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Christian Budde

Publications and source records attributed to Christian Budde.

14 recordsLinked to original sources
arXiv2023

A critical review of existing and new population stability testing procedures in credit risk scoring

Credit scorecards are models used for the modelling of the probability of default of clients. The decision to extend credit to an applicant, as well as the price of the credit, is often based on these models. In order to ensure that scorecards remain accurate over time, the hypothesis of population stability is tested periodically; that is, the hypothesis that the distributions of the attributes of clients at the time when the scorecard was developed is still representative of these distributions at review is tested. A number of measures of population stability are used in practice, with several being proposed in the recent literature. This paper provides a critical review of several testing procedures for the mentioned hypothesis. The widely used population stability index is discussed alongside two recently proposed techniques. Additionally, the use of classical goodness-of-fit techniques is considered and the problems associated with large samples are investigated. In addition to the existing testing procedures, we propose two new techniques which can be used to test population stability. The first is based on the calculation of effect sizes which does not suffer the same problems as classical goodness-of-fit techniques when faced with large samples. The second proposed procedure is the so-called overlapping statistic. We argue that this simple measure can be useful due to its intuitive interpretation. In order to demonstrate the use of the various measures, as well as to highlight their strengths and weaknesses, several numerical examples are included.

stat.AP↗
arXiv2023

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

math.FA↗
arXiv2022

Well-posedness of non-autonomous transport equation on metric graphs

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

math.AP↗
arXiv2022

A monotone convergence theorem for strong Feller semigroups

For an increasing sequence $(T_n)$ of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps the constant 1 function to a continuous function. This is instrumental in the study of elliptic operators on $\mathbb{R}^d$ with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients. We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.

math.FA↗
arXiv2022

A Lumer-Phillips type generation theorem for bi-continuous semigroups

The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $A\colon D(A)\subseteq X\rightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is dissipative and the range of $\lambda-A$ is surjective in $X$ for some $\lambda>0$. In this paper, we establish a version of this result for bi-continuous semigroups and apply the latter amongst other examples to the transport equation as well as to flows on infinite networks.

math.FA↗
arXiv2021

Non-autonomous Desch-Schappacher perturbations

We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on $\mathrm{L}^p$-spaces.

math.FA↗
arXiv2021

Positive Desch-Schappacher perturbations of bi-continuous semigroups on $\mathrm{AM}$-spaces

In this paper, we consider positive Desch-Schappacher perturbations of bi-continuous semigroups on $\mathrm{AM}$-spaces with an additional property concerning the additional locally convex topology. As an examples, we discuss perturbations of the left-translation semigroup on the space of bounded continuous function on the real line and on the space of bounded linear operators.

math.FA↗
arXiv2021

Bi-continuous semigroups for flows on infinite networks

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $\\mathrm{L}^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

math.AP↗
arXiv2021

Semigroups for flows on limits of graphs

We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study flows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory.

math.AP↗
arXiv2018

A Desch-Schappacher perturbation theorem for bi-continuous semigroups

We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra $\mathscr{L}(E)$ of bounded, linear operators on a Banach space $E$, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem

math.FA↗
arXiv2017

Intermediate and extrapolated spaces for bi-continuous semigroups

We discuss the construction of the full Sobolev (Hölder) scale for non-densely defined operators on a Banach space with rays of minimal growth. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille-Yosida operators on Saks spaces.

math.FA↗
arXiv2015

A bounded transform approach to self-adjoint operators: Functional calculus and affiliated von Neumann algebras

Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.

math.OA↗