We investigate the different notions of solutions to the double-phase equation - div ( | D u | p - 2 D u + a ( x ) | D u | q - 2 D u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the A H ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions of nonlinear potential theory and then show that A H ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.
We consider the fully nonlinear equation with variable-exponent double phase type degeneraciesUnder some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques, we obtain the sharp local C 1,α regularity of viscosity solutions to such equations.
We consider the nonlocal double phase equation $$\begin{align*} \textrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,\textrm{d}y\\ &+\textrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,\textrm{d}y=0, \end{align*}$$where $1<p\leq q$ and the modulating coefficient $a(\cdot ,\cdot )\geq 0$. Under some suitable hypotheses, we first use the De Giorgi–Nash–Moser methods to derive the local Hölder continuity for bounded weak solutions and then establish the relationship between weak solutions and viscosity solutions to such equations.
We consider the nonlocal double phase equationwhere 1 < p ≤ q and the modulating coefficient a(•, •) ≥ 0. Under some suitable hypotheses, we first use the De Giorgi-Nash-Moser methods to derive the local Hölder continuity for bounded weak solutions, and then establish the relationship between weak solutions and viscosity solutions to such equations.
Basel Committee on Banking Supervision published Standards on Interest Rate Risk in Banking Book in April 2016. Apart from others, it proposed a standardized framework under which banks should identify core and noncore deposits within their stable nonmaturity deposits (NMD) and determine appropriate cash flow slotting for the NMD. This paper proposed a unique solution to slot Core NMD into repricing time buckets to address Basel requirements on NMD. The proposed solution was based on pass-through rate model under ECM (error correction model) framework. The solution itself showed interesting mathematical property to form a generalized Fibonacci sequence with converged partial sum. What is more, this paper proposed a model-neutral back testing scheme to make objective comparison of performance across different NMD repricing behavior models. The contents of this paper are expected to be useful for practitioners due to lack of quantitative modeling and model validation methodologies on this topic in the industry while, at the same time, to motivate academic discussion on the best practice and further enhancement of the modeling approach for the industry.
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