Download My Angel Dog by Masanobu Takeyama PDF

By Masanobu Takeyama

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5. 4 below, assumption (iii) will be fulfilled, since the event under P will be empty for all n ∈ N, R ∈ [0, ∞[. For a case where assumption (iii) is more difficult to check, we refer to [Kry99, Section 1]. 2. 1. The proof is based on Euler’s method. Fix n ∈ N and define the processes X (n) (t), t ∈ [0, ∞[, iteratively by setting X (n) (0) := X0 50 3. Stochastic Differential Equations in Finite Dimensions and for k ∈ N ∪ {0} and t ∈ k k+1 n, n by X (n) (t) =X (n) k n t b s, X (n) + k n k n t σ s, X (n) ds + k n k n dW (s).

An } and n τ= aj 1Aj j=0 where 0 aj < aj+1 T and Aj = {τ = aj } ∈ Faj . In this way we get that 32 2. Stochastic Integral in Hilbert Spaces 1]τ,T ] Φ is an elementary process since k−1 1]τ,T ] (s)Φ(s) = Φm 1]tm ,tm+1 ]∩]τ,T ] (s) m=0 k−1 n 1Aj Φm 1]tm ,tm+1 ]∩]aj ,T ] (s) = m=0 j=0 k−1 n = 1Aj Φm m=0 j=0 1]tm ∨aj ,tm+1 ∨aj ] (s) Ftm ∨aj -measurable and concerning the integral we are interested in, we obtain that t t 0 t Φ(s) dW (s) − 1]0,τ ] (s)Φ(s) dW (s) = 0 1]τ,T ] (s)Φ(s) dW (s) 0 k−1 Φm W (tm+1 ∧ t) − W (tm ∧ t) = m=0 k−1 n 1Aj Φm W (tm+1 ∨ aj ) ∧ t − W (tm ∨ aj ) ∧ t − m=0 j=0 k−1 Φm W (tm+1 ∧ t) − W (tm ∧ t) = m=0 k−1 n 1Aj Φm W (tm+1 ∨ τ ) ∧ t − W (tm ∨ τ ) ∧ t − m=0 j=0 k−1 Φm W (tm+1 ∧ t) − W (tm ∧ t) = m=0 k−1 Φm W (tm+1 ∨ τ ) ∧ t − W (tm ∨ τ ) ∧ t − m=0 k−1 Φm W (tm+1 ∧ t) − W (tm ∧ t) = m=0 − W (tm+1 ∨ τ ) ∧ t − W (tm ∨ τ ) ∧ t k−1 t∧τ Φm W (tm+1 ∧ τ ∧ t) − W (tm ∧ τ ∧ t) = = m=0 Φ(s) dW (s).

Then G is a Dynkin system and therefore PT = σ(K) = D(K) ⊂ G as K ⊂ G. 30 2. Stochastic Integral in Hilbert Spaces Step 4: Finally the so-called localization procedure provides the possibility to extend the definition of the stochastic integral even to the linear space NW (0, T ; H) := Φ : ΩT → L02 Φ is predictable with T P Φ(s) 0 2 L02 ds < ∞ =1 . For simplicity we also write NW (0, T ) or NW instead of NW (0, T ; H) and NW is called the class of stochastically integrable processes on [0, T ]. The extension is done in the following way: For Φ ∈ NW we define t τn := inf t ∈ [0, T ] Φ(s) 0 2 L02 ds > n ∧ T.

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