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24) with o(1) = (ε1/2 + βε )| ln ε|1/2 → 0. Proof. We introduce the stopping time ε . θ = θηε,N := inf{t ≥ t0 : |Δεt | ≥ η} ∧ τ ∧ σN Put ε Γ := {||δ ε ||t0 ,θ ≤ δ, |Δεt0 | ≤ γ, σN ≥ t0 }. We have: P (||Δε ||t0 ,τ ≥ Δ) ≤ P (||Δε ||t0 ,τ ≥ η) ε ε < τ ) + P (||Δε ||t0 ,τ ≥ η, σN ≥ t0 ) ≤ P (σN ε ε < τ ) + P (|Δεt0 | > γ, σN ≥ t0 ) ≤ P (σN ε ε ε ≥ t0 ) + P (IΓ |Δεθ | ≥ η) + P (||δ ||t0 ,θ > δ, |Δt0 | ≤ γ, σN ε ε < τ ) + P (|Δεt0 | > γ, σN ≥ t0 ) + P1 + P2 = P (σN where P1 and P2 are the third and the fourth terms on the left-hand side of the last inequality.

To avoid awkward expressions like ||||y||T ||p we shall use the alternative notation for ||y||T , namely, yT∗ := sups≤T |ys |, which is a standard one in the literature on stochastic calculus. 1). 21) where the constant Cp depends only on p. Proof. 20) the parameter λ = 1/μ2 , where μ2 := 12M 2 /γ. Then 2 2 Ee||y||T /μ ≤ 2γT and the claim follows from the lemma below. 8 Let ξ be a scalar random variable such that 2 Eeξ ≤ K. 22) √ ln K . 23) Proof. Without loss of generality we may assume that p is even integer.

20) that on the set {σN ≥ t0 } E(||δ ε ||2t0 ,τ ∧t |Ft0 ) ≤ C |δtε0 | + E(||Δε ||2t0 ,τ |Ft0 ) + t t0 E(||δ ε ||2t0 ,τ ∧s |Ft0 )ds and by the Gronwall–Bellman lemma E(||δ ε ||2t0 ,τ |Ft0 ) ≤ CeCT |δtε0 |2 + E(||Δε ||2t0 ,τ |Ft0 ) . ε < t0 } is trivial. 4. For h > 0 let ρN (h) := sup ∂Fj ∂Fj (t, x, y1 ) − (t, x, y2 ) ∂yi ∂yi the sup is taken over all t ∈ [0, T ] and x, y1 , y2 such that |x| + |y1 | + |y2 | ≤ N and |y1 − y2 | ≤ h. Let φN (h) := sup ∂Fj ∂Fj (t, x1 , ϕ(t, x1 )) − (t, x2 , ϕ(t, x2 )) , ∂yi ∂yi the sup is taken over all t ∈ [0, T ], and x1 , x2 such that |x1 | ≤ N , |x2 | ≤ N , and |x1 − x2 | ≤ h.

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CALCULUS OF VARIATIONS by A. R. Forsyth


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