Some Properties of a Solution to a Family of Inhomogeneous Linear Ordinary Differential Equations

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

Rational functions with maximal radius of absolute monotonicity

We study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$. Such functions arise in the application of implicit $s$-stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

Analyticity of Poisson-driven stochastic systems

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ, E λ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.

Analyticity of Poisson-driven stochastic systems

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ,Eλ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.