Homework Assignment 1
Assigned Mon 08/30. Due Mon 09/13.
1. Let , be two stopping times.
(a) Show that , , + are also stopping times.
(b) If almost surely, then show F F .
2. (a) Let cfw_Fn | n N be a decreasing sequence of -algebras (i.e. Fn Fn+1 ), and
Martingales in Continuous Time
Let Ft be a filtration, M adapted and E[|Mt|]< for all t. If the following hold (almost surely,
of course) for all s<t then M is called a
1. Martingale if E[Mt|Fs]=Ms
2. Submartingale if E[Mt|Fs]Ms
3. Supermartingale if E[Mt
Stopping Times and Sigma Algebras
Remark. You can actually characterize martingales by processes that when stopped have the
same expectation. See Chapter 5 Theorem 3.11 of inlar (Graduate Texts 261 Proability and
Stochastics) or problem 3.26 in Karatzas a
880 Stochastic Calculus: Final.
Dec 15th , 2011
This is a closed book test.
You have 3 hours. The exam has a total of 7 questions and 30 points.
You may use without proof any result that has been proved in class or on the
homework.
You may (or may not
A first construction of Brownian Motion
This construction is not very easy, but it does help build some intuition.
Intuitively we are trying to build a continuous-time random walk.
What do we really mean by this?
Let i be i.i.d., for convenience with mean
Quadratic Variation
As stated last time, we claim that M2 is complete under the norm, and that M2c is a closed
subspace.
Our strategy: Just prove for M2c, the proof carries over naturally to the other case (and we don't
really care about the other case ei
Finishing Construction of quadratic variation
Recall that last time we did all of our proofs with MMc bounded. Last time we considered the
discrete quadratic variation:
Xt=1n(Xtti+1Xtti)2
We then demonstrated that as |0 that MM in L2(,L).
Recall that the
880 Stochastic Calculus: Midterm.
Wed 10/19
This is a closed book test.
You have 90 mins. The exam has a total of 5 questions and 50 points.
You may use without proof any result that has been proved in class or on the
homework.
You may (or may not) nd
880 Stochastic Calculus: Midterm.
Oct 17th
This is a closed book test. No calculators or computational aids are allowed.
You have 80 mins. The exam has a total of 4 questions and 20 points.
You may use without proof any result that has been proved in c
A better way to construct Brownian Motion
Gautam doesn't know where this comes from, no print source that he knows of, but it is well
known.
Last time:
1. Kolmogorov-Daniell, consistency theorem gives us an adapted process, with X0=0 a.s.,
and normally di
Constructing Stochastic Integrals: Ito's construction
Our goal: Fix MM2c. We want to define t0XsdM. Now, we can't define using limits of
partitions:
lim|0Xti(Mti+1tMtit)
because we don't have bounded variation for M. We can further note that if X isn't ad
Density of Simple Processes
We had a lemma from last time that we still needed to prove.
Recall from last time that we denoted:
L0=cfw_Simple Processes that are Measurable, Adapted, Bounded,
Piecewise constant in time (L.C.R.L.)
We defined an Ito integral
Defining Integrals for Local Martingales
Recall, we assumed MM2c and (dMdt) (remembering that this second condition could
be replaced with a condition on our processes X).
Now we define
XL(M):=cfw_X|Xadapted, measurable and Et0|Xs|2dMs<
Now if MMc,loc the
Proving Ito's Formula
Today we'll try to prove Ito's formula and then do a brief application to pde.
Recall that for fC1,2 (f:[0,)RdR), with X a continuous local semi-martingale then we
have that: f(X) is a continuous local semi-martingale, and we have th
Assignment 6: Assigned Wed 11/20. Due Wed 12/04
1. Let d N, b : Rd [0, ) Rd be bounded, Borel measurable and : Rd
2
[0, ) Rd be bounded and uniformly Lipschitz. Suppose further there exists
> 0 such that for all t 0 and x, y Rd we have
(i,k)
t
(j,k)
(x)
880 Stochastic Calculus: Final.
Dec 11th , 2013
This is a closed book test. No calculators or computational aids are allowed.
You have 3 hours. The exam has a total of 7 questions and 35 points.
You may use any result from class or homework PROVIDED it
Stochastic Processes
We start with the standard probability notation: is our sample space, F is our sigma algebra
and P is our probability measure.
Definition. A Stochastic Process is a family of Random Variables indexed on t.
Definition. For a fixed we c
Homework Assignment 2
Assigned Thu 10/06. Due Wed 10/26.
1. Let B be a standard 1D Brownian motion. Show that each of the following are also standard 1D
Brownian motions:
(a) cfw_Bt t
(b)
0
1
cfw_ Bt t 0 ,
for any > 0.
(c) Wt = tB 1 for t > 0, and W0 = 0.
Homework Assignment 4
Assigned Thu 11/10. Due Mon 11/28.
W
1. Does there exist a process Y adapted to cfw_Ft t
1
0
1
2
Ws
Ys dWs =
0
1
0
with E
0
Ys2 ds < such that
1
2
ds?
If yes, nd Y . If no, prove it.
2. For any b R, dene (b) = infcfw_s | Ws s = b. S
Homework Assignment 3
Assigned Wed 10/06. Due Wed 10/27.
Questions 2(a) and 3(c) are a little harder, and you can probably nd a hint an almost any book
on Stochastic Calculus, as theyre standard results. However I recommend you try them on your own
rst.
1
Homework Assignment 5
Assigned Wed 11/17. Due Fri 12/03.
1. (Ornstein-Uhlenbeck process) Find an explicit solution of the SDE
dXt = Xt dt + dWt
where , R, and W is a 1D Wiener process. Also compute EXt and Var(Xt ).
2. (a) (Brownian bridge) Let a, b R, W
Homework Assignment 2
Assigned Thu 09/16. Due Wed 10/06.
X
1. Say Xt is a process with independent increments. Show that if s < t, Xt Xs is independent of Fs .
X
[Recall Ft = (s
t (Xs ).
Also, X has independent increments means that for any nite sequence
Homework Assignment 4
Assigned Wed 10/27. Due Wed 11/17.
1. (a) Let be a complete metric space, and (n ) be a sequence of Borel probability measures on
w
. If (n ) , show that lim sup n (K) (K) for all compact sets K .
(b) Let be a Polish space, and A be
880 Stochastic Calculus: Final.
Mon Dec, 17th , 2012
This is a closed book test. No calculators or computational aids are allowed.
You have 3 hours. The exam has a total of 7 questions and 70 points.
You may use without proof any result that has been p
880 Stochastic Calculus: Midterm.
Oct 17th
This is a closed book test. No calculators or computational aids are allowed.
You have 80 mins. The exam has a total of 4 questions and 20 points.
You may use without proof any result that has been proved in c