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Quantitative Finance · Glosarium

Apa itu Viscosity solution?

Definition 9.10 Quantitative Methods · Bab 9 — Stochastic Control

A continuous VV is a viscosity solution of F(x,V,DV,D2V)=0F(x, V, DV, D^2V) = 0 (FF nonincreasing in its last argument) if at every point where a smooth ϕ\phi touches VV from above (V−ϕV - \phi has a local maximum), F(x,V,Dϕ,D2ϕ)≤0F(x, V, D\phi, D^2\phi) \le 0, and at every point where it touches from below, F≥0F \ge 0.

Solutions of - u + |u'| = 1 on (-1, 1) with u(±1) = 0, namely 1 - |x| + (e-1/ - e-|x|/ ): smooth for every > 0, converging to the kinked value function 1 - |x|, the viscosity solution of |u'| = 1.
Figure 9.3. Solutions of −εu′′+∣u′∣=1-\varepsilon u^{\prime\prime} + |u'| = 1 on (−1,1)(-1, 1) with u(±1)=0u(\pm1) = 0, namely 1−∣x∣+ε(e−1/ε−e−∣x∣/ε)1 - |x| + \varepsilon(e^{-1/\varepsilon} - e^{-|x|/\varepsilon}): smooth for every ε>0\varepsilon > 0, converging to the kinked value function 1−∣x∣1 - |x|, the viscosity solution of ∣u′∣=1|u'| = 1.
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