Optimal Investment under Dual Risk Model and Markov Modulated Financial Market ()
1. Introduction
The classical surplus process of an insurer is given by
(1)
where x > 0 is the initial surplus, c is the positive constant premium income rate, Nt is Poisson process with parameter
, which denotes the total number of claims up to time t. Denote the time of arrival of the ith claim by Ti and the size of the ith claim by Yi. More details about the surplus process can be found in Asmussen and Albrecher [1] , Rolski et al. [2] . As pointed out by Albrecher et al. [3] , its dual process may also be relevant for companies whose inherent business involves a constant flow of expenses while revenues arrive occasionally due to some contingent events (e.g. discoveries, sales). For instance, pharmaceutical or petroleum companies are prime examples of companies for which it is reasonable to model their surplus process as
(2)
The past decade has witnessed an increasing attention on the research of dual risk model. For example, see Albrecher et al. [3] for optimal dividend problem, see Cheung and Drekic [4] for dividend approximation and dual risk model with perturbation, see Yao et al. [5] for optimal dividend and equity issuance, see Zhu and Yang [6] for ruin probability under a Markov modulated dual risk model.
As we all know, investment is an important element in the financial agent for which can bring them potential profit. Thus, optimal investment for insurers has drawn great attentions in recent years, for example, see the works of Bai and Guo [7] , Browne [8] , Fleming and Hernández [9] , Hipp and Plum [10] , Li et al. [11] , Zhang and Siu et al. [12] . However, to our best knowledge, there are few papers concentrate on the optimal investment of agent with dual risk process. This is the main contributions of this paper.
Usually, the coefficients of the dynamics of the financial market are assumed to be constant. However, in reality, the returns from the risky assets might not be constants. So, it would be of practical relevance and importance to consider asset pricing models with non-constant coefficients, which can incorporate the feature of non- stationary returns. Among all kinds of stochastic coefficients models, Markov-modulated risky model has been recognized recently as an important feature to asset price models. There is much literature documenting such models in assets returns, such as French et al. [13] . Meanwhile, since Markov-modulated risky model contains several very important stochastic volatility models, thus can be seen as an explanation of many well-known empirical findings, such as the volatility smile, the volatility clustering, and the heavy-tailed nature of return distributions (c.f. Fleming and Hernández [9] , Pham [14] , Zariphopoulou [15] [16] , and references therein). In this paper, the optimal investment problem of an agent with dual risk process under the Markov modulated financial market is studied. By dynamic programming principle, we obtained the HJB equations satisfied by the value function and finished the corresponding verification theorem. A solid example is presented to illustrate how to solve the HJB equation when the claims are exponential distribution. This rest of this paper is organized as follows. In Section 2, the model and problem are introduced. The HJB equation associated with our control problem and the verification theorem for optimal control are investigated in Section 3. In Section 4, we focus on the exponential utility function and closed form expression for optimal investment is obtained. In Section 5, we listed the highlights of this paper and conclusions from the results.
2. Formulation of the Problem
Let
be a complete probability space which carries all random variables to appear in this paper. To proceed our discussion, we introduce the following variables and notations. Let
and
are two standard Brownian motions, which describe the perturbations of the insurer and the financial market and
is the augmented filtration generated by aforementioned stochastic process, i.e.
![](//html.scirp.org/file/8-1490303x12.png)
and satisfying the usual conditions. For simplicity,
,
,
and
are assumed to be mutually independent.
Assume that there are two kinds of asset available for investors, one risky asset and one risk free asset. The risky asset is assumed to be
(2.1)
where
,
are the stochastic investment return rate and volatility of the risky market respectively. The dynamic of the external factor is specified by the solution to the following stochastic differential equation (SDE for short)
(2.2)
where
, and
.
are correlated Brownian motions with the correlation coefficient
. Model (2.2) covers many Markov modulated risk models, such as the Heston model and a special CIR model. Our model also includes a risk-free asset governed by
(2.3)
where
is the interest rate function. We interpret the process Zt as the behavior of some economic factor that has an impact on the dynamics of the risky asset and the risk-free asset price. In this paper, we allow the company takes an investment strategy into account when making decisions. Then if Xt is the company’s wealth, and let Kt denote the amount invested into the risky asset at time t. The remained reserve
is invested into the risk-free asset, then the wealth process of the insurer can be written as following equation. To clarify the impact of investment policy, we adopt
as the wealth process of the insurer, then
(2.4)
where
is the initial surplus of the insurer and c the positive real constant premium rate. Moreover, if at time
the wealth of the insurer is x and the external factor is z. Then the wealth process satisfies
![]()
with the convention that
.
Definition 1 We say that the strategy
is admissible if it satisfies the following conditions
1) The strategies Kt has to be measurable and predictable with respect to the filtration
;
2) There is a constant CK which may depend on the strategies K such that
![]()
We denote the set of admissible strategies as
.
Suppose that the company is interested in maximizing the expected utility of wealth at time T. Without loss of generality, we can define the utility function
to be a twice continuously differentiable function, with
and
, then our goal is the following value function:
(2.5)
We say that an admissible combined strategies
is optimal if
![]()
Hypothesis 1 1) The functions
,
and
are such that there is a strong solution for SDE (2.1), (2.2) for example the functions fulfil Lipschitz and linear growth conditions.
2) The function
is continuous, positive and
for all
.
3. Properties of Value Function and the Verification Theorem
In this section we embed the problem of maximizing the expected utility from terminal wealth on a finite horizon
in the framework of stochastic control theory by dynamic programming method. Then the HJB equation associated with the control problem (2.5) is given by
(3.1)
with terminal condition
, where
![]()
The following verification theorem shows that under some proper conditions, a solution to previous HJB equation provides us the optimal investment policy.
Theorem 2 (The Verification Theorem) Suppose that there is a smooth solution
to the HJB Equation (3.1) with terminal condition
. Assume also that for each ![]()
(3.2)
(3.3)
(3.4)
Then for each ![]()
![]()
Suppose further that there exist two bounded measurable functions
,
such that
![]()
then
defines a pair of optimal strategy and
![]()
Proof. Let
, by Itô’s Lemma, it follows that for ![]()
(3.5)
where
is the Poisson random measure on
defined by
![]()
Compensate (3.5) by
![]()
we have
(3.6)
Assumptions (3.3) and (3.4) mean that
![]()
are martingales. Assumption (2.8) implies that
![]()
is a martingale. Then, by taking the expectation on both sides of (2.11) yields that
(3.7)
Note that f is a smooth solution to HJB equation (2.6), we have
![]()
That is to say for any
, ![]()
(3.8)
Taking
in (2.13), it follows that
(3.9)
The proof of the second part of this theorem follows in a similar manner. If we plug
back into (2.12), for all
we obtain
(3.10)
By taking the supremum over all
in (2.15), we obtain the inequality
![]()
By considering (2.13) for all
, we deduce that
![]()
Letting
in the last equality, we have
![]()
Moreover, by recalling (2.14), it is easy to find that
![]()
This completes the proof. ![]()
Remark 1 Classical method of applying HJB equation for solving optimal control problems is pre-assume (or find) that there exist a smooth solution to the HJB equation, and then finish the argument by verification theorem. However, the HJB equations do not always admit classical solution, and thus the verification theorem invalid. In this case, viscosity solution will be introduced to cover the connections between the optimal control problem and the HJB equation. However, in next section, we exploit a closed representation of the solution to the HJB Equation (3.1) when the utility function is an exponential type. By this results, we further find the closed from optimal investment policy and the expressions of value function. As to very general utility function, it is difficult to find closed form solutions to HJB equation and we leave it as future research.
4. Existence of a Optimal Pair of Solutions under the Exponential Utility Function
In this section, we devote to the existence and uniqueness of the solution of the HJB Equation (3.1) when the preferences of the insurer are exponential, i.e., the utility function is governed by
(4.1)
In order to get a linear PDE, in the remainder of this paper we consider only the case where the correlation coefficient is equal to zero
. Besides Hypothesis 1, we make the following assumptions.
Hypothesis 2 1)
is constant;
2) g is uniformly Lipschitz and bounded;
3)
bounded with a bounded first derivative;
Considering the form of the utility function, We speculate the following function as a solution to the HJB Equation (3.1)
(4.2)
where
will be governed below by a solution to a Cauchy problem. From the definition of
, we obtain
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
Plugging these partial derivatives of f into the HJB Equation (3.1), we obtain
(4.8)
For simplicity of presentation let us introduce the following notation
![]()
It is trivial to see that the supremum is achieved at
![]()
Indeed, by a measurable selection theorem, we may find a pair of bounded progressively measurable processes
satisfied the supremum in (4.8). By substituting
into (4.8), we obtain the following Cauchy problem:
(4.9)
The following theorem asserts the existence and uniqueness of aforementioned Cauchy equation (4.9).
Theorem 3 Assume that
(4.10)
Then the Cauchy problem given by (4.9) has a unique solution, which satisfies the following conditions:
(4.11)
(4.12)
where C1 and C2 are constants.
Proof. The theorem will be proved if we can show that the Cauchy problem given by (4.9) satisfies the conditions of the Theorem A.1. So we just need to check them.
・ Since
is constant, naturally, it is Lipschitz continuous, Hölder continuous, and the operator
is uniformly elliptic.
・ Considering Hypothesis 2, we know immediately that
is bounded and uniformly Lipschitz continuous.
・ Now we show that
![]()
is bounded and uniformly Hölder continuous in compact subsets of
.
In fact, by Hypothesis 2, it is clear that the first term of
is bounded. The second term is bounded by
. Note that for
we have
![]()
Thus
is bounded. Next we prove that
is uniformly Hölder continuous in compact subsets of
. Denote
![]()
Noting that
is bounded with a bounded first derivative by
,
and Hypothesis 2, then it follows from Lemma A.1 that
is uniformly Hölder continuous with exponent
, i.e., for all ![]()
![]()
For the second term of
, combining the mean value theorem and the Hypothesis 2 and Definition 1, we have that for all
,
such that
![]()
Then
is uniformly Lipschitz in
. Therefore
is uniformly Hölder continuous in the compact set
. For the third term of
, first, a routine computation gives rise to the following derivatives
![]()
Then by the mean value theorem of bivariate functions, we know that there is
such that
![]()
where
and
mean, for instance,
,
. In the last line, we used
. So we obtain
![]()
By (4.10), we obtain that
is uniformly Lipschitz continuous in
, and then
is uniformly Hölder continuous in compact subsets of
.
Since the Cauchy problem (4.9) is homogeneous with a constant terminal condition, then the right-hand side of (4.9) satisfies the property of linear growth and continuous. Finally, the conditions of Theorem A.1, it is easy to find that the Cauchy problem (4.9) has a unique solution
which satisfies (4.11) and (4.12). The proof of the theorem is now complete. ![]()
The aim of the next theorem is to relate the value function V in the form (4.2) to the HJB Equation (3.1) in the form of the Cauchy problem (4.9).
Theorem 4 If (4.10) are satisfied, then the value function defined by (2.5) has the form:
![]()
where
is the unique solution of the Cauchy problem (4.8), In addition, if
![]()
then the investment strategy
is optimal, When
, we get
![]()
and
![]()
Proof. We have already verified that
![]()
is a smooth solution of the HJB Equation (3.1). To prove that
really copies the value function, we need to verify that Assumptions (3.2)-(3.4) of the Theorem 2 are satisfied by
.
Firstly, we consider the case in which
. Let
be a pair of admissible strategies, then by (4.11) and the fact that Y is independent of
and
, we have
![]()
In the last line, we used
for
and
. To get condition (3.2), it suffices to obtain an estimate of
![]()
We find that
![]()
In the last inequality, we used Hölder inequality (
, where
and
). By considering Hypothesis 2.2 and Theorem A.2, we know that
![]()
where C is a positive constant. Moreover, by the Minkovski inequality
(
, where
), one will find that
(4.13)
i.e.,
Then it is enough to estimate
Denote
![]()
One should note that
![]()
and
![]()
Recall that K is a pair of admissible strategy, by Hölder inequality, we have
![]()
Since
is a martingale, it follows that
![]()
This indicates that
, i.e., (3.2) holds for the case
. In order to prove conditions (3.3) and (3.4), by (4.11) and (4.12) we have
![]()
and
![]()
Evidently, (3.3) and (3.4) are easily seen to hold with
. For the case in which the interest rate
, let
By Itô’s formula it is easy to see that
satisfies the following SDE
(4.14)
By
for all
in the Hypothesis 1, we obtain that
(4.15)
This case is dealt with the same arguments by suitable modification to the first part of the proof. First, we get
![]()
![]()
and
![]()
To accomplish the proof, it is sufficient to prove that
(4.16)
Note that
![]()
and the fact that
from the first part of the proof, the proof is reduced to showing
In fact, by applying (4.15) and with similar arguments to the first part of the proof we have
![]()
Similarly, since K is a pair of admissible strategy and
is a martingale, we also have
![]()
This completes the proof. ![]()
5. Highlights and Summary
The main contributions of this paper include:
・ Both stochastic coefficients financial model and dual risk model are taken into account.
・ Rigorous proof of verification theorem for optimal policy is provided and closed form expressions for optimal policies and value function are derived.
・ A solid example is presented to illustrate how to solve the HJB equation.
As a result, we find that the optimal investment policy is a function of the state of the external Modulate Markov process. When there is no modulated process, the model considered in this paper is reduced to the optimal investment problem under the risky market with stationary coefficient and our results cover those existing results (see Bai and Guo [7] or Li et al. [11] ). One should note that when the coefficients are not sensitivity to the changes of the external Markov process, i.e. when the external Markov process changes, the coefficients of the risky market do not oscillate greatly, then our optimal investment policies seem to be very conservative because the optimal investment amount is near to a constant.
Acknowledgements
The authors are very grateful to anonymous referees’ detailed comments and suggestions, which makes this paper much better. Lin Xu would like to acknowledge the support of the National Natural Science Foundation of China (Grant No. 11201006). Zhu Dongjin would like to acknowledge the support of Major Projects of Colleges and Universities in Anhui Province Natural Science Foundation (KJ2012ZD01).
Appendix. Parabolic Partial Differential Equations
To illuminate the expression of our research problem, now we introduce and summarize some important results on parabolic PDEs, which play a key role in the proof of Theorem 4.1 (existence and uniqueness theorem), and some terminology and definitions are introduced. We believe that this work will be useful in the development of this paper.
Definition 5 Let ![]()
1) We say that E is uniformly elliptic, if there is
such that
![]()
for all
and all ![]()
2) A function f on
is called Hölder continuous in x with exponent
, uniformly with respect to t in compact subsets of
, if for each compact set
there is a constant
such that
![]()
3) f is said to be uniformly Hölder continuous in
in compact subsets of
if for each compact set
there is a constant C such that
![]()
Theorem A.1 (Friedman, 1975). We consider the following Cauchy problem:
(A.1)
where L is given by
![]()
If the Cauchy problem (A.1) satisfies the following conditions:
1) The coefficients of L are uniformly elliptic;
2) The functions
are bounded in
and uniformly Lipschitz continuous in
in compact subsets of
;
3) The functions
are Hölder continuous in x, uniformly with respect to
in
;
4) The function
is bounded in
and uniformly Hölder continuous in
in compact subsets of
;
5) The function
is continuous in
, uniformly Hölder continuous in x with respect to
) and;
6) The function
is continuous in
and
with
; then there is a unique solution u of the Cauchy problem (A.1) satisfying
and
Lemma A.1 Let f be a real positive bounded function with bounded derivative, then f is uniformly Hölder continuous with exponent
, i.e.,
![]()
Proof. By the mean value theorem and using that
is bounded,we have:
![]()
where K is a constant. By f is positive,we have
![]()
i.e.
![]()
The proof of this Lemma is now complete. ![]()
Theorem A.2 (Pham, 1998). Let
be a stochastic processes defined by the following SDE:
![]()
with a standard Brownian motion Bs. We assume that for some
, the coefficients satisfy:
![]()
(A.2)
for all
. Let T > 0 and
. Then, there is
such that for all
we have: