By Masanobu Takeyama
ISBN-10: 1905200714
ISBN-13: 9781905200719
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But τ (n) (R) ∧ τ (R) P T, |X (n) (t)| sup r(R) t∈[0,τ (n) (R)∧τ (R)] τ (n) (R) P T, sup |X (n) (t)| r(R), τ (n) (R) τ (R) t∈[0,τ (n) (R)] T, τ (n) (R) > τ (R)}) + P ({τ (R) and limR→∞ P ({τ (R) T }) = 0. So, also assumption (iii) holds when τ (n) (R) is replaced by τ (n) (R)∧τ (R). We may thus assume that τ (n) (R) τ (R), hence u(R) for all t, R ∈ [0, ∞[, n ∈ N. 10) Fix R ∈ [0, ∞[ and define t (n) |p(n) (s)| Ks (R) ds, λt (R) := t ∈ ]0, ∞[, n ∈ N. 10) it follows that (n) lim E λT ∧τ (n) (R) (R) = 0 for all R, T ∈ [0, ∞[.
For every t ∈ [0, T ], these maps restricted to [0, t] × V × Ω are B([0, t]) ⊗ B(V ) ⊗ Ft -measurable. As usual by writing A(t, v) we mean the map ω → A(t, v, ω). Analogously for B(t, v). We impose the following conditions on A and B: (H1) (Hemicontinuity) For all u, v, x ∈ V, ω ∈ Ω and t ∈ [0, T ] the map R λ→ V∗ A(t, u + λv, ω), x V is continuous. (H2) (Weak monotonicity) There exists c ∈ R such that for all u, v ∈ V 2 V ∗ A(·, u) − A(·, v), u − v c u−v 2 H V + B(·, u) − B(·, v) 2 L2 (U,H) on [0, T ] × Ω.
E. 14) + σ(s, X (n) (s) + p(n) (s)) − σ(s, X (m) (s) + p(m) (s)) (n,m) − 2Ks (R)|X (n) (s) − X (m) (s)|2 ds + MR 2 (t), (n,m) where MR (t), t ∈ [0, ∞[, is a continuous local (Ft )-martingale with (n,m) (0) = 0. e. 9) and assumption (i) in the last step. e. for t ∈ [0, γ (n,m) (R)] |X (n) (t) − X (m) (t)|2 ψt (R) (n) (m) 4(1 + R)(λt (R) + λt (n,m) (R)) + MR (t). 15) γ (n,m) (R) and (Ft )-stopping times Hence for any (Ft )-stopping time τ (n,m) σk ↑ ∞ as k → ∞ so that MR (t ∧ σk ), t ∈ [0, ∞[, is a martingale for all 48 3.
My Angel Dog by Masanobu Takeyama
by Joseph
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