Selected problems (from TK) and exercises (from Schweder ("S")).

Note: From TK, the list refers to the "problems", not "exercises". Numbers are the same in 4th as in 3rd edition (except 4th ed. uses Arabic numerals for chapters).

For 04.02: TK II problems 1.7, 1.9, 1.10, 2.1, 3.4, 4.2. S: 1, 3, 4.

For 11.02: TK III: 1.1, 1.4, 2.1, 2.4, 2.5, 4.7, 4.17, 7.1, 7.4, 8.3
Notes: 2.1 will be simpler to solve after ch 4. Give it a try without -- I will not review it, though. For 7.1, invert the matrix for a suitable number of states, and then use the pattern to find the inverse for the infinity-by-infinity matrix. 7.4 I have not done myself, and you might want to try "last step analysis" -- i.e., given that you were just absorbed, what did just happen? 8.3 requires you to read a bit about branching processes, and is a bonus problem I do not think I will review.

For 18.02: TK IV: 1.3, 1.10, 1.12, 3.2, 4.1, 4.4. Problem 3.2 contains theory -- you should note the result whether or not you are able to prove it.
In addition, go through the reducible case treated in the Feb11 note (i.e., «reconstruct each step» for yourself).

For 25.02: TK V problems
3.4, should be easy.
4.2: requires you to read section V.4!
4.7: material covered later in this course will give this result fairly easy, but give it a shot,
6.3: hard! I solved it by putting v(x) = sum G^{(n)}(x) (sum from 0 to infinity) and deducing the difference equation v(x+1) = v(x) + constant),
6.10: even harder! Let V be the winning bid and try E[E[V|T]]. I did not get any closed-form expression for the constant threshold, but I did for the time-dependent threshold.
In addition, I will cover problem 4.1 from chapter IV -- notice the error in the solution as is now, it should be n-1 ones.

For 11.03: TK II p5.5, III p8.4, p9.1, VI p4.1, 6.1, 6.3, and from Schweder: 11,12.

For 18.03: TK VIII p1.1, p1.3, p1.7, p2.3, p2.4, p2.5, p2.6

For 25.03:

From TK/PK: Problems 8.4.1, 8.4.6, 8.4.7, 8.4.10.
Let X(t) = exp(a t + q B(t)) be a geometric Brownian motion, and let m(t) = E X(t).
1. For which values of a and q, will
  m(t) --> 0 as t grows?
  m(t) --> infinity?
  What happens otherwise?
2. Use the law of the iterated logarithm to decide:
  when will X(t) --> 0 (with probability 1) as t grows?
  when will X(t) --> infinity?
  What happens otherwise?
3. Compare the answers in 1 and 2. Puzzling?
Some linear algebra:
1. Diagonalise the matrix A =
  1 3 4
  3 1 0
  4 0 1
  Then calculate exp A.
2. Calculate exp B, where B =
  6 -9
  4 -6
  Then show that it cannot be diagonalised.

For 08.04 and 29.04:

See note. For the 29th, do also the following:

Let dX = q(t^-) dY, where q is a stochastic process which is non-anticipative. Show that if the integrability condtion holds, then X is a martingale under each of the following condition:

Y is a Brownian motion
Y = dN - E[N(1)] dt
Y is a martingale.

Hint: use the limit-of-sums definition of the Itô integral, not the one I gave in the lectures.