By Seppalainen T.

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**Extra info for Basics of Stochastic Analysis**

**Example text**

Let ε > 0. 2. Quadratic variation 49 Borel measure µ = Λ[X] + Λ[Y ] + |Λ[X,Y ] | on [0, T ]. Above, Λ[X] is the positive Lebesgue-Stieltjes measure of the function t → [X]t (for the fixed ω under consideration), same for Λ[Y ] , and |Λ[X,Y ] | is the positive total variation measure of the signed LebesgueStieltjes measure Λ[X,Y ] . 16 we can choose step functions g and h so that |g| ≤ C, |h| ≤ C, and |g − g| + |h − h| dµ < ε . 2C On the one hand [0,T ] [0,T ] |g − g| d|Λ[X,Y ] | + C ≤C gh − g h d|Λ[X,Y ] | g h d[X, Y ]t ≤ gh d[X, Y ]t − [0,T ] [0,T ] |h − h| d|Λ[X,Y ] | ≤ ε.

Instead, try to say something useful about the class of sets for which a countable L exists. (b) Let R[0,∞) be the space of all functions x : [0, ∞) → R, with the product σ-algebra generated by the projections x → x(t), t ∈ [0, ∞). Show that the set of continuous functions is not measurable. 5. (a) Let E1 , . . , En be collections of measurable sets on (Ω, F, P ), each closed under intersections (if A, B ∈ Ei then A ∩ B ∈ Ei ). Suppose P (A1 ∩ A2 ∩ · · · ∩ An ) = P (A1 ) · P (A2 ) · · · P (An ) for all A1 ∈ E1 , .

Let A be an event such that 0 < P (A) < 1, and A = {∅, Ω, A, Ac }. 29) E(X|A)(ω) = E(1A X) E(1Ac X) · 1A (ω) + · 1Ac (ω). 26) for A. 29). Then E(1A X) E(1A X) 1A (ω) P (dω) = 1A (ω) P (dω) Y dP = P (A) A A A P (A) = E(1A X) = X dP. A A similar calculation checks Ac Y dP = Ac X dP , and adding these together gives the integral over Ω. ∅ is of course trivial, since any integral over ∅ equals zero. 7 for a generalization of this. The next theorem lists the main properties of the conditional expectation.

### Basics of Stochastic Analysis by Seppalainen T.

by Donald

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