-
Asmussen, S., Schmidli, H. & Schmidt, V. (1999).
- Tail probabilities
for non-standard risk and queueing processes with
subexponential jumps. Adv. in
Appl. Probab. 31, 422-447.
(Abstract)
-
Barndorff-Nielsen,
O.E. & Schmidli, H. (1995).
- Saddlepoint approximations for the probability of ruin in finite
time. Scand. Actuarial J., 169-186.
(Abstract)
-
Bata, K.
& Schmidli, H. (2020).
- Optimal Capital Injections and Dividends with Tax in a Risk
Model in Discrete Time. European Actuarial J. 10, 235-259.
(Abstract)
-
Brachetta, M. & Schmidli, H. (2020).
- Optimal reinsurance and investment in a diffusion
model. Decisions in Economics and Finance 43, 341-361.
(Abstract)
-
Brinker, L.V. & Schmidli, H. (2022).
- Optimal discounted drawdowns in a diffusion approximation under
proportional reinsurance J. Appl. Probab. 59, 527-540.
(Abstract)
-
Brinker, L.V. & Schmidli, H. (2023).
- Optimisation of drawdowns by generalised reinsurance in the classical risk model Decisions in Economics and Finance forthcoming.
(Abstract)
-
Christensen,
C.V. & Schmidli, H. (2000).
- Pricing catastrophe insurance products based on actually reported
claims. Insurance Math. Econom. 27, 189-200.
(Abstract)
-
Eisenberg, J. &
Schmidli, H. (2009).
- Optimal control of capital injections by reinsurance
in a diffusion approximation. Blätter DGVFM
30, 1-13.
(Abstract)
-
Eisenberg, J. &
Schmidli, H. (2011).
- Minimising expected discounted capital injections by
reinsurance in a classical risk model. Scand.
Actuarial J., 155-176.
(Abstract)
-
Eisenberg, J. &
Schmidli, H. (2011).
- Optimal Control of Capital Injections by Reinsurance with
Riskless Rate of Interest. J. Appl. Probab. 48,
733-748.
(Abstract)
-
Embrechts, P.,
Grandell, J. & Schmidli, H. (1993).
- Finite-time Lundberg inequalities in the Cox case. Scand.
Actuarial J., 17-41.
(Abstract)
-
Embrechts, P.
& Schmidli, H. (1994).
- Ruin estimation for a general insurance risk model. Adv. in
Appl. Probab. 26, 404-422.
(Abstract)
-
Embrechts, P.
& Schmidli, H. (1994).
- Modelling of extremal events in insurance and finance. Z.
Oper. Res., 1-33.
(Abstract)
-
Furrer, H.J. & Schmidli, H. (1994).
- Exponential inequalities for ruin probabilities of risk
processes perturbed by diffusion. Insurance Math.
Econom. 15, 23-36.
(Abstract)
-
Grandell, J. & Schmidli, H. (2011).
- Ruin probabilities in a diffusion
environment. J. Appl. Probab. 48A, 39-50.
(Abstract)
-
Hald, M.
& Schmidli, H. (2004).
- On the maximisation of the adjustment coefficient under
proportional reinsurance. ASTIN Bull. 34,75-83.
(Abstract)
-
Hipp, C.
& Schmidli, H. (2004).
- Asymptotics of ruin probabilities for controlled risk processes
in the small claims case. Scand. Actuarial J., 321-335.
(Abstract)
-
Kulenko, N. & Schmidli, H. (2008).
- Optimal Dividend Strategies in a Cramér-Lundberg Model with
Capital Injections. Insurance Math. Econom. 43, 270-278.
(Abstract)
-
Mishura, Y. & Schmidli, H. (2012).
- Dividend barrier strategies in a renewal risk model with
generalized Erlang interarrival times. North Amer. Actuarial
J. 16 (4), 493-512.
(Abstract)
-
Scheer, N. & Schmidli, H. (2011).
- Optimal Dividend Strategies in a Cramér-Lundberg Model with
Capital Injections and Administration Costs. European Actuarial
Journal 1, 57-92.
(Abstract)
-
Schmeck, M.D. & Schmidli, H. (2020).
- Mortality Options: the Point of View of an Insurer.
Insurance Math. Econom., 96 98-115.
(Abstract)
-
Schmidli, H. (1994).
- Diffusion approximations for a risk process with the possibility
of borrowing and investment. Comm. Statist. Stochastic
Models 10, 365-388.
(Abstract)
-
Schmidli, H. (1994).
- Risk theory in an economic environment and Markov processes.
Schweiz. Verein. Versicherungsmath. Mitt., 51-70.
(Abstract, Zusammenfassung, Résumé)
-
Schmidli, H. (1994).
- Corrected diffusion approximations for a risk process with the
possibility of borrowing and investment. Schweiz. Verein.
Versicherungsmath. Mitt., 71-82.
(Abstract, Zusammenfassung, Résumé)
-
Schmidli, H. (1994).
- Estimation of the abscissa of convergence of the moment
generating function. Research Report No. 276, Dept. Theor.
Statist., Aarhus University.
(Abstract)
-
Schmidli, H. (1995).
- Cramér-Lundberg approximations
for ruin probabilities of risk processes perturbed by diffusion.
Insurance Math. Econom. 16, 135-149.
(Abstract)
-
Schmidli, H. (1996).
- Lundberg inequalities for a Cox model with a piecewise constant
intensity. J. Appl. Probab. 33, 196-210.
(Abstract)
-
Schmidli, H. (1996).
- Martingales and insurance risk. In: Obretenov A. (ed.)
Lecture notes of the 8-th
international summer school on probability and mathematical
statistics (Varna). Science Culture Technology
Publishing, Singapore, 155-188.
(Abstract)
-
Schmidli, H. (1997).
- Estimation of the Lundberg coefficient for a Markov modulated
risk model. Scand. Actuarial J., 48-57.
(Abstract)
-
Schmidli, H. (1997).
- An extension to the renewal theorem and an application to risk
theory. Ann. Appl. Probab. 7, 121-133.
(Abstract)
-
Schmidli, H. (1999).
- On the distribution of the surplus prior and at ruin.
ASTIN Bull. 29, 227-244.
(Abstract)
-
Schmidli, H. (1999).
- Compound sums and subexponentiality. Bernoulli
5, 999-1012.
(Abstract)
-
Schmidli, H. (1999).
- Perturbed risk processes: a review.
Theory of Stochastic Processes 5, 145-165.
(Abstract)
-
Schmidli, H. (2000).
- Characteristics of ruin probabilities in classical risk models
with and without investment, Cox risk models and perturbed risk
models. Memoirs No 15, Dept. Theor. Statist., Aarhus
University. (Abstract)
-
Schmidli, H. (2001).
- Distribution of the first ladder height of a stationary
risk process perturbed by α-stable Lévy motion. Insurance
Math. Econom., 28, 13-20.
(Abstract)
-
Schmidli, H. (2001).
- Optimal proportional reinsurance policies in a dynamic
setting. Scand. Actuarial J., 55-68.
(Abstract)
-
Schmidli, H. (2002).
- On minimising the ruin probability by investment and
reinsurance. Ann. Appl. Probab. 12, 890-907.
(Abstract)
-
Schmidli, H. (2001).
- Risk processes conditioned on ruin. Working paper
176, Laboratory of Actuarial Mathematics,
University of Copenhagen.
(Abstract)
-
Schmidli, H. (2004).
- Asymptotics of ruin probabilities for risk processes under optimal
reinsurance and investment policies: the large claim
case. Queueing Syst. Theory Appl. 46,
149-157.
(Abstract)
-
Schmidli, H. (2002).
- Asymptotics of ruin probabilities for risk processes under optimal
reinsurance policies: the small claim case. Working paper
180, Laboratory of Actuarial Mathematics,
University of Copenhagen.
(Abstract)
-
Schmidli, H. (2003).
- Modelling PCS options via individual indices. Working paper
187, Laboratory of Actuarial Mathematics,
University of Copenhagen.
(Abstract)
-
Schmidli, H. (2004).
- Diffusion Approximations. In: Teugels, J.L. and Sundt, B. (ed.)
Encyclopedia of Actuarial Sciences, Vol. 1.
J. Wiley and Sons, Chichester, 519-522.
(Abstract)
-
Schmidli, H. (2004).
- Filtrations. In: Teugels, J.L. and Sundt, B. (ed.)
Encyclopedia of Actuarial Sciences, Vol. 2.
J. Wiley and Sons, Chichester, 671-672.
(Abstract)
-
Schmidli, H. (2004).
- Martingales. In: Teugels, J.L. and Sundt, B. (ed.)
Encyclopedia of Actuarial Sciences, Vol. 2.
J. Wiley and Sons, Chichester, 1096-1101.
(Abstract)
-
Schmidli, H. (2004).
- Surplus Process. In: Teugels, J.L. and Sundt, B. (ed.)
Encyclopedia of Actuarial Sciences, Vol. 3.
J. Wiley and Sons, Chichester, 1645-1650.
(Abstract)
-
Schmidli, H. (2005).
- On optimal investment and subexponential claims. Insurance
Math. Econom. 36, 25-35.
(Abstract)
-
Schmidli, H. (2005).
- Controlled risk processes and subexponential claims. In: Kolev,
N. and Morettin, P. (ed.) Proceedings of the Second Brazilian
Conference on Statistical Modelling in Insurance and Finance,
55-63. (Abstract)
-
Schmidli, H. (2005).
- Discussion of the Article: Gerber, H.U. and Shiu, E.S.W. The
time value of ruin in a Sparre Andersen Model.
N. Am. Actuar. J. 9, 69-70.
-
Schmidli, H. (2006).
- Optimisation in non-life insurance. Stochastic Models
22, 689-722.
(Abstract)
-
Schmidli, H. (2008).
- Stochastic Control for Insurance Companies. In: Melnick,
E.L. and Everitt, B.S. (ed.)
Encyclopedia of Quantitative Risk Analysis and
Assessment, Vol. 4.
J. Wiley and Sons, Chichester.
(Abstract)
-
Schmidli, H. (2008).
- On Cramér-Lundberg approximations for ruin probabilities
under optimal excess of loss reinsurance. Journal of
Numerical and Applied Mathematics 96, 198-205.
(Abstract)
-
Schmidli, H. (2009).
- Mathematics: Curse or blessing. Annals of Actuarial
Science 4, 173-176.
-
Schmidli, H. (2010).
- On the Gerber-Shiu Function and Change of
Measure. Insurance Math. Econom. 46, 3-11.
(Abstract)
-
Schmidli, H. (2010).
- Conditional Law of Risk Processes Given that Ruin
Occurs. Insurance Math. Econom. 46, 281-289.
(Abstract)
-
Schmidli, H. (2010).
- Accumulated claims. In: Cont, R. (ed.). Encyclopedia of
Quantitative Finance, J. Wiley and Sons, Chichester, 4-6.
(Abstract)
-
Schmidli, H. (2013).
- A note on Gerber-Shiu functions with an application. In:
Silvestrov, D. and Martin-Löf, A. (eds.) Modern Problems in
Insurance Mathematics. Springer-Verlag, Cham, 21-36.
(Abstract)
-
Schmidli, H. (2015).
- Extended Gerber-Shiu Functions in a Risk Model with
Interest. Insurance Math. Econom. 61, 271-275.
(Abstract)
-
Schmidli, H. (2016).
- On Capital Injections and Dividends with Tax in a Classical
Risk Model. Insurance Math. Econom. 71, 138-144.
(Abstract)
-
Schmidli, H. (2017).
- Dividends with tax and capital injection in a spectrally negative
Lévy risk model. Theory of Probability and Mathematical
Statistics
96, 171-183.
(Abstract)
-
Schmidli, H. (2017).
- On Capital Injections and Dividends with Tax in a Diffusion
Approximation. Scand. Actuarial J. 9, 751-760.
(Abstract)
-
Schmidli, H. (2022).
- Dividends and Capital Injections in a Renewal Model with Erlang
Distributed Inter-Arrival Times. Scand. Actuarial J.
2022, 49-63.
(Abstract)
-
Vierkötter, M. & Schmidli, H. (2017).
- On optimal dividends with exponential and linear penalty
payments.
Insurance Math. Econom. 72, 265-270.
(Abstract)
Abstracts
A well-known result on the distribution tail of the maximum of a random walk
with heavy-tailed increments is extended to
more general stochastic processes. Results are given
in different settings, involving, e.g., stationary increments
and regeneration. Several examples and counterexamples illustrate that
the conditions of the theorems can easily be verified in
practice and are in part necessary. The examples include Poisson processes,
superimposed renewal processes,
Markovian arrival processes, semi-Markov input and Cox processes
with piecewise constant intensities.
KEY WORDS: Ruin probability; stationary waiting time
distribution; random walk; ergodic inter-occurrence times; integrated tail
distribution; regenerative surplus process; regular variation; subexponential
distribution.
1991 Mathematical subject classification: primary 60G70; secondary
60J15, 60K25
Saddlepoint techniques are applied to obtain approximations for the probability
of ruin both in finite and in infinite time for the classical
Cramér-Lundberg model. The resulting approximations are compared to
exact values.
KEY WORDS: Ruin probability; saddlepoint techniques; Laplace transform;
cumulant function; Cramér-Lundberg model.
We consider a risk model in discrete time with dividends and capital
injections. The goal is to maximise the value of a dividend strategy. We show
that the optimal strategy is of barrier type. That is, all capital above a
certain threshold is paid as dividend. A second problem add tax to the
dividends but an injection leads to an exemption from tax. We show that the
value function fulfils a Bellman equation. As a special case, we consider the
case of premia of size one. In this case we show that the optimal strategy is
a two barrier strategy. That is, there is a barrier if a next dividend of size
one can be paid without tax and a barrier if the next dividend of size one
will be taxed. In both models, we illustrate the findings by de Finetti's example.
KEY WORDS: discrete risk model; optimal dividend problem;
capital injections; tax; Bellman equation; two barrier strategy; de Finetti
model
2010 Mathematical subject classification: primary 91B30; secondary
60G42, 60K30, 60J10.
We consider a diffusion approximation to an insurance risk model where an
external driver models a stochastic environment. The insurer can buy reinsurance.
Moreover, investment in a financial market is possible. The financial market
is also driven by the environmental process. Our goal is to maximise terminal
expected utility. In particular, we consider the case of SAHARA utility
functions. In the case of proportional and excess-of-loss reinsurance, we
obtain explicit results.
KEY WORDS: Optimal reinsurance; optimal investment;
Hamilton-Jacobi-Bellman equation; SAHARA utility; proportional reinsurance;
excess-of-loss reinsurance
2010 Mathematical subject classification: primary 91B30; secondary
60G44, 60J60, 93E20.
A diffusion approximation to a risk process under dynamic proportional
reinsurance is considered. The goal is to minimise the discounted time in
drawdown; that is, the time where the distance of the present surplus to the
running maximum is larger than a given level $d > 0$. We calculate the value
function and determine the optimal reinsurance strategy. We conclude that the
drawdown measure stabilises process paths but has a drawback
as it also prevents surpassing the initial maximum. That is, the insurer
is, under the optimal strategy, not interested in any more profits. We
therefore suggest to use optimisation criteria not avoiding future profits.
KEY WORDS: drawdown; diffusion approximation; optimal
proportional reinsurance; Hamilton-Jacobi-Bellman equation
2020 Mathematical subject classification: primary 91G05; secondary
93E20, 60G44, 60J60.
We consider a Cramér-Lundberg model representing the surplus of an insurance
company under a general reinsurance control process. We aim to minimise the
expected time during which the surplus is bounded away from its own running
maximum by at least d >0 (discounted at a preference rate δ>0) by
choosing a reinsurance strategy. By analysing the drawdown process (i.e. the absolute distance of the controlled surplus model to its maximum) directly, we prove that the value function fulfils the
corresponding Hamilton-Jacobi-Bellman equation and show how one can
calculate the value function and the optimal strategy. If the initial drawdown is critically large, the problem corresponds to the maximisation of the Laplace transform of a passage time. We show that a constant retention level is optimal. If the drawdown is smaller than d, the problem can be expressed as an element of a set of Gerber-Shiu optimisation problems. We show how these problems can be solved and that the optimal strategy is of feedback form. We illustrate the
theory by examples of the cases of light and heavy tailed claims.
KEY WORDS: drawdowns; general optimal reinsurance;
classical risk model; Hamilton-Jacobi-Bellman equation
2020 Mathematical subject classification: primary 91B05; secondary
60G44, 60J25.
This article deals with the problem of pricing a financial product
relying on an index of reported claims from catastrophe insurance.
The problem of pricing such products is that, at a fixed time in
the trading period, the total claim amount from the
catastrophes occurred is not known. Therefore one has to price these products
solely from knowing the aggregate amount of the reported claims at the fixed
time point. This article will propose a way to handle this problem, and will
thereby extend the existing pricing models for products of this kind.
KEY WORDS: Insurance futures; derivatives; claims-process;
catastrophe insurance; mixed Poisson model; IBNR techniques; change of
measure; expected utility; approximations.
1991 Mathematical subject classification: primary 62PO5; secondary
60G55, 90A43, 62E17
In this paper we consider a diffusion approximation to a classical risk
process, where the claims are reinsured by some reinsurance with deductible
b ∈[0,˜b], where b=˜b means "no reinsurance"
and b=0 means "full reinsurance". The cedent can choose an adapted
reinsurance strategy (bt)t ≥ 0, i.e. the
deductible can be changed continuously. In addition, the cedent has to inject
fresh capital in order to keep the surplus positive. The problem is to
minimise the expected discounted cost over all admissible reinsurance
strategies. We find an explicit expression for the value function and the
optimal strategy using the Hamilton-Jacobi-Bellman approach. Some examples
illustrate the method.
KEY WORDS: optimal control; stochastic control; Hamilton-Jacobi-Bellman
equation; dividend; reinvestment; classical risk model; barrier strategy.
1991 Mathematical subject classification: primary 60K10; secondary
91B30, 60J60
In this paper we consider a classical continuous time risk model, where the
claims are reinsured by some reinsurance with retention level
b ∈[0,˜b], where b=˜b means "no reinsurance"
and b=0 means "full reinsurance". The insurer can change the retention
level continuously. To prevent negative surplus the insurer has to inject
additional capital. The problem is to minimise the expected discounted cost
over all admissible reinsurance strategies. We show, that an optimal reinsurance
strategy exists. For some special cases we will be able to give the optimal
strategy explicitly. In other cases the method will be illustrated only
numerically.
KEY WORDS: optimal control; stochastic control; Hamilton-Jacobi-Bellman
equation; capital injections; classical risk model; reinsurance.
2010 Mathematical subject classification: primary 60K10; secondary
93E20, 91B30
We consider a classical risk model and its diffusion approximation, where the
individual claims are reinsured by a reinsurance treaty with deductible
b ∈ [0,˜b]. Here b=˜b means "no
reinsurance" and b=0 means "full reinsurance". In addition to that the
insurer is allowed to invest into a riskless asset with some constant interest
rate m>0. The cedent can choose an adapted reinsurance strategy {bt}t≥ 0, i.e. the parameter can be changed continuously.
If the surplus process becomes negative, the cedent has to inject additional
capital. Our aim is to minimise the expected discounted capital injections
over all admissible reinsurance strategies. We find an explicit expression for
the value function and the optimal strategy using the Hamilton-Jacobi-Bellman
approach in the case of a diffusion approximation. In the case of the
classical risk model, we show the existence of a "weak" solution and calculate
the value function numerically.
KEY WORDS: optimal control; stochastic control; Hamilton-Jacobi-Bellman
equation; capital injections; classical risk model; riskless interest rate.
2010 Mathematical subject classification: primary 60K10; secondary
91B30, 60J65
We consider an insurance model, where the underlying
point process is a Cox process. Using a martingale approach, applied to
piecewise-deterministic Markov processes, finite-time Lundberg inequalities
are obtained.
KEY WORDS: Ruin probability; Lundberg inequality; Cox process;
piecewise-deterministic Markov process; martingale methods.
The theory of piecewise-deterministic Markov processes is used in order to
investigate insurance risk models where borrowing, investment and inflation are
present.
KEY WORDS: Risk theory; ruin; martingales; piecewise-deterministic
Markov processes.
1991 Mathematical subject classification: primary 60K30; secondary
62P05
Extremal events play an increasingly important role in stochastic modelling in
insurance and finance. Over many years, probabilists and statisticians have
developed techniques for the description, analysis and prediction of such
events. In the present paper, we review the relevant theory which may also be
used in the wider context of OR. Various applications from the field of
insurance and finance are discussed. Via an extensive list of references, the
reader is guided towards further material related to the above problem
areas.
by Hans-Jörg Furrer & Hanspeter Schmidli
A class of diffusion processes following
locally a vector field is constructed and the extended generator is computed
for a subset of the domain of the generator. Using this theory, martingales for
risk processes perturbed by diffusion are obtained. This leads to exponential
bounds for the ruin probability in infinite as well as in finite time.
KEY WORDS: Ruin probability; Lundberg inequality; risk
theory; martingale methods; diffusion.
We consider an insurance model, where the underlying point process is
a Cox process. Using a martingale approach applied to diffusion
processes, finite-time Lundberg inequalities are obtained. By change
of measure techniques Cramér-Lundberg approximations are derived.
KEY WORDS: ruin probability; Lundberg inequality; Cox process;
diffusion process; martingale methods; change of measure techniques.
2010 Mathematical subject classification: primary 60K37, 60J60;
secondary 60G44, 60F10
by Morten Hald &
Hanspeter Schmidli
In this note we consider how to maximise the adjustment coefficient in the case
of proportional reinsurance. This complements some work of
Waters (1983), where it was shown that there is a unique retention
level maximising the adjustment coefficient. The advantage of our method is
that only one implicit equation has to be solved.
KEY WORDS: Adjustment coefficient; proportional reinsurance;
Cramér-Lundberg risk model; Sparre-Andersen risk model; Markov modulated
risk model.
2000 Mathematical subject classification: primary 91B30; secondary
60J25
by Christian Hipp & Hanspeter Schmidli
We consider a risk process with the possibility of investment into
a risky asset. The aim of the paper is to obtain the asymptotic behaviour of
the ruin probability under the optimal investment strategy in the small
claims case. In addition we prove convergence of the optimal investment
level as the initial capital tends to infinity.
KEY WORDS: Ruin probability; change of
measure; optimal control; Cramér-Lundberg approximation; adjustment
coefficient; Lundberg bounds; martingale; geometric Browninan motion.
2000 Mathematical subject classification: primary 60F10; secondary
60J25, 91B30
by Natalie Kulenko &
Hanspeter Schmidli
We consider a classical risk model with dividend payments and capital
injections. Thereby, the surplus has to stay positive. Like in the classical
de Finetti problem, we want to maximise the discounted dividend payments minus
the penalised discounted capital injections. We derive the
Hamilton-Jacobi-Bellman equation for the problem
and show that the optimal strategy is a barrier strategy. We explicitly
characterise when the optimal barrier is at 0 and find the solution for
exponentially distributed claim sizes.
KEY WORDS: optimal control, stochastic control, Hamilton-Jacobi-Bellman
equation, dividend, capital injection, classical risk model, barrier
strategy.
2000 Mathematical subject classification: primary 93E20; secondary
49J22, 60G44
We consider a renewal risk model with generalised Erlang distributed
interarrival times. We assume that the phases of the interarrival time can be
obeserved. In order to solve de Finetti's dividend problem, we first
consider phase-wise barrier strategies and look for the optimal barriers when
the initial capital is 0. For exponentially distributed claim sizes, we show
that the barrier strategy is optimal among all admissible strategies. For the
special case of Erlang(2) interarrival times, we calculate the value function
and the optimal barriers.
KEY WORDS: Insurance portfolio; dividend process; barrier strategy;
optimal dividend strategy; homogeneous Markov chain with finite phase space;
generalised Erlang distribution; exponentially distributed claim sizes.
2010 Mathematical subject classification: primary 91B30; secondary
34B60, 60K30
by Natalie Scheer &
Hanspeter Schmidli
In this paper, we consider a classical risk model with dividend payments and
capital injections in the presence of both fixed and proportionals
administration costs. Negative surplus or ruin is not allowed. We measure the
value of a strategy by the discounted value of the dividends minus the
costs. It turns out, capital injections are only made if the claim process
falls below zero. Further, the company may at the time of an injection not
only inject the deficit, but inject additional capital C ≥ 0 to prevent
future capital injections. We derive the associated Hamilton-Jacobi-Bellman
equation and show that the optimal strategy is of band type. By using
Gerber-Shiu functions, we derive a method to determine numerically the
solution to the integro-differential equation and the unknown value C.
KEY WORDS: stochastic control; Hamilton-Jacobi-Bellman equation;
dividend; capital injection; classical risk model; administration costs.
2010 Mathematical subject classification: primary 93E20; secondary
91B30, 60G44
In a discrete time framework we consider a life insurer who is able to buy a
securitisation product to hedge mortality. Two cohorts are considered: one
underlying the securitisation product and one for the portfolio of the
insurer. In a general setting, we show that there exists a unique strategy
that maximises the insurer's expected utility from terminal wealth. We then
numerically illustrate our findings: in a Gompertz-Makeham model, where the
realized survival probabilities can fluctuate moderately within an ε-corridor,
as well as in a toy model for mortality shocks. In both examples the insurer
can hedge longevity risk by trading in a survival bond.
KEY WORDS: mortality option; optimal strategy; maximal utility;
exponential utility
2010 Mathematical subject classification: primary 91B30; secondary
93E20
by Hanspeter Schmidli
A diffusion approximation is constructed for an insurance risk model which was
considered by Embrechts and Schmidli (1994), where the
company is allowed to borrow money if needed and to invest money for large
surpluses. Besides the weak convergence of a sequence of such processes to a
diffusion, the convergence of the finite and infinite time ruin probabilities
is shown. The ruin probabilities of the diffusion are calculated and, for two
examples, compared with the exact values. The convergence of the corresponding
infinite time ruin probabilities for a diffusion approximation for the
classical process is also shown.
by Hanspeter Schmidli
Abstract: Two models of the collective theory of risk, one introduced by
Gerber (1971) and the other by Dassios and Embrechts (1989), where
borrowing is allowed, are extended with the possibility of investment
above a certain level. An application of Davis' method of piecewise
deterministic Markov processes (Davis, 1984) yields the Laplace transform of
the ruin probabilities as well as a `Lundberg exponent' for the model. For the
case of equal interest rates for invested and borrowed money an explicit
inversion formula is given.
Zusammenfassung: Zwei Modelle der kollektiven
Risikotheorie (das eine geht zurück auf Gerber (1971), das andere auf
Dassios und Embrechts (1989)), in denen die Möglichkeit von Geldanleihe
besteht, werden durch die Möglichkeit von Investition des Kapitals
über einer gewissen Schranke erweitert. Durch Anwendung der Methode der
von Davis (1984) eingeführten stückweise deterministischen Markov
Prozesse lässt sich die Laplace-Transformierte der
Ruinwahrscheinlichkeiten sowie ein `Lundberg Exponent' bestimmen. Für den
Fall von gleichen Zinsraten für geliehenes und investiertes Kapital
lässt sich eine explizite Umkehrformel herleiten.
Résumé: On propose une extension
de deux modèles de la théorie du risque collectif (l'un introduit
par Gerber (1971), l'autre par Dassios et Embrechts (1989)) incluant la
possibilité d'investir le capital supérieur à une certaine
limite. La méthode des processus de Markov déterministes par
morceaux, introduite par Davis (1984), permet d'obtenir la transformée
de Laplace de la probabilité de ruine ainsi qu'un `exposant de
Lundberg'. Dans le cas où les taux d'intérêt du capital
emprunté et du capital investi sont égaux on donne une formule
d'inversion explicite.
by Hanspeter Schmidli
Abstract: In many situations in insurance risk theory one has the
problem that exact values for the ruin probability are hard to obtain.
Therefore approximations are called for. It turns out that to get crude
estimates for the ruin probability the so called diffusion approximations are
useful. Recently a method to combine diffusion approximations for the classical
Cramér-Lundberg model and diffusion approximations for more general
models was worked out (Schmidli, 1994). In the present
paper it is shown that
the method can be improved by using corrected diffusion approximations.
Zusammenfassung: In der Risikotheorie steht man
oft vor dem Problem, dass exakte Werte für die Ruinwahrscheinlichkeit nur
sehr schwer zu erhalten sind. Man versucht sich deshalb mit
Approximationsmethoden auszuhelfen. Um grobe Schätzwerte für die
Ruinwahrscheinlichkeiten zu erhalten stellen die sogenannten
Diffusionsapproximationen eine nützliche Methode dar. Kürzlich wurde
eine Methode entwickelt (Schmidli, 1994), die es erlaubt,
Diffusionsapproximationen für das klassische Cramér-Lundberg Modell
und Diffusionsapproximationen für allgemeinere Prozesse zu kombinieren. In
diesem Artikel wird gezeigt, dass eine Verbesserung der Approximation erreicht
werden kann, indem man korrigierte Diffusionsapproximationen benutzt.
Résumé: Dans la théorie du
risque on est souvent confronté au problème, que les valeur
exactes de la probabilité de ruine sont très difficiles à
obtenir. On cherche alors à utiliser des méthodes
d'approximation. Les approximations par processus de diffusion se
révellent être une méthode efficace pour obtenir des
estimations grossières de la probabilité de ruine. Une
méthode développée récemment (Schmidli, 1994)
permet de combiner des approximations par procuessus de diffusion pour le
modèle classique de Cramér-Lundberg et des approximations par
processus de diffusion pour des processus plus généraux. Dans ce
papier on montre que l'approximaion peut être améliorée en
utilisant des approximations par processus de diffusion corrigées.
by Hanspeter Schmidli
Let F be the distribution function of a positive random variable
X and assume that (1-F(x))-1 (1-F(x+y)) converges to a
strictly positive value as x → ∞. It is shown that the
right end point R of the interval where the moment generating function
exists is finite and that the distribution function of erX is
regularly varying with coefficient -R/r. Hence Hill's estimator is
proposed for estimation of R.
KEY WORDS: Divergence point; regular variation; parameter estimation;
extreme value theory; order statistics.
1991 Mathematical subject classification: primary 62G05; secondary
62G30, 60E10
by Hanspeter Schmidli
In the present paper risk processes perturbed by diffusion are considered. By
exponential tilting the processes are inbedded in an exponential family of
stochastic processes, such that the type of process is preserved. By change of
measure techniques asymptotic expressions for the ruin probability are
obtained. This proves that the coefficients obtained by Furrer and Schmidli (1994) are the adjustment
coefficients.
KEY WORDS: Ruin probability; Cramér-Lundberg approximation; risk
theory; martingale methods; change of measure; diffusion; exponential
family.
1991 Mathematical subject classification: primary 60F10; secondary
60J25, 60K10
by Hanspeter Schmidli
A Cox risk process with a piecewise constant intensity is considered where the
sequence (Li)
of successive levels of the intensity forms a Markov
chain. The duration σi of the level
Li is assumed to be only dependent via
Li. In the small-claim case a Lundberg inequality is
obtained via a martingale approach. It is shown furthermore by a Lundberg bound
from below that the resulting adjustment coefficient gives the best possible
exponential bound for the ruin probability. In the case where the stationary
distribution of Li contains a discrete component, a
Cramér-Lundberg approximation can be obtained. By way of example we
consider the independent jump intensity model (Björk and Grandell, 1988)
and the risk model in a Markovian environment (Asmussen, 1989).
KEY WORDS: Cox model; ruin probability; Lundberg inequality;
Cramér-Lundberg approximation; risk theory; martingale methods; change
of measure.
1991 Mathematical subject classification: primary 60H99; secondary
60G44, 60J99
by Hanspeter Schmidli
The fundamental work in risk theory mainly due to Harald Cramér and
Filip Lundberg is based on the theory of random walks. The main results, such
as the Laplace transform of the ruin probabilities, the Cramér-Lundberg
approximation or the Lundberg inequality, were obtained by complicated
computations based on inversion of Laplace-Stieltjes transforms. In 1973 Gerber
introduced martingale methods into risk theory (see also
Gerber (1979)). He was able to obtain the Lundberg inequality in a very
elegant way. Since then martingale techniques became a standard tool in
actuarial mathematics. We shall review some of the martingale approaches in
insurance risk theory.
We will start with the definition of the classical Cramér-Lundberg model
(Section 1). Then we review Gerber's work (Section 2) and
a nice work by Delbaen and Haezendonck (Section 3). A useful tool in
order to find martingales for risk processes is the theory of the so-called
piecewise deterministic Markov processes introduced by Davis (1984). We
give an intuitive definition of
these processes (Section 4) and two examples (Sections 5
and 6). And finally, in Sections 7 - 10, we review
some martingale techniques not based on Markov process theory.
by Hanspeter Schmidli
For a Cox risk model with a piecewise constant intensity some random variables
with an exponential tail are constructed and an estimation procedure for the
Lundberg exponent (adjustment coefficient) is proposed. It is shown that in the
case of a Markov modulated risk model the estimator is strongly consistent.
KEY WORDS: Cox model; Lundberg exponent; adjustment coefficient; Hill's
estimator; tail probabilities; ruin probability.
1991 Mathematical subject classification: primary 62M05; secondary
62G20, 60J25
by Hanspeter Schmidli
In applied probability one is often interested in the asymptotic behaviour of a
certain quantity. If a regenerative phenomena can be imbedded then one has the
problem that the event of interest may have occured but cannot be observed at
the renewal points. In this paper an extension to the renewal theorem is proved
which shows that the quantity of interest converges. As an illustration an open
problem in risk theory is solved.
KEY WORDS: Renewal theorem, limit theorems, large deviations, risk
theory, ruin probabilities.
1991 Mathematical subject classification: primary 60K05; secondary
60F10, 62P05
by Hanspeter Schmidli
Consider a classical compound Poisson model. The safety loading can be
positive, negative or zero. Explicit expressions for the
distributions of the surplus prior and at ruin are given in terms of the ruin
probability. Moreover, the asymptotic behaviour of these distributions as the
initial capital tends to infinity are obtained. In particular, for positive
safety loading the Cramér case, the case of subexponential distributions
and some intermediate cases are discussed.
KEY WORDS: ruin, asymptotic distribution, change of measure, Laplace
transform, subexponential distribution, Cramér condition, generalized
Pareto distribution, maximum domain of attraction, Gumbel distribution.
1991 Mathematical subject classification: primary 60J30; secondary
90A46, 60K05
by Hanspeter Schmidli
We investigate compound distributions, for example compound mixed Poisson
distributions in the case where the summands, the mixing distribution or the
number of summands are subexponential. It is shown that such a distribution is
subexponential. As an illustration the tail of the maximum of a certain
stochastic process is obtained.
KEY WORDS: subexponential distribution, compound distribution, mixed
Poisson distribution, extreme value theory, integrated tail distribution.
1991 Mathematical subject classification: primary 60G70; secondary
60E99, 60K25
by Hanspeter Schmidli
To a risk model an independent perturbation process is added. If the
perturbation process is Brownian motion, Lundberg inequalities and
Cramér-Lundberg approximations can be proved. Also the asymptotic
behaviour of the ruin probability in the case of heavy claims can be
obtained. If, the perturbation is a Lévy process, ladder epochs and
ladder heights can be defined. In the stationary case, the distribution of the
ladder height are obtained.
KEY WORDS: Perturbed risk model, ruin probability, Lévy process,
Brownian motion, Lévy motion, compound Poisson process,
asymptotics.
1991 Mathematical subject classification: primary 60K30; secondary
60G44, 60J30, 60G10
by Hanspeter Schmidli
We consider a risk model described by an ergodic stationary marked point
process. The model is perturbed by a Lévy process with no downward
jumps. The (modified) ladder height is defined as the first epoch where an
event of the marked point process leads to a new maximum. Properties of the
process until the first ladder height are studied and results of Dufresne and
Gerber (1991), Furrer (1998), Asmussen and Schmidt (1995) and Asmussen, Frey,
Rolski and Schmidt (1995) are generalized.
KEY WORDS: Perturbed risk model, risk theory, ruin probability, compound
Poisson model, Lévy process, ladder heights, Campbell's formula, marked
point process, palm distribution, Markov modulated risk model.
1991 Mathematical subject classification: primary 60G10; secondary
60E07, 60G17
by Hanspeter Schmidli
We consider dynamic proportional reinsurance strategies and derive the optimal
strategies in a diffusion setup and a classical risk model. Optimal is meant
in the sense of minimizing the ruin probability. Two basic examples are
discussed.
KEY WORDS: optimal control, stochastic control, ruin probability,
diffusion, Hamilton-Jacoby-Bellman equation, proportional reinsurance,
subexponential distribution.
1991 Mathematical subject classification: primary 93E20; secondary
90A46, 60G99
by Hanspeter Schmidli
In this survey article we consider different aspects of ruin theory. We start
considering the surplus prior and at ruin in a classical risk model, without
the assumption of positive safety loading. Next we review the application of
Markov process methods in ruin theory, including diffusion
approximations. Further types of risk models are risk models perturbed by
Lévy processes and Cox risk processes. For the latter we also consider
the question of how to estimate the adjustment coefficient. Moreover, we
discuss the question of when a compound sum is subexponential. Finally,
optimal reinsurance policies are constructed.
KEY WORDS: adjustment coefficient, borrowing, compound sum, Cox
process, diffusion approximation, Hamilton-Jacoby-Belmann equation, interest,
ladder height, Markov processes, martingales, optimal policy, perturbed risk
model, piecewise constant intensity, renewal theorem, ruin probability,
severity of ruin, subexponentiality, surplus at ruin.
1991 Mathematical subject classification: primary 91B30; secondary
60J25
by Hanspeter Schmidli
We consider a classical risk model and allow investment into a risky asset
modelled as a Black-Scholes model as well as (proportional) reinsurance. Via
the Hamilton-Jacobi-Bellman approach we find a candidate for the optimal
strategy and develop a numerical procedure to solve the HJB equation. We prove a
verification theorem in order to show that any increasing
solution to the HJB equation is bounded and solves the optimisation
problem. We prove that an increasing solution to the HJB equation
exists. Finally two numerical examples are discussed.
KEY WORDS: optimal control, stochastic control, ruin probability,
Hamilton-Jacobi-Bellman equation, Black-Scholes model, reinsurance
1991 Mathematical subject classification: primary 93E20; secondary
60G99, 91B30
by Hanspeter Schmidli
A risk process that can be Markovised is conditioned on ruin. We prove that
the process remains a Markov process. If the risk process is a PDMP, it is
shown that the conditioned process remains a PDMP. For many examples the
asymptotics of the parameters in both the light-tail case and the heavy-tailed
case are discussed.
KEY WORDS: Markov process, generator, absorbing state, ruin, diffusion
process, jump process, weak convergence, piecewise deterministic Markov
process (PDMP), change of measure, Cramer condition, subexponential
distribution
1991 Mathematical subject classification: primary 60J25; secondary
60J75, 60J35
by Hanspeter Schmidli
In a classical risk process reinsurance and investment can be chosen at any
time. We find the Lundberg exponent and the Cramér-Lundberg approximation
for the ruin probability under the optimal strategy in the case where no
exponential moments for the claim size distribution exist. We also show that
the optimal strategies converge.
KEY WORDS: ruin probability, optimal control, Cramér-Lundberg
approximation, adjustment coefficient, heavy tails, subexponential
distributions, geometric Brownian motion
1991 Mathematical subject classification: primary 60F10; secondary
60G35, 65K10
by Hanspeter Schmidli
We consider a classical risk model with the possibility of
reinsurance. Moreover, in one of the models also investment into a risky asset
is possible. The insurer follows the optimal strategy. In this paper we find
the Cramér-Lundberg approximation in the small claim case and prove
that the optimal strategy converges to the asymptotically optimal strategy as the
capital increases to infinity.
KEY WORDS: ruin probability, optimal control, Cramér-Lundberg
approximation, adjustment coefficient, light tails, martingale methods, change
of measure, geometric Brownian motion, Hamilton-Jacobi-Bellman equation
1991 Mathematical subject classification: primary 60F10; secondary
60G35, 65K10
by Hanspeter Schmidli
A model for the PCS index is introduced and it is shown how to price a PCS
option. It is discussed how to approximate option prices.
KEY WORDS: PCS option, inhomogeneous Poisson process, change of measure,
geometric Brownian motion, approximations, Girsanov's theorem
2000 Mathematical subject classification: primary 91B24; secondary
91B30, 91B28
by Hanspeter Schmidli
We consider a classical risk model with the possibility of investment.
We find the asymptotics of the ruin probability under the optimal investment
strategy in the case where the claim sizes are subexponentially
distributed. As a side result we obtain the rate at which A(x) tends to
infinity.
KEY WORDS: ruin probability, subexponential claims, regular variation,
optimal control, geometric Brownian motion, Hamilton-Jacobi-Bellman equation
2000 Mathematical subject classification: primary 60F10; secondary
60G35, 60J60
by Hanspeter Schmidli
In this article we review the ideas and techniques behind diffusion
approximations.
KEY WORDS: functional central limit theorems, Gaussian process, classical
risk model, ruin probability, interest, Markov process
by Hanspeter Schmidli
In this article we give the basic definition of a filtration.
KEY WORDS: σ-algebra,
stochastic process, stopping time, first entrance time
by Hanspeter Schmidli
An important concept in probability are martingales, stochastic
processes that can be considered as fair games. Many results on stochastic
processes can be proved via the optional stopping theorem and the
martingale convergence theorem. We define in this article the martingale,
state the important results, and illustrate applications of the theory by
way of examples.
KEY WORDS: martingale, stopping theorem, convergence theorem, Brownian
motion, Lévy process, Poisson process, stochastic integral, bounded
variation
by Hanspeter Schmidli
In this article we discuss the ideas behind modelling actuarial surplus
processes in continuous time. Several models that have been treated
successfully in the literature are reviewed.
KEY WORDS: risk process, classical model, Sparre-Anderson model, Markov
modulated model, Cox model, Marked point process, perturbed model, Lévy
process
by Hanspeter Schmidli
Recently, there was increased interest in risk processes where the insurer can
control the surplus process. One can show that under quite mild conditions
there exists an optimal strategy such that the ruin probability becomes
minimal. We focus in this talk on the case with heavy claim size distributions
and show the asymptotic behaviour of the optimal ruin probabilities as well as
the asymptotic behaviour of the optimal strategies.
KEY WORDS: ruin probability, subexponential claims, regular variation,
optimal control, geometric Brownian motion, Hamilton-Jacobi-Bellman equation
2000 Mathematical subject classification: primary 60F10; secondary
60G35, 60J60
by Hanspeter Schmidli
In this paper we review two optimisation problems. The first problem is to
minimise the ruin probability by proportional reinsurance. The results are
complemented by some asymptotic considerations of the minimal ruin probability
and the optimal strategy. The second problem is to maximise the expected
discounted dividend payout. Both problems are solved via the
Hamilton-Jacobi-Bellman approach.
KEY WORDS: optimal control; stochastic control; ruin probability;
Hamilton-Jacobi-Bellman equation; proportional reinsurance; dividends; change
of measure; subexponential distribution; small claims; Cramér-Lundberg
approximation
2000 Mathematical subject classification: primary 93E20; secondary
90B30, 60G99
by Hanspeter Schmidli
We consider some (simple) optimization problems in insurance. In particular,
we want to minimize the ruin probability by dynamic decisions, maximize the
value of future dividends, and keep the contribution rate and the size of a
pension fund close to some predefined values. The method is stochastic control
theory. We illustrate how a Hamilton-Jacobi-Bellman equation can be obtained. We
also show how to solve the equation and how the optimal strategies can be
obtained.
KEY WORDS: Hamilton-Jacobi-Bellman equation; verification theorem;
classical Cramér-Lundberg model; minimal ruin
probability; reinsurance; optimal new business; optimal dividend strategy;
pension fund; singular control problem
by Hanspeter Schmidli
We consider a classical risk model with the possibility of excess of loss
reinsurance. The insurer follows the optimal strategy. In this paper we find
the Cramér-Lundberg approximation. We prove that the optimal strategy
converges to the asymptotically optimal strategy as the capital increases to
infinity. This extends the results of Vogt (2004) and Schmidli (2002).
KEY WORDS: Ruin probability, heavy tail, excess of loss reinsurance,
optimal stochastic control, Cramér-Lundberg approximation, adjustment
coefficient, Hamilton-Jacobi-Bellman equation.
2000 Mathematical subject classification: primary 60F10; secondary
60G35, 65K10
by Hanspeter Schmidli
We consider several models for the surplus of an insurance company mainly
under some light-tail assumptions. We are interested in the expected
discounted penalty at ruin. By a change of measure we remove the discounting,
which simplifies the expression. This leads to (defective) renewal equations
as they had been found by different methods in the literature. If we use the
change of measure such that ruin becomes certain, the renewal equations
simplify to ordinary renewal equations. This helps to discuss the asymptotics
as the initial capital goes to infinity. For phase-type claim sizes, explicit
formulae can be derived.
KEY WORDS: Expected discounted penalty function; change of measure;
Laplace transform; Sparre--Andersen risk model; Markov-modulated risk model;
Björk-Grandell risk model; perturbed risk model; lump sum premia
2000 Mathematical subject classification: primary 91B30; secondary
60G44, 60J75, 60F10
by Hanspeter Schmidli
A risk process that can be Markovised is conditioned on ruin. We prove that
the process remains a Markov process. If the risk process is a PDMP, it is
shown that the conditioned process remains a PDMP. For many examples the
asymptotics of the parameters in both the light-tailed case and the heavy-tailed
case are discussed.
KEY WORDS: Markov process; generator; absorbing state; ruin; diffusion
process; jump process; weak convergence; piecewise deterministic Markov
process (PDMP); change of measure; Cramér condition; subexponential
distribution
2000 Mathematical subject classification: primary 60J25; secondary
60J75, 60J35
by Hanspeter Schmidli
We consider models for the accumulated claim sizes, as it can for instance be
used for modeling operational risk. A natural model are the
compound distributions. These models turn out to be difficult to
handle. Modeling the accumulated claim sizes directly leads to simple
approximations such as the normal approximation or the translated Gamma
approximation. Finally, numerical methods turn out to be successful for
discrete compound distributions that are in the Panjer class of
distributions.
KEY WORDS: compound distribution; normal approximation; translated
Gamma approximation; Panjer recursion; compound Poisson distribution; compound
negative binomial distribution; compound binomial distribution; compound mixed
Poisson distribution; Pareto distribution
by Hanspeter Schmidli
We consider a classical compound Poisson risk model. The Laplace
transform of the discounted penalty function is inverted, giving an
explicit formula. We apply this formula to obtain the value of the discounted
capital injections. Finally, we derive the asymptotic behaviour of the value
as the initial capital tends to infinity in the light and heavy tail case as
well as for some intermediate cases.
KEY WORDS: discounted penalty function; Craméer-Lundberg model;
discounted capital injections; light tails; subexponential distributions;
intermediate cases
2010 Mathematical subject classification: primary 91B30; secondary
60G44, 60F10
by Hanspeter Schmidli
We consider a compound Poisson risk model with interest. The Gerber-Shiu
discounted penalty function is modified with an additional penalty for
reaching a level above the initial capital. We show that the problem can be
split into two independent problems; an original Gerber-Shiu function and a
first passage problem. We also consider the case of negative
interest. Finally, we apply the results to a model considered by Embrechts and Schmidli (1994).
KEY WORDS: discounted penalty function; interest; classical risk
process; shot noise process; first passage time
2010 Mathematical subject classification: primary 91B30; secondary
60J75; 60G44
by Hanspeter Schmidli
Consider the classical risk model with dividends and capital injections. In
addition to the model considered by Kulenko and Schmidli
(2008), tax has to be paid for dividends. Capital injections yield tax
exemptions. We calculate the value function and derive the optimal dividend
strategy.
KEY WORDS: dividends; capital injections; tax; barrier strategy;
Hamilton--Jacobi--Bellman equation
2010 Mathematical subject classification: primary 91B30; secondary
60G44; 60K30; 60J25
by Hanspeter Schmidli
We consider a risk model driven by a spectrally negative Lévy process. From
the surplus dividends are paid and capital injections have to be made in order
to keep the surplus positive. In addition, tax has to be paid for dividends,
but injections lead to an exemption from tax. We generalise the results from
Schmidli (2016, 2017) and show that the optimal dividend strategy is a
two barrier strategy. The barrier depends on whether an immediate dividend
would be taxed or not. For a risk process perturbed by diffusion with
exponentially distributed claim sizes we show how the value function and the
barriers can be determined.
KEY WORDS: Lévy risk model; dividends; capital injections; tax;
barrier strategy; Hamilton-Jacobi-Bellman equation; perturbed risk model
2010 Mathematical subject classification: primary 91B30; secondary
60G44; 60K30
by Hanspeter Schmidli
We consider a diffusion approximation to a risk process with dividends and
capital injections. Tax has to be paid on dividends, but capital injections
lead to an exemption from tax. We solve the problem and show that the optimal
dividend strategy is a barrier strategy.
KEY WORDS: diffusion approximation; dividends; capital injections; tax;
barrier strategy
2010 Mathematical subject classification: primary 91B30; secondary
60G44; 60J60
by Hanspeter Schmidli
We consider a renewal risk model with general Erlang distributed inter arrival
times. We treat this as a Markov modulated risk model and assume, for
simplicity, that the states are observable. The insurer can pay dividends and
has to inject capital in order to keep the surplus positive. We determine the
optimal dividend/capital injection strategy.
KEY WORDS: renewal model; optimal dividend strategy; capital injections;
barrier strategy; Hamilton-Jacobi-Bellman equation
2020 Mathematical subject classification: primary 91B05; secondary
93E20; 60G44; 49K45
by Matthias Vierkötter & Hanspeter
Schmidli
We study the optimal dividend problem where the surplus process of an
insurance company is modelled by a diffusion process. The insurer is not
ruined when the surplus becomes negative, but penalty payments occur,
depending on the level of the surplus. The penalty payments shall avoid that
losses can rise above any number and can be seen as a preference measure or
costs for negative capital. As examples, exponential and linear penalty
payments are considered. It turns out that a barrier dividend strategy is
optimal.
KEY WORDS: Optimal dividends; penalty payments; barrier strategy;
diffusion process
2010 Mathematical subject classification: primary 91B30; secondary
60K10; 60G44
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