Chapter 9: The transition to continuous time

Question 1

probability space

continuous time,

Answer 1

probability space (Ω,F,P)
 continuous time, meaning that our time will be indexed as t ∈ [0,∞)

Question 2

DEF 9.0.1

stochastic process in continuous time

Answer 2

A stochastic process (in continuous time) is a family of random variables
(Xt)^∞ _t=0. We think of t ∈ [0,∞) as time.

As in discrete time, we will usual write (X_t) = (X_t)^∞_t=0.
We will also sometimes write simply X instead of (Xt)

Question 3

DEF 9.0.2

stochastic process (Xt) is continuous

Answer 3

We say that a stochastic process (Xt) is continuous (or, equivalently, has continuous paths) if, for almost all ω ∈ Ω, the function t → Xt(ω) is continuous.

continuous stochastic process ≈ random continuous function.

Question 4

EXAMPLE

if A,B and C are i.i.d. N(0,1) random variables

Answer 4

Xt = At^2 +Bt+C for t ∈ [0,∞) is a random continuous (quadratic) function

we need filtrations to track how info over time. As time increases larger amount of information

Question 5

DEF 9.0.3 filtration

Answer 5

We say that a family (F_t) of σ-fields is a (continuous time) filtration if
F_u ⊆ F_t whenever u ≤ t.

A stochastic process (Xt) is adapted to the filtration (Ft) if Mt ∈ mFt for all t ≥ 0.

Question 6

continuous time filtered space

Answer 6

work over a filtered space (Ω,F,(Ft),P) where (Ω,F,P) is a probability space and (Ft) is a filtration.

Question 7

generated/natural filtration for continuous time

Answer 7

If we are given a stochastic process (Xt), implicitly over some probability space, the generated (or natural) filtration of the stochastic process is Ft = σ(X_u); u ≤ t).

ie X_u info generated before time t

Question 8

DEF 9.0.4]

continuous time martingale

Answer 8

A (continuous time) stochastic process (Mt) is a martingale wrt F_t if

1. (Mt) is adapted
2. Mt ∈ L^1 for all t
3. E[M_t|F_u] = Mu for all 0 ≤ u ≤ t.

(3 is fairness property)

(or X_t instead of M_t)

We say that (Mt) is a submartingale if, instead of 3, we have E[Mt|Fu] ≥ Mu almost surely.

We say that (Mt) is a supermartingale if, instead of 3, we have E[Mt|Fu] ≤ Mu almost surely.

for all u ≤ t.