Chapter 30

Continuous Stochastic ProcessesThe change here is that the stochastic process will depend on t ∈ I an interval rather thann ∈ N. Everything becomes much more technical.

30.1 Fundamental Definitions and PropertiesHere E will be a separable Banach space and B (E) will be the Borel sets of E. Let(Ω,F ,P) be a probability space and I will be an interval of R. A set of E valued randomvariables, one for each t ∈ I, {X (t) : t ∈ I} is called a stochastic process. Thus for each t,X (t) is a measurable function of ω ∈ Ω. Set X (t,ω) ≡ X (t)(ω) . Functions t → X (t,ω)are called trajectories. Thus there is a trajectory for each ω ∈Ω. A stochastic process, Y iscalled a version or a modification of a stochastic process X if for all t ∈ I,

X (t,ω) = Y (t,ω) a.e. ω

There are several descriptions of stochastic processes.

1. X is measurable if X (·, ·) : I×Ω→ E is B(I)×F measurable. Note that a stochasticprocess X is not necessarily measurable.

2. X is stochastically continuous at t0 ∈ I means: for all ε > 0 and δ > 0 there existsρ > 0 such that

P([∥X (t)−X (t0)∥ ≥ ε])≤ δ whenever |t− t0|< ρ, t ∈ I.

Note the above condition says that for each ε > 0,

limt→t0

P([∥X (t)−X (t0)∥ ≥ ε]) = 0.

3. X is stochastically continuous if it is stochastically continuous at every t ∈ I.

4. X is stochastically uniformly continuous if for every ε,δ > 0 there exists ρ > 0 suchthat whenever s, t ∈ I with |s− t|< ρ, it follows

P([∥X (t)−X (s)∥ ≥ ε])≤ δ .

5. X is mean square continuous at t0 ∈ I if

limt→t0

E(∥X (t)−X (t0)∥2

)≡ lim

t→t0

∫Ω

∥X (t)(ω)−X (t0)(ω)∥2 dP = 0.

6. X is mean square continuous in I if it is mean square continuous at every point of I.

7. X is continuous with probability 1 or continuous if t→ X (t,ω) is continuous for allω outside some set of measure 0.

8. X is Hölder continuous if t→ X (t,ω) is Hölder continuous for a.e. ω.

Lemma 30.1.1 A stochastically continuous process on [a,b]≡ I is uniformly stochasti-cally continuous on [a,b]≡ I.

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