Modeling Optimal Investment and Reinsurance in Ambiguity Markets explores the complexities of investment strategies under market uncertainty. The research employs stochastic control theory to derive a Hamilton-Jacobi-Bellman equation, focusing on maximizing utility for insurers. It provides numerical solutions using finite difference methods, offering insights into optimal investment proportions and consumption strategies. This work is essential for finance professionals and researchers interested in risk management and asset allocation in ambiguous environments.

Key Points

  • Explores optimal investment and reinsurance strategies in uncertain markets
  • Utilizes stochastic control theory to derive Hamilton-Jacobi-Bellman equations
  • Provides numerical solutions through finite difference methods
  • Analyzes the impact of market parameters on investment strategies
RITHIK SHARMA
16 pages
Language:English
Type:Research Paper
Marketing
RITHIK SHARMA
16 pages
Language:English
Type:Research Paper
Marketing
399
/ 16
https://doi.org/10.62836/emi.v5i1.0005
Modeling and Numerical Solution of Optimal Investment
and Reinsurance Problems in Ambiguity Markets
Xuezhen Liang
1
, Xinyue Zhang
2,
*
, Dongping Wei
1,
*
and Ruihong Ma
3
1
Institute of Applied Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, China
2
Human Resources Department, Guangdong University of Petrochemical Technology,
Maoming 525099, China
3
Guangming School Affiliated to Shenzhen University, Shenzhen 518107, China
Abstract: In a framework of market incompleteness induced by ambiguity, this paper employs stochastic
processes and stochastic analysis to formulate the decision-making problem concerning investment,
consumption, and proportional reinsurance in an ambiguous market as a one-dimensional stochastic optimal
control problem over a finite time horizon. Specifically, the insurer aims to maximize the utility of terminal
wealth through dynamic optimal strategies, which is inherently a forward – backward stochastic differential
equation system. The ambiguity in the model is captured by the Chen – Epstein multiple-priors framework,
leading to a fully nonlinear Hamilton – Jacobi – Bellman equation that is generally analytically intractable. To
address this, an implicit finite difference scheme is designed to numerically solve the value function as well as
the optimal investment proportion, consumption strategy, and retention ratio. Furthermore, a systematic analysis
is conducted to examine the quantitative impact of key market parameter variations on these optimal strategies.
The findings provide theoretical support and numerical decision-making references for insurance institutions in
asset allocation and risk management under complex and uncertain environments.
Keywords: reinsurance; ambiguity; stochastic differential utility; finite difference method; HJB equation
1. Introduction
The problem of investment, consumption and reinsurance is a special kind of portfolio problem, which is
paid attention by domestic and international scholars. Since the reinsurance is an effective approach for risk-
management, it is of practical importance to derive the optimal strategies.
In this paper, we focus on an investment, consumption and reinsurance problem in ambiguity markets. Our
objective is to explore the optimal strategies to maximize the utility function of consumption and terminal wealth
for the insurer. A significant amount of portfolio problems have been studied under different market assumptions.
The foundational framework for continuous-time portfolio selection was established by [1], whose application of
stochastic control theory yielded landmark analytical solutions. After the work of Merton, different generalizations
of this classical model have been studied by different directions and different techniques. For instance, refs. [2,3]
studied the optimal investment problem of minimizing the ruin probability under different conditions. Besides,
consideration of the reinsurance is another natural extension to the classic investment and consumption framework [4].
Article
EMI
Economics & Management Information
https://ojs.sgsci.org/journals/emi
Received: 6 January 2026; Accepted: 2 February 2026.
* Corresponding: Xinyue Zhang (370141996@qq.com); Dongping Wei (wdp@szpu.edu.cn)
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Xuezhen L. et al. Econ. & Mgmt. Info. 2026, 5(1)
Schmidli considered dynamic proportional reinsurance strategy and studied the minimum ruin probability under the
classical risk model and the diffusion risk model respectively [5]. On the other hand, some scholars also consider
this kind of problems based on the maximization of utility. For example, Cao and Wan applied the stochastic technique
to study the investment and proportional reinsurance problem based on maximizing the utility and get the explicit
solution [6].
The above results were almost achieved under the assumption of complete markets. In reality, there are
many uncertainties and diversities in financial markets, which make the investment theory become increasingly
complicated. However, in order to make the results more realistic, it is significant to explore the investment
problem in incomplete markets.
Actually, the utility maximization problem in ambiguity markets has been attracted considerable research
attention recently. Chen and Epstein proposed the multiple-priors utility model to discuss the ambiguity
problem [7]. It is assumed that the investor does not know the probability measure of the market, rank the
uncertain prospects according to a multiple-priors model, and chose a worst case from a family continuous
probability measures. Based on the robust utility theory, there are many specific problems are solved. For a
consumption-investment problem, Schied focused on maximizing the utility of both terminal wealth and
intertemporal consumption by duality theory under model uncertainty and obtain it’ s explicit formulas for the
optimal strategy and value function [8]. Similar problems can be referred to the paper of [9]. In addition, it is
also important to study the investment and reinsurance problem in ambiguity markets. Zhang and Siu
formulated an reinsurance and investment problem as two-player, zero-sum, stochastic differential games
between the insurer and the market under model uncertainty [10]. They derived the closed-form solution to
maximize the expected utility or minimize the discounted penalty of ruin. Lin et al. also used the game-theory
approach to discuss an optimal portfolio selection problem for the insurer in a jump-diffusion model under
model uncertainty [11]. They only considered an exponential utility of terminal surplus, and obtained the closed-
form solutions in both the jump-diffusion risk process and its diffusion approximation.
In our paper, we study an investment, consumption and reinsurance problem by stochastic control theory
and dynamic programming principle in ambiguity market. However, different from the previous literatures, the
utility function of the insurer is formulated as an additive of consumption and terminal wealth, which is more
sophisticated and difficult to derive it’s explicit solution in ambiguity markets.
Due to the inherent complexity of the fully nonlinear HJB equations arising from model ambiguity,
analytical closed-form solutions are seldom attainable, necessitating the implementation of robust numerical
algorithms. Boulbrachene applied finite element method to discuss the numerical solution for Hamilton-Jacobi-
Bellman equations [12]. Kushner and Dupuis used the finite difference method for stochastic control problems
in continuous time [13]. In fact, many practices show that the finite difference method is a better one, thus, we
apply the finite difference method in this paper.
The significant innovation distinguishing our paper from the others is that we apply a multiple-priors model
to explore the investment, consumption and reinsurance problem in ambiguity markets. As for the utility
function of the insurer, we assume that the insurer concerned with not only the terminal wealth but also the
consumption. The general idea of this paper is that we formulate the problem as an optimal stochastic control
problem of forward-backward stochastic differential equation (FBSDE) via the stochastic control theory, and
apply the dynamic programming principle to obtain the HJB equation for the value function, which is a fully
nonlinear second-order partial differential equation.
It is difficult to obtain its analytical solution due to its complexity. However, we succeed in obtaining the
numerical solution by the finite difference method and providing economic explanations to analyse the effects of
market parameters on the value function and the optimal strategies. In fact, a numerical solution is also practical
in financial markets. In this paper, we overcome two difficulties. Firstly, we formulate a complicated problem as
a FBSDE by stochastic control theory. Secondly, our numerical results converge to the partial differential
equation. It is a challenge to compute a fully nonlinear second-order partial differential equation, but we do it.
The rest of this paper is organized as follows. In Section 2, we formulate the problem. In Section 3, we
apply the finite difference method to obtain the numerical results for the value function and the optimal
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Xuezhen L. et al. Econ. & Mgmt. Info. 2026, 5(1)
strategies. In Section 4, we provide analysis for the effects of the market parameters on the value function and
the optimal strategies. In Section 5, we investigate the effects form different level of ambiguity. Finally we
conclude this paper in Section 6.
2. Formulation of the Model
In general, consider a finite investment duration [0, T] defined on a complete filtered probability space
(ΩF
t
{F
t
}
0 £ t £ T
P)
, where the filtration
F
t
encapsulates all market information available up to time t,
P
is
described as the risk-neutral probability measure. In reality, the financial market is incomplete, the insurer is
uncertain about the market, and rank the uncertain prospects according to a multiple-priors model, which was
initially introduced by [7]. The probability space set
Θ
is formulated as follows
Θ
{
Q:
dQ
dP
= exp (-
1
2
∫
0
T
 |ξ
s
|
2
ds -
∫
0
T
 ξ
s
dW
s
)
}
for ξ ∈ Ξ, where Ξ is a set of
F
t
adaptive stochastic process with a values on a range
C
=[-
-
k k
ˉ
]
, and
-
-
k k
ˉ
are
non-negative constants, besides, W is a standard Brownian motion on the probability space
(ΩF
t
{F
t
}
0 £ t £ T
P)
.
2.1. The Surplus Process and Proportional Reinsurance
Assume that R
t
stands for the surplus process. From the classical risk model, the surplus process could be
governed by the following stochastic differential equations.
d R
t
= adt - dQ
t
(1)
where t ≥ 0, a is a constant insurance premium rate, and Q
t
=
(
∑
i = 1
N
t
 Y
i
)
. It is worth mentioning that N
t
follows a
Poisson process with a constant intensity λ, known as the number of claims in the time interval [0, t]. The claim
sizes {Y
i
, i ≥ 1} are independent and identically distributed non-negative random variables with E
P
(
Y
i
)
=
μ
1
E
P
(
Y
2
i
)
= μ
2
. N
t
and Y
i
are adapted to a filtered probability space
(ΩF
t
{F
t
}
0 £ t £ T
P)
. In addition, N
t
, Y
i
and W
t
are independent with each other. According to the premium principle, a positive constant safety loading η of
insurer will be set to satisfy
a = (1 + η)λμ
1
.
From the view of [14], the dQ
t
can be approached by the diffusion model
dQ
t
= λμ
1
dt - λμ
2
dW
1
t
(2)
where
W
1
t
is a standard Brownian motion and independent with W. Without loss of generalization, we suppose
{F
t
}
0 £ t £ T
is a complete natural filtration generated by W and
W
1
t
. In order to avoid catastrophe risk, the insurer
will consider to purchase proportional reinsurance to transfer part of potential risk. Suppose that the self-
retention proportion of the insurer is
q
t
Î[01]
and
1 - q
t
is the reinsurance proportion. In order to purchase this
reinsurance contract, the insurer has to pay the premium at a rate of a
1
= (1 + θ)
(
1 - q
t
)
λμ
1
where
1 ³ θ > η ³ 0
is
the safety loading of the reinsurance business.
Referring to the paper of [5], the surplus process of the insurer after purchasing the proportional reinsurance is
dR
t
=
[
(1 + η)λμ
1
- (1 + θ)
(
1 - q
t
)
λμ
1
]
dt -
(
λμ
1
q
t
dt - λμ
2
q
t
dW
1
t
)
= λμ
1
(
θq
t
+ η - θ
)
dt + λμ
2
q
t
dW
1
t
.
(3)
Remark: There are three parts in surplus process dR
t
, including premium income adt, reinsurance premium
expenses a
1
dt, and the claim cost
(
λμ
1
q
t
dt - λμ
2
q
t
dW
1
t
)
.
2.2. The Financial Investment Market
Suppose that the financial investment market is composed of one risk-free asset and one risky asset which
are traded continuously over [0, T]. The price process B
t
of the risk-free
dB
t
= r(t)B
t
dt
(4)
where r(t) is assumed to be a deterministic and continuous function, and represents the interest rate. The price of
the risky asset is driven by a geometric Brownian motion
dS
t
= μ(t)S
t
dt + σ(t)S
t
dW
t
(5)
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FAQs

What methodologies are used in this research on investment and reinsurance?
The research employs stochastic control theory to formulate the investment and reinsurance problem as a Hamilton-Jacobi-Bellman (HJB) equation. This approach allows for the analysis of optimal strategies in ambiguous markets. The study also utilizes finite difference methods to numerically solve the HJB equation, providing practical insights into the decision-making processes of insurers.
What are the main findings regarding optimal investment strategies?
The findings indicate that optimal investment strategies are significantly influenced by market parameters such as interest rates and volatility. As wealth increases, the proportion of investment in risky assets tends to rise, while higher volatility generally leads to more conservative investment choices. The research highlights the importance of understanding these dynamics for effective risk management.
How does ambiguity affect investment decisions in this study?
Ambiguity plays a crucial role in shaping investment decisions, as it introduces uncertainty about market conditions. The study utilizes a multiple-priors model to capture this ambiguity, allowing insurers to evaluate different scenarios and make informed decisions. This framework helps in understanding how varying levels of ambiguity can impact optimal investment and reinsurance strategies.
What implications does this research have for insurance professionals?
The research provides valuable insights for insurance professionals by highlighting the importance of incorporating market ambiguity into investment strategies. It emphasizes the need for robust decision-making frameworks that account for uncertainty, thereby aiding in effective asset allocation and risk management. These findings can guide insurers in developing strategies that maximize utility while navigating complex market conditions.
What is the significance of the numerical solutions obtained in the study?
The numerical solutions derived from the finite difference method offer practical applications for insurers in real-world scenarios. By providing concrete strategies for investment and consumption, these solutions help professionals understand the quantitative impacts of market parameters. This significance lies in its ability to translate theoretical models into actionable insights for effective financial decision-making.

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