# Proportional Reinsurance for Models with Stochastic Cash Reserve Rate

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value function, HJB equation, The optimal policy

Pan,Zhaoxin

Supervisor and department: Tahir, Choulli Department of Mathematical and Statistical Sciences

Examining committee member and department: Abel Cadenillas Mathematical and Statistical Sciences Valentina Galvani Economics Alexander Melnikov Mathematical and Statistical Sciences Tahir, Choulli Mathematical and Statistical Sciences

Department: Department of Mathematical and Statistical Sciences

Specialization: Mathematical Finance

Date accepted: 2017-01-10T13:37:35Z

Graduation date: 2017-06:Spring 2017

Degree: Master of Science

Degree level: Master's

Abstract: This thesis investigates a problem of risk control for a financial corporation. Precisely, the thesis considers the case of proportional reinsurance for an insurance company. The objective is to find the optimal policy, that consists of risk control, which maximizes the total expected discounted value of cash reserve up to the bankruptcy time.The models for the cash reserve process, considered in this thesis, have stochastic drifts per unit time that we call stochastic cash reserve rate hereafter and constant volatility. These models extend the literature on proportional reinsurance, to the case of stochastic cash reserve rate that is either fully or partially observed. Precisely, I address three principal models. The first model deals with the case when the cash reserve rate is time dependent but deterministic. The second model assumes that the cash reserve rate process has an observable noise, while the third model assumes that the cash reserve rate is stochastic and is not observable.Thanks to the Bellman-s principle, for each of these three models, I derive the Hamilton-Jacobi-Bellman equation that corresponds to the stochastic control problem. Then I solve these equations as explicitly as possible. Afterwards, I describe the optimal policy for each model in terms of the obtained optimal value function, and I state the verification theorem. Finally, I consider the case where the insurance company pays liability at a constant rate per unit time.

Language: English

DOI: doi:10.7939-R34747416

Rights: This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.

Autor: ** Pan,Zhaoxin **

Fuente: https://era.library.ualberta.ca/

## Introducción

Proportional Reinsurance for Models with Stochastic
Cash Reserve Rate
by
Zhaoxin Pan
A thesis submitted in partial fulﬁllment of the thesis requirement for the degree of
Master of Science
in
Mathematical Finance
Department of Mathematical and Statistical Science
University of Alberta
c Zhaoxin Pan 2017
Abstract
This thesis investigates a problem of risk control for a ﬁnancial corporation.

Precisely,
the thesis considers the case of proportional reinsurance for an insurance company.

The
objective is to ﬁnd the optimal policy, that consists of risk control, which maximizes the
total expected discounted value of cash reserve up to the bankruptcy time.
The models for the cash reserve process, considered in this thesis, have stochastic
drifts per unit time (that we call stochastic cash reserve rate hereafter) and constant
volatility.

These models extend the literature on proportional reinsurance, to the case of
stochastic cash reserve rate that is either fully or partially observed.

Precisely, I address
three principal models.

The ﬁrst model deals with the case when the cash reserve rate is
time dependent but deterministic.

The second model assumes that the cash reserve rate
process has an observable noise, while the third model assumes that the cash reserve rate
is stochastic and is not observable.
Thanks to the Bellman’s principle, for each of these three models, I derive the HamiltonJacobi-Bellman equation that corresponds to the stochastic control problem.

Then I solve
these equations as explicitly as possible.

Afterwards, I describe the optimal policy for each
model in terms of the obtained optimal value function, and I state the veriﬁcation theorem.
Finally, I consider the case where the insurance company pays liability at a constant rate
per unit time.
ii
Acknowledgements
I feel much indebted to many people who have instructed and favored me in the course
of writing this thesis.

First of all, I would like to express my deepest gratitude t...