Stochastic Calculus, Problems and Solutions

Question 1

Define a field F

Answer 1

A field F is a collection (or family) F of subsets of omega satisfying:

1. Null set in F
2. A in F => A^C in F
3. F is closed under finite unions

Question 2

Define a sigma-algebra F

Answer 2

A sigma-algebra F is a collection (or family) of subsets of omega satisfying:

1. Null set in F
2. A in F => A^C in F
3. F is closed under countable unions

Question 3

Define a measurable space and a probability space

Answer 3

Measurable Space: pair (omega, F)

Probability Space: triple (omega, F, P)

Question 4

Define a probability measure P

Answer 4

A probability measure P on a measurable space (omega, F) is a function P -> [0, 1] s.t.:

1. P(null set) = 0
2. P(omega) = 1
3. P(union A_i) = Sum(P(A_i))

Question 5

Define a filtration F

Answer 5

A collection of sigma-algebras F_t where each set F_s in F_t

Question 6

Define a real valied R.V. X

Answer 6

A real valued R.V. X is a function X: omega -> R s.t. (w in omega, X(w) < x) in F for each x in R
X is F measurable

Question 7

State the partial averaging property

Answer 7

For every set A in G:

Int_A E[X|G]dP = int_A XdP

Question 8

State the formal definition of the conditional expectation

Answer 8

E[X|G] is G measurable

For every A in G, the partial averaging property is satisfied

Question 9

Compare the Markov Property and the Strong Markov Property

Answer 9

Markov: conditional distribution W_t given filtration F_s depends only on W_s
Strong Markov: Markox, and W_t+s - W_t is independent of F_t

Question 10

State the requirements for a Standard Wiener Process

Answer 10

1. W_0 = 0 and has continuous sample paths
2. W_t+s - W_t = N(0, s)
3. W_t+s - W_t independent of W_t

Question 11

State the solution for the Feynman-Kac formula for a one dimensional process

Answer 11

V(X_t, t) = E[phi(X_T)*exp(-int_t^T r(u)du|F_t|)

Question 12

Define trading strategy

Answer 12

A pair of stochastic processes which are adapted to the filtration F_t to yield a portfolio
Pi_t = phi_tS_t + psi_tB_t

Question 13

Define self financing trading strategy

Answer 13

dPi_t = phi_tdS_t + psi_tdB_t

Question 14

Define admissable trading strategy

Answer 14

P(Pi_t >= -alpha) = 1

There must be a lower bound on profit, almost surely

Question 15

Define a simple contingent claim

Answer 15

A state claim that only pays off in one state

Question 16

Define a contingent claim

Answer 16

A collection of simple contingent claims

Question 17

Define attainable contingent claim

Answer 17

A contingent claim is attainable if there exists an admissable strategy Pi_t = Psi(S_T) at exercise T

Question 18

State the three required conditions for arbitrage

Answer 18

1. Pi_0 = 0
2. P(Pi_T >= 0) = 1
3. P(Pi_T > 0) = 0

Question 19

Define complete markets

Answer 19

A market is complete if every contingent claim is attainable

The number if attainable simple contingent claims equals the number of possible states