Previously, we found the strongly continuous semigroups governing ∂u/∂t= cx^{2a} ∂^2 u/∂x^2 + kx^a ∂u/∂x for x, t ≥ 0 and a = 0, 1. In this paper, we do this for a variant of the above equation where a = 1/2. We also deal with nonautonomous versions having governing operators such as L_{α(t),θ(t),r(t)}u(x) :=α(t) xu''(x) + (1/2 alpha(t) + theta (t) x^{1/2})u'(x)-r(t)u(x). Here α, θ, and r are real-valued continuous functions in [0,+∞), α(t) >0, θ(t) ≥ 0, r(t) ≥ 0, for any t ≥ 0. When θ = 0 = r on [0,∞), the corresponding equation reduces to a nonautonomous version of the Cox–Ingersoll–Ross (CIR) bond equation.
On a class of nonautonomous problems related to financial mathematics
Gisele Ruiz Goldstein;Jerome A. Goldstein;Silvia Romanelli
2024-01-01
Abstract
Previously, we found the strongly continuous semigroups governing ∂u/∂t= cx^{2a} ∂^2 u/∂x^2 + kx^a ∂u/∂x for x, t ≥ 0 and a = 0, 1. In this paper, we do this for a variant of the above equation where a = 1/2. We also deal with nonautonomous versions having governing operators such as L_{α(t),θ(t),r(t)}u(x) :=α(t) xu''(x) + (1/2 alpha(t) + theta (t) x^{1/2})u'(x)-r(t)u(x). Here α, θ, and r are real-valued continuous functions in [0,+∞), α(t) >0, θ(t) ≥ 0, r(t) ≥ 0, for any t ≥ 0. When θ = 0 = r on [0,∞), the corresponding equation reduces to a nonautonomous version of the Cox–Ingersoll–Ross (CIR) bond equation.File in questo prodotto:
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