Chapter 72

The Hard Ito FormulaRecall the following definition of stochastically continuous.

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

72.1 Predictable And Stochastic ContinuityDefinition 72.1.1 Let Ft be a filtration. The predictable sets consists of those sets whichare in the smallest σ algebra which contains the sets E ×{0} for E ∈F0 and E× (a,b]where E ∈Fa. Thus every predictable set is a progressively measurable set.

First of all, here is an important observation.

Proposition 72.1.2 Let X (t) be a stochastic process having values in E a complete metricspace and let it be Ft adapted and left continuous where Ft is a normal filtration. Thenit is predictable. If t → X (t,ω) is continuous for all ω /∈ N,P(N) = 0, then (t,ω)→X (t,ω)XNC (ω) is predictable. Also, if X (t) is stochastically continuous and adapted on[0,T ] , then it has a predictable version. If X ∈C ([0,T ] ;Lp (Ω;F)) , p≥ 1 for F a Banachspace, then X is stochastically continuous.

Proof: First suppose X is continuous for all ω ∈Ω. Define

Im,k ≡ ((k−1)2−mT,k2−mT ]

if k ≥ 1 and Im,0 = {0} if k = 1. Then define

Xm (t) ≡2m

∑k=1

X(T (k−1)2−m)X((k−1)2−mT,k2−mT ] (t)

+X (0)X[0,0] (t)

Here the sum means that Xm (t) has value X (T (k−1)2−m) on the interval

((k−1)2−mT,k2−mT ].

Thus Xm is predictable because each term in the formal sum is. Thus

X−1m (U) = ∪2m

k=1(X(T (k−1)2−m)X((k−1)2−mT,k2−mT ]

)−1(U)

= ∪2m

k=1((k−1)2−mT,k2−mT ]×(X(T (k−1)2−m))−1

(U) ,

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