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Simon Hochgerner

Publications and source records attributed to Simon Hochgerner.

At least 19 recordsLinked to original sources
arXiv2026

On the market-consistent valuation of health insurance liabilities

We are concerned with the market-consistent valuation of lifelong health insurance products, which are subject to adjustments derived from the actuarial equivalence principle and driven by (medical) inflation. Such products are well-established in the European national markets, and the dynamics of the adjustment mechanism is well-understood from an actuarial perspective. However, the question of market-consistent valuation (as is necessary for Solvency II reporting) has not previously been addressed. This gap has led to a situation where some practitioners use stochastic models while others rely on deterministic methods to assign market-consistent values (Best Estimates) to the same type of health insurance liabilities. The purpose of this note is to fill this gap by showing that the Best Estimate of a lifelong health insurance policy depends on the choice of model for the interest and inflation rates. That is, the Best Estimate is not uniquely determined by the currently prevailing term structures of nominal and real spot rates, whence a deterministic calculation is theoretically unjustified. Furthermore, we construct a valuation portfolio such that the Best Estimate valuation decouples into calculations of 1.) deterministic coefficients derived from policy data and 2.) the prices of basis financial instruments that are independent of the individual policy data. Using this decomposition, the policies do not have to be tracked individually along each generated stochastic path. This allows for a more efficient evaluation of the Best Estimate for a large stock of policies with a stochastic model.

q-fin.MF↗
arXiv2023

Mean-field Libor market model and valuation of long term guarantees

Existence and uniqueness of solutions to the multi-dimensional mean-field Libor market model (introduced by [7]) is shown. This is used as the basis for a numerical asset-liability management (ALM) model capable of calculating future discretionary benefits in accordance with Solvency~II regulation. This ALM model is complimented with aggregated life insurance data to perform a realistic numerical study. This yields numerical evidence for heuristic assumptions which allow to derive estimators of lower and upper bounds for future discretionary benefits. These estimators are applied to publicly available life insurance data.

q-fin.RM↗
arXiv2023

Nonlinear feedback, double bracket dissipation and port control of Lie-Poisson systems

Methods from controlled Lagrangians, double bracket dissipation and interconnection and damping assignment -- passivity based control (IDA-PBC) are used to construct nonlinear feedback controls which (asymptotically) stabilize previously unstable equilibria of Lie-Poisson Hamiltonian systems. The results are applied to find an asymptotically stabilizing control for the rotor driven satellite, and a stabilizing control for Hall magnetohydrodynamic flow.

math-ph↗
arXiv2021

A mean-field extension of the LIBOR market model

We introduce a mean-field extension of the LIBOR market model (LMM) which preserves the basic features of the original model. Among others, these features are the martingale property, a directly implementable calibration and an economically reasonable parametrization of the classical LMM. At the same time, the mean-field LIBOR market model (MF-LMM) is designed to reduce the probability of exploding scenarios, arising in particular in the market-consistent valuation of long-term guarantees. To this end, we prove existence and uniqueness of the corresponding MF-LMM and investigate its practical aspects, including a Black '76-type formula. Moreover, we present an extensive numerical analysis of the MF-LMM. The corresponding Monte Carlo method is based on a suitable interacting particle system which approximates the underlying mean-field equation.

math.PR↗
arXiv2021

Estimation of future discretionary benefits in traditional life insurance

In the context of life insurance with profit participation, the future discretionary benefits ($FDB$), which are a central item for Solvency~II reporting, are generally calculated by computationally expensive Monte Carlo algorithms. We derive analytic formulas to estimate lower and upper bounds for the $FDB$. This yields an estimation interval for the $FDB$, and the average of lower and upper bound is a simple estimator. These formulae are designed for real world applications, and we compare the results to publicly available reporting data.

stat.ME↗
arXiv2020

Symmetry actuated closed-loop Hamiltonian systems

This paper extends the theory of controlled Hamiltonian systems with symmetries due to [9, 10, 6, 7, 11] to the case of non-abelian symmetry groups $G$. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the $G$-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9, 10]).

math-ph↗
arXiv2020

Probabilistic representation of helicity in viscous fluids

It is shown that the helicity of three dimensional viscous incompressible flow can be identified with the overall linking of the fluid's initial vorticity to the expectation of a stochastic mean field limit. The relevant mean field limit is obtained by following the Lagrangian paths in the stochastic Hamiltonian interacting particle system of [S. Hochgerner, Proc. R. Soc. A 474:20180178].

physics.flu-dyn↗
arXiv2020

Feedback control of charged ideal fluids

The theory of controlled mechanical systems of [6, 3, 4] is extended to the case of ideal incompressible fluids consisting of charged particles in the presence of an external magnetic field. The resulting control is of feedback type and depends on the Eulerian state of the controlled system. Moreover, the control is set up so that the corresponding closed loop equations are Lie-Poisson. This implies that the energy-momentum method of [1, 11] can be used to find a stabilizing control. As an example the case of planar parallel shear flow with an inflection point is treated. A state dependent feedback control is constructed which stabilizes the system for an arbitrarily long channel.

math-ph↗
arXiv2020

A Hamiltonian interacting particle system for compressible flow

The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean-Vlasov equation associated to the mean field limit yields the barotropic Navier-Stokes equation with density dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

physics.flu-dyn↗
arXiv2019

Analytical Validation Formulas for Best Estimate Calculation in Traditional Life Insurance

Within the context of traditional life insurance, a model-independent relationship about how the market value of assets is attributed to the best estimate, the value of in-force business and tax is established. This relationship holds true for any portfolio under run-off assumptions and can be used for the validation of models set up for Solvency~II best estimate calculation. Furthermore, we derive a lower bound for the value of future discretionary benefits. This lower bound formula is applied to publicly available insurance data to show how it can be used for practical validation purposes.

q-fin.RM↗
arXiv2018

A Hamiltonian Mean-Field System for the Navier-Stokes Equation

We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean-Vlasov equation yields the incompressible Navier Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.

math-ph↗
arXiv2012

Geometry of non-holonomic diffusion

We study stochastically perturbed non-holonomic systems from a geometric point of view. In this setting, it turns out that the probabilistic properties of the perturbed system are intimately linked to the geometry of the constraint distribution. For $G$-Chaplygin systems, this yields a stochastic criterion for the existence of a smooth preserved measure. As an application of our results we consider the motion planning problem for the noisy two-wheeled robot and the noisy snakeboard.

math-ph↗
arXiv2012

Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems

Let $Q$ be a Riemannian $G$-manifold. This paper is concerned with the symmetry reduction of Brownian motion in $Q$ and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions we discuss various versions of the stochastic Hamilton-Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum free particle reduction as described by Feher and Pusztai. As an application we use this reduction scheme to derive examples of quantum Calogero-Moser systems from a stochastic setting.

math.PR↗
arXiv2010

Stochastic Chaplygin systems

We mimic the stochastic Hamiltonian reduction of Lazaro-Cami and Ortega [17, 18] for the case of certain non-holonomic systems with symmetries. Using the non-holonomic connection it is shown that the drift of the stochastically perturbed $n$-dimensional Chaplygin ball is a certain gradient of the density of the preserved measure of the deterministic system.

math-ph↗
arXiv2009

$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball

Via compression ([11, 7]) we write the $n$-dimensional Chaplygin sphere system as an almost Hamiltonian system on $T^*SO(n)$ with internal symmetry group $SO(n-1)$. We show how this symmetry group can be factored out, and pass to the fully reduced system on (a fiber bundle over) $T^*S^{n-1}$. This approach yields an explicit description of the reduced system in terms of the geometric data involved. Due to this description we can study Hamiltonizability of the system. It turns out that the homogeneous Chaplygin ball, which is not Hamiltonian at the $T^*SO(n)$-level is Hamiltonian at the $T^*S^{n-1}$-level. Moreover, the 3-dimensional ball becomes Hamiltonian at the $T^*S^2$-level after time reparametrization, whereby we re-prove a result of [4, 5] in symplecto-geometric terms. We also study compression followed by reduction of generalized Chaplygin systems.

math-ph↗
arXiv2009

Chaplygin systems associated to Cartan decompositions of semi-simple Lie algebras

We relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process called truncation. Furthermore, a criterion for Hamiltonizability of the system on the so-called ultimate reduced level is given. As important special cases we find the Chaplygin ball rolling on a table and the rubber ball rolling over another ball.

math.DG↗
arXiv2008

Singular cotangent bundle reduction and spin Calogero-Moser systems

We develop a bundle picture for the case that the configuration manifold has only a single isotropy type, and give a formula for the reduced symplectic form in this setting. Furthermore, as an application of this bundle picture we consider Calogero-Moser systems with spin associated to polar representations of compact Lie groups.

math.SG↗