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Wir stellen nun oft benutzte Eigenschaften des bedingten Erwartungswerts zusammen. Unter der Voraussetzung der Existenz der auftretenden bedingten Erwartungswerte gilt: (i) E(X|G) ≥ 0 f¨ ur X ≥ 0. (ii) E(αX1 + βX2 |G) = αE(X1 |G) + βE(X2 |G) f¨ ur alle α, β ∈ IR. (iii) Ist Z G-messbar, so folgt E(ZX|G) = ZE(X|G). (iv) Sind X und G stochastisch unabh¨angig, d. h. X und 1G stochastisch unabh¨angig f¨ ur alle G ∈ G, so gilt E(X|G) = EX. (v) F¨ ur G ⊆ F gilt E(E(X|F)|G) = E(X|G). 12 Jensensche Ungleichung Es seien X eine Zufallsgr¨oße mit Werten in (a, b), −∞ ≤ a < b ≤ ∞ und ϕ : (a, b) → IR konvex mit E|X| < ∞, E|ϕ(X)| < ∞.

Geben Sie limn→∞ E(X|An ) an, falls X zus¨atzlich stetig wert unter An f¨ ist. 2. 5 Seien (Ω, A, P ) ein Wahrscheinlichkeitsraum, X eine integrierbare Zufallsgr¨oße und G eine Unter-σ-Algebra von A. Zeigen Sie, dass f¨ ur jede beschr¨ankte G-messbare Zufallsgr¨oße Z gilt: ZXdP = ZE(X|G)dP und E(ZX|G) = ZE(X|G). 6 Seien X1 , X2 , . . stochastisch unabh¨angige Zufallsgr¨oßen mit EXi = 0 und EXi2 = c < +∞ f¨ ur i = 1, 2, . .. Sei die Filtration gegeben durch An = σ(X1 , . . , Xn ). Zeigen Sie, dass n Xi )2 − nc)n∈IN (( i=1 ein Martingal ist.

M − 1) i=jk+1 (j+1)k m−1 Xi ≤ k − 1) P( = j=0 i=jk+1 m−1 ≤ P ((Xjk+1 , . . , X(j+1)k ) = (1, . . , 1)) j=0 = (1 − pk )m . 2. Stochastische Grundlagen diskreter M¨arkte 52 F¨ ur j > k gilt j k) k j ≤ (1 − pk )[ k ] P (τ > j) ≤ P (τ > 1 ≤ (1 − pk )−1 ((1 − pk ) k )j , 1 also mit geeignetem a > 0 und γ = (1 − pk ) k P (τ > j) ≤ aγ j . ✷ Daraus folgt sofort P (τ = ∞) = 0, Eτ = ∞ P (τ ≥ j) < ∞, j=1 also 1 = P (τ < ∞) = P (Sτ = G) + P (Sτ = −C) und P (Sτ = G) = 1 − P (Sτ = −C). (ii) Anwendung des Optional-Sampling-Theorems: Betrachtet werde nun das Martingal n (Xi − EXi ), n ≥ 1, M0 = 0.

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A. F. Lavriks truncated equations by Kaufman R. M.

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