By A. R. Forsyth

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Neoliberalismus: Analysen und Alternativen by Christoph Butterwegge, Bettina Lösch, Ralf (Eds.) Ptak, Ralf PDF

Der Neoliberalismus hat in den letzten Jahren weite Bereiche unserer Gesellschaft geprägt. Es ist ihm gelungen, zumindest in einem Großteil der medialen Öffe- lichkeit die Legitimität des grundgesetzlich geschützten Sozialstaates – erstmals nach 1945 – zu erschüttern und dessen Säulen ins Wanken zu bringen.

Extra resources for CALCULUS OF VARIATIONS

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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.