.t. E[Int( (K_s)^2 dA_s ; [0,inf]] finite
• P_M is a Hilbert space when endowed with (K,K’)_M:= E[ Int( K_s*K’_s dA_s ; [0,inf] )

pg 82

Question 56

Define semimartingale.

Define quadratic variation of semimartingale.

Answer 56

A process (Z_t)_{t>=0} is a cont. semimart. in the filtered prob. space (Omega, F,(F_t)_t,P) if it can be written as Z_t=M_t+V_t, where:

• M is a local martingale in this filtered prob. space
• V is the difference between two adapted cont. non-decr. processes V^+ and V^- started from 0.

The quadratic variation of a semimart. Z_t=M_t +V_t is the quadratic variation of its mart. M_t

Question 57

What holds for the integral of a cont. adapted process w.r.t the quadr. var. of a semimartingale?
What is the corollary of this?
State lemmas/cor

Answer 57

lemmas and cor pg 85-88

Question 58

State and give idea of proof of Ito’s formula for 1-d.

State the observations after the statement, i.e. interpretation of the summands.

Answer 58

page 88, theorem 4.7

Question 59

State Ito in higher dimensions.

Answer 59

page 89, theorem 4.9

Question 60

Prove the applications of optimal stopping theorem
What is the prob. of a d dim BM leaving unit sphere?
What is the prob of a 1 dim BM leaving interval (a,b) with 0 in (a,b)?

Answer 60

in particular that the stopping time is 1 (eg 1)

UPDATE

Question 61

Solve exercise 3, sheet 9

Answer 61

..

Question 62

Solve exercise 4, sheet 9

Maybe do and check one before doing all….

Answer 62

..

Question 63

Prove exercises 5 from 9 and 6 from 10

Answer 63

..

Question 64

Given a local martingale (M_t)_t define a local “exponential” martingale and give and prove two equations relating them. (IMPORTANT)

State generalization after proof.

Answer 64

page 90, proposition 4.11

Same proof F in C^2 on R^2 s.t. d_a F + d^2_x F/2=0, then (F(M_t,QV(M)t)){t>=0} is a local martingale as soon as M is.

Question 65

Give Lévy’s characterization of Brownian motion in 1.d
Prove.
Give for n-d.

Answer 65

page 93, corollary 4.13

also give page 94, corollary 4.14 if time allows

Question 66

How can one construct a BM from a local martingale?

Answer 66

page 91, corollary 4.15

Question 67

Apply Ito to BM.

What happens when F is harmonic?

Answer 67

Theorem 4.17

Question 68

Prove that if a cont. solution to Laplace(H(x))=2alpha(x)H(x) on D (open subset of R^d) with H=f in C^2 on dD (open subset of R^d) exists, then it is unique and it is equal to the function U defined for all x in D by: U(x)=E_x[f(B_T)exp(-Int( alpha(B_t) dt; [o,T] ))], where B is a BM started from x and T denotes its exit time from D.

Answer 68

page 93, proposition 4.18

Question 69

Define the Heat equation problem.

Prove Prop:
If the solution to the heat equation exists, then it is unique and it is equal to U(x,t):=E_x[f(B_t)*1{T>t}], where B is BM started from x and T denotes its exit time from D.

Answer 69

page 93

Question 70

State both theorems about strong solutions to stoch. diff. equations.
Prove the uniqueness part of Theorem 5.2 (page 100)

Answer 70

state gronwall’s lemma
proof:
page 97

Recall Borel-Cantelli and Markov’s inequality (prob the one on wiki)

Question 71

State and prove the Yamada-Watanabe criteria for (pathwise) uniqueness.

Answer 71

page 100, theorem 5.8

Question 72

State the theorem about weak solutions.

Answer 72

page 105, theorem 5.12

Question 73

State the corollary with the definition of squared Bessel processes.

Define Bessel processes.

List 3 properties of Bessel processes. And which theorems are used to prove them in the exercises.

Answer 73

page 106, corollary 5.14

page 106, definition 5.15

page 106 (see ex sheets and old script!)

Question 74

Explain Tanaka’s example

Answer 74

page 104, 5.3.2

UPDATE QUESTION TO BE MORE SPECIFIC!!

Question 75

Prove:
Proposition 5.11. Suppose that F is a continuous function on D that is C^2 in D, such that
F = f on ∂D and such that LF = 0 in D. Then necessarily,
F (x 0 ) = E[f (X_τ )].

Answer 75

page 102

Question 76

UPDATE

As text in this years skript in warm-up:

Prove for D_t:=Radon-Nikodym der. of Q w.r.t (Omega,F_t), where Q abs. cont. w.r.t. P. such that for any A ∈ F_t ,
Q(A) = E[1_A*D_t ]:
Proposition 6.2. This process (D_t ) t≥0 is a uniformly integrable martingale (in the filtered
probability space (Ω, F, (F t ) t≥0 , P )) that converges (almost surely and in L 1 ) to the non-negative
random variable D_∞ . Furthermore, for any stopping time T , the Radon-Nikodym derivative of Q
with respect to P on F T is the random variable D T .

Answer 76

page 109, proposition 6.2

Question 77

State the result on Cameron-Martin space for BM.

Answer 77

page 110, proposition 6.4

Question 78

State the general case of the result on the Cameron-Martin space.

Answer 78

page 111, proposition 6.7

Question 79

State Girsanov’s theorem.

Answer 79

page 111, theorem 6.8

Question 80

State Girsanov’s theorem for BM.

Answer 80

page 110, theorem 6.10

Question 81

Prove Girsanov’s theorem.

Answer 81

page 113

Question 82

Exercise 1 after Girsanov’s BM theorem, there are examples after thm..
Show that B_t - h(t) for h in C is a BM for some prob measure.

Generalize this to a random process.

Answer 82

UPDATE

Question 83

Solve ex. sheet 10

Answer 83

UPDATE

Question 84

State Novikov’s and Kazamaki’s criteria

Answer 84

page 114

Question 85

What results follow from Girsanov’s theorem regarding SDEs?

Answer 85

page 117, proposition 6.12

page 118, Proposition 6.14

Question 86

Prove the strong Markov property.

Answer 86

pg. 31, prop 4.12

Question 87

Name some Harmonic functions.

How can this come up in an exam?

Answer 87

const or affine polys (each coordinates degree is less than two),
real & imaginary parts of holomorphic functions,
ln(x^2+y^2), x/[r(r+z)], x/[r^2-z^2], ||x||^{2(1-n/2)} for n geq 2 on D=IR\0

A domain could be given, and a probability that a Brownian motion leaves/visits this domain could be asked for.

Question 88

Show that if F is a cont. function on the completion of D that is C^2 in D, s.t. F=f on the boundry of D and LF=0 in D. Then necessarily, F(x0)=E[F(X_tau)]

Answer 88

..

Question 89

Construct BM. (vaguely, at least explain the steps)

Answer 89

..

Question 90

Give weak solutions via Girsanov.

Answer 90

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