Michael Plum

Optimal investment for insurers

Abstract We consider a risk process modelled as a compound Poisson process. The ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index. The optimal strategy is computed using the Bellman equation. We prove the existence of a smooth solution and a verification theorem, and give explicit solutions in some cases with exponential claim size distribution, as well as numerical results in a case with Pareto claim size. For this last case, the optimal amount invested will not be bounded.

Optimal investment for investors with state dependent income, and for insurers

Abstract. An optimal control problem is considered where a risky asset is used for investment, and this investment is financed by initial wealth as well as by a state dependent income. The objective function is accumulated discounted expected utility of wealth, where the utility function is nondecreasing and bounded. This problem is investigated for constant as well as for stochastic discount rate, where the stochastic model is a time homogeneous finite state Markov process. We prove that the Bellman equation to this optimization problem has a classical solution and give a verification argument. Based on this we deal with the problem of optimal investment for an insurer with an insurance business modelled by a compound Poisson or a compound Cox process, under the presence of constant as well as (finite state space Markov) stochastic interest rate.

Explicit H2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems

Abstract It is well known that the H 2 -norm and the C 0 -norm of a function u ∈ H 2 (Ω) (where Ω ⊂ R n is a bounded domain, n ⩽ 3) can be estimated in terms of a given uniformly elliptic second-order differential operator L and some boundary operator B applied to u , if certain regularity assumptions are satisfied. If these bounds shall be used for numerical purposes, the constants occurring in the estimates must be known explicitly . The main goal of the present article is the computation of such explicit constants. For simplicity of presentation, we restrict ourselves to the case where L [ u ] = − Δu + c ( x ) u . As an application, we prove an existence and inclusion result for nonlinear boundary value problems.

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

Abstract We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.

Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method

SummaryInclusion intervals for the firstN eigenvalues of a second-order ordinary differential operator with boundary conditions of Sturm-Liouville or of periodic type are derived by a combination of “elementary” estimates, an appropriate numerical procedure and a homotopy algorithm.

ON THE SPECTRUM OF SECOND-ORDER DIFFERENTIAL OPERATORS WITH COMPLEX COEFFICIENTS

The main objective of this paper is to extend the pioneering work of Sims on second-order linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh–Weyl theory and classification. The generalization considered exposes interesting features not visible in the special case in Sims paper from 1957. An m-function is constructed (which is either unique or a point on a ‘limit-circle’), and the relationship between its properties and the spectrum of underlying m-accretive differential operators analysed. The paper is a contribution to the study of non–self–adjoint operators; in general, the spectral theory of such operators is rather fragmentary, and further study is being driven by important physical applications, to hydrodynamics, electro–magnetic theory and nuclear physics, for instance.

Computer-assisted existence proofs for two-point boundary value problems

For (scalar) nonlinear two-point boundary value problems of the form−U″+F(x, U, U′)=0, B0[U]=B1[U]=0, with Sturm-Liouville or periodic boundary operatorsB0 andB1, we present a method for proving the existence of a solution within a “close”C1-neighborhood of an approximate solution.ZusammenfassungFür (skalare) nichtlineare Zweipunkt-Randwertprobleme der Form−U″+F(x, U, U′)=0,B0[U]=B1[U]=0 mit Sturm-Liouville-oder periodischen RandoperatorenB0,B1 wird eine Methode vorgestellt, mit der die Existenz einer Lösung innerhalb einer “kleinen”C1-Umgebung einer Näherungslösung bewiesen werden kann.

Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems

We propose a numerical method for computing enclosures for continua of solutions of nonlinear elliptic boundary value problems depending on a real parameter, at the same time proving the existence of a smooth solution-continuum. The method is extended to turning-point problems by change of parameters.

Bounds for eigenvalues of second-order elliptic differential operators

SummaryWe derive bounds for the firstN eigenvalues of a linear second-order elliptic differential operator on a bounded domain, subject to mixed boundary conditions. The results are achieved by a combination of (a generalized version of) Katos estimates and a homotopy algorithm.

Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems

For elliptic boundary value problems of the form −ΔU+F(x, U, Ux)=0 on Ω,B[U]=0 on ϖΩ, with a nonlinearityF growing at most quadratically with respect to the gradientUx and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H1,4(Ω)-neighborhood of some approximate solution ω∈H2(Ω) satisfying the boundary condition, provided that the defect-norm ∥−Δω +F(·, ω, ωx)∥2 is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.ZusammenfassungGegeben sei ein elliptisches Randwertproblem der Form −ΔU+F(x, U, Ux)=0 auf Ω,B[U]=0 auf ϖΩ, mit einer NichtlinearitätF, die einer quadratischen Wachstumsbedingung bezüglich des GradientenUx genügt, und mit einem linearen RandoperatorB von gemischtem Typ. Es wird eine numerische Methode vorgestellt, mit deren Hilfe sich die Existenz einer Lösung innerhalb einer “kleinen”H1,4(Ω)-Umgebung einer Näherungslösung ω∈H2(Ω), die die Randbedingung erfüllt, nachweisen läßt, sofern die Defektnorm ∥−Δω +F(·, ω, ωx)∥2 hinreichend klein ist und ferner die Linearisierung des gegebenen Problems in ω auf einen invertierbaren OperatorL führt. Die wesentlichen Hilfsmittel sind explizite Sobolevsche Einbettungen und Eigenwertschranken fürL oderL*L. Jegliche Monotonie- und Inverspositivitätsbedingungen werden vermieden.