arXiv subjects

Julia Eisenberg

Publications and source records attributed to Julia Eisenberg.

13 recordsLinked to original sources

arXiv2025

Optimal risk mitigation by deep reinsurance

We consider an insurance company which faces financial risk in the form of insurance claims and market-dependent surplus fluctuations. The company aims to simultaneously control its terminal wealth (e.g. at the end of an accounting period) and the ruin probability in a finite time interval by purchasing reinsurance. The target functional is given by the expected utility of terminal wealth perturbed by a modified Gerber-Shiu penalty function. We solve the problem of finding the optimal reinsurance strategy and the corresponding maximal target functional via neural networks. The procedure is illustrated by a numerical example, where the surplus process is given by a Cramér-Lundberg model perturbed by a mean-reverting Ornstein-Uhlenbeck process.

q-fin.RM↗

arXiv2025

A Bayesian Spatio-Temporal Model of Temperature- and Humidity-Related Mortality Using High-Resolution Climate Data

In this study, we introduce a novel and comprehensive extension of a Bayesian spatio-temporal disease mapping model that explicitly accounts for gender-specific effects of meteorological exposures. Leveraging fine-scale weekly mortality and high-resolution climate data from Austria (2002 to 2019), we assess how interactions between temperature, humidity, age, and gender influence mortality patterns. Our approach goes beyond conventional modelling by capturing complex dependencies through structured interactions across space-time, space-age, and age-time dimensions, allowing us to capture complex demographic and environmental dependencies. The analysis identifies district-level mortality patterns and quantifies climate-related risks on a weekly basis, offering new insights for public health surveillance.

stat.AP↗

arXiv2022

Some Optimisation Problems in Insurance with a Terminal Distribution Constraint

In this paper, we study two optimisation settings for an insurance company, under the constraint that the terminal surplus at a deterministic and finite time $T$ follows a normal distribution with a given mean and a given variance. In both cases, the surplus of the insurance company is assumed to follow a Brownian motion with drift. First, we allow the insurance company to pay dividends and seek to maximise the expected discounted dividend payments or to minimise the ruin probability under the terminal distribution constraint. Here, we find explicit expressions for the optimal strategies in both cases: in discrete and continuous time settings. Second, we let the insurance company buy a reinsurance contract for a pool of insured or a branch of business. To achieve a certain level of sustainability (i.e. the collected premia should be sufficient to buy reinsurance and to pay the occurring claims) the initial capital is set to be zero. We only allow for piecewise constant reinsurance strategies producing a normally distributed terminal surplus, whose mean and variance lead to a given Value at Risk or Expected Shortfall at some confidence level $α$. We investigate the question which admissible reinsurance strategy produces a smaller ruin probability, if the ruin-checks are due at discrete deterministic points in time.

q-fin.MF↗

arXiv2021

Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve $α(t)$ can be found to determine the optimal strategy at time $t$. In a second setting we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach as well as using backward stochastic differential equations.

math.OC↗

arXiv2020

Optimal Dividends Paid in a Foreign Currency for a Lévy Insurance Risk Model

This paper considers an optimal dividend distribution problem for an insurance company where the dividends are paid in a foreign currency. In the absence of dividend payments, our risk process follows a spectrally negative Lévy process. We assume that the exchange rate is described by a an exponentially Lévy process, possibly containing the same risk sources like the surplus of the insurance company under consideration. The control mechanism chooses the amount of dividend payments. The objective is to maximise the expected dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. Via the corresponding Hamilton--Jacobi--Bellman equation we find the necessary and sufficient conditions for optimality of a single dividend barrier strategy. A number of numerical examples illustrate the theoretical analysis.

q-fin.MF↗

arXiv2019

Transforming public pensions: A mixed scheme with a credit granted by the state

Birth rates have dramatically decreased and, with continuous improvements in life expectancy, pension expenditure is on an irreversibly increasing path. This will raise serious concerns for the sustainability of the public pension systems usually financed on a pay-as-you-go (PAYG) basis where current contributions cover current pension expenditure. With this in mind, the aim of this paper is to propose a mixed pension system that consists of a combination of a classical PAYG scheme and an increase of the contribution rate invested in a funding scheme. The investment of the funding part is designed so that the PAYG pension system is financially sustainable at a particular level of probability and at the same time provide some gains to individuals. In this sense, we make the individuals be an active part to face the demographic risks inherent in the PAYG and re-establish its financial sustainability.

q-fin.RM↗

arXiv2019

Maximising with-profit pensions without guarantees

Currently, pension providers are running into trouble mainly due to the ultra-low interest rates and the guarantees associated to some pension benefits. With the aim of reducing the pension volatility and providing adequate pension levels with no guarantees, we carry out mathematical analysis of a new pension design in the accumulation phase. The individual's premium is split into the individual and collective part and invested in funds. In times when the return from the individual fund exits a predefined corridor, a certain number of units is transferred to or from the collective account smoothing in this way the volatility of the individual fund. The target is to maximise the total accumulated capital, consisting of the individual account and a portion of the collective account due to a so-called redistribution index, at retirement by controlling the corridor width. We also discuss the necessary and sufficient conditions that have to be put on the redistribution index in order to avoid arbitrage opportunities for contributors.

q-fin.RM↗

arXiv2019

Suboptimal Control of Dividends under Exponential Utility

We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time homogeneity of the considered problem. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a non-linear barrier. In the related article by granditz et al., it has been observed that standard numerical methods break down in certain parameter cases and no close form solution has been found. For these reasons, we offer a new method allowing to estimate the distance of an arbitrary smooth enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies - payout on the maximal rate, and constant barrier strategies - by measuring the distance of its performance function to the value function.

q-fin.RM↗

arXiv2018

An Exponential Cox-Ingersoll-Ross Process as Discounting Factor

We consider an economic agent (a household or an insurance company) modelling its surplus process by a deterministic process or by a Brownian motion with drift. The goal is to maximise the expected discounted spendings/dividend payments, given that the discounting factor is given by an exponential CIR process. In the deterministic case, we are able to find explicit expressions for the optimal strategy and the value function. For the Brownian motion case, we offer a method allowing to show that for a small volatility the optimal strategy is a constant-barrier strategy.

q-fin.MF↗

arXiv2016

The Impact of Negative Interest Rates on Optimal Capital Injections

In the present paper, we investigate the optimal capital injection behaviour of an insurance company if the interest rate is allowed to become negative. The surplus process of the considered insurance entity is assumed to follow a Brownian motion with drift. The changes in the interest rate are described via a Markov-switching process. It turns out that in times with a positive rate, it is optimal to inject capital only if the company becomes insolvent. However, if the rate is negative it might be optimal to hold a strictly positive reserve. We establish an algorithm for finding the value function and the optimal strategy, which is proved to be of barrier type. Using the iteration argument, we show that the value function solves the Hamilton--Jacobi--Bellman equation, corresponding to the problem.

q-fin.MF↗

arXiv2016

Deterministic Income with Deterministic and Stochastic Interest Rates

We consider an individual or household endowed with an initial capital and an income, modeled as a deterministic process with a continuous drift rate. At first, we model the discounting rate as the price of a zero-coupon bond at zero under the assumption of a short rate evolving as an Ornstein-Uhlenbeck process. Then, a geometric Brownian motion as the preference function and an Ornstein-Uhlenbeck process as the short rate are taken into consideration. It is assumed that the primal interest of the economic agent is to maximise the cumulated value of (expected) discounted consumption from a given time up to a finite deterministic time horizon $T\in\R_+$ or, in a stochastic setting, infinite time horizon. We find an explicit expression for the value function and for the optimal strategy in the first two cases. In the third case, we have to apply the viscosity ansatz.

math.OC↗

arXiv2016

A Note on the Optimal Dividends Paid in a Foreign Currency

We consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.

q-fin.MF↗

arXiv2016

Unrestricted Consumption under a Deterministic Wealth and an Ornstein-Uhlenbeck Process as a Discount Rate

We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein-Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.

math.OC↗