By Shigeo Kusuoka, Toru Maruyama

ISBN-10: 4431548335

ISBN-13: 9784431548331

ISBN-10: 4431548343

ISBN-13: 9784431548348

A lot of financial difficulties may be formulated as restricted optimizations and equilibration in their recommendations. a number of mathematical theories were providing economists with fundamental machineries for those difficulties coming up in financial concept. Conversely, mathematicians were prompted through a number of mathematical problems raised via financial theories. The sequence is designed to collect these mathematicians who're heavily drawn to getting new demanding stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her research.

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Is convex on E for every (t, x) ∈ [0, 1] × E. Let us consider the value function 1 VJ (τ, x) = sup f ∈SZ1 J (t, uτ,x,f (t), f (t))dt , (τ, x) ∈ [0, η1 [×E τ where uτ,x,f is the trajectory solution on [τ, 1] associated the control f ∈ SZ1 starting from x ∈ E at time τ to ⎧ ⎪ ⎪ u¨ τ,x,f (t) + γ u˙ τ,x,f (t) = f (t), t ∈ [τ, 1] ⎨ m−2 (SODE) ⎪ u (τ ) = x, u (1) = αi uτ,x,f (ηi ). ⎪ τ,x,f ⎩ τ,x,f i=1 Then VJ satisfies a viscosity property: For any ϕ ∈ C 1 ([0, 1] × E) such that VJ reaches a local maximum at (t0 , x0 ) ∈ [0, η1 [×E, then 1 ∂G ∂ϕ t0 (t0 , x0 )+max{J (t0 , x0 , z)}+δ ∗ (∇ϕ(t0 , x0 ), e˙t0 ,x0 (t0 )+ (t0 , s)Zds) ≥ 0.

J. Nonlinear Convex Anal. 5(1), 131–152 (2004) 14. : Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer Academic, Dordrecht (2004) 15. : Some limit results for integrands and Hamiltonians with application to viscosity. J. Nonlinear Convex Anal. 6(3), 465–485 (2005) 16. : Some variational convergence results with application to evolution inclusions. Adv. Math. Econ. 8, 33–73 (2006) 17. : On a optimal control problem governed by the sweeping process (2013).

Here E is a separable Hilbert space. 3. Let C : [0, T ] → ck(E) be a convex compact valued LLipschitzean mapping: |d(x, C(t)) − d(y, C(τ ))| ≤ L|t − τ | + ||x − y||, ∀x, y ∈ E × E, ∀t, τ ∈ [0, T ] × [0, T ]. Let Z be a convex compact subset in E and SZ1 is the set of all integrable mappings f : [0, T ] → Z. ) is convex for every (t, x) ∈ [0, T ] × E. Let us consider the value function T VJ (τ, x) = sup f ∈SZ1 J (t, uτ,x,f (t), f (t)) dt , (τ, x) ∈ [0, T ] × E τ where uτ,x,f is the trajectory solution on [τ, T ] associated the control f ∈ SZ1 starting from x ∈ E at time τ to the sweeping process (PSW)(C; f ; x) − u˙ τ,x,f (t) − f (t) ∈ NC(t) (uτ,x,f (t)), t ∈ [τ, T ] uτ,x,f (τ ) = x Optimal Control Problems Governed by a Second Order Ordinary .

### Advances in mathematical economics. Vol.18 by Shigeo Kusuoka, Toru Maruyama

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