ABSTRACT

Based on the mean-variance portfolio selection under multi-period criterion, this paper focuses on the study of the uncertain time horizon and the regime-switching market including the bankruptcy state, where the conditional distribution of exit time is followed by the market state. When the market enters the bankruptcy state, investors are assumed to get back $\delta$ part of the wealth from the bankrupt company, where $\delta$ refers to the retrieval rate. By introducing the Lagrange multiplier $\lambda$, we create an innovative expression for the wealth process and the iterative representation of the value function to obtain the analytical expression of the optimal strategy and the corresponding efficient frontier. Besides, some special cases and numerical examples are presented to demonstrate the effects of state-dependent exit probability and bankruptcy state on the investment strategy.

Keywords:

Portfolio Selection, Bankruptcy State, State-Dependent Exit Probability, Dynamic Programming, Regime-Switching

1. Introduction

Since the precursory research of Markowitz [1] , the mean-variance (MV) criterion has become the most commonly used theoretical assumption and basis in portfolio selection and has been extended to many different kinds of applications. For example, Merton [2] studied the portfolio selection under a continuous time setting. By the embedding techniques, Li and Ng [3] and Zhou and Li [4] transformed the original portfolio selection problem into the auxiliary problem which is then solved by the dynamic programming approach.

However, previous works focused only on the static market state. In reality, the market state includes different states. For example, when the market is at the bullish state, the market response may be enthusiastic and the market may be profitable beyond the expectation of the public. When the state is bearish, investors will have pessimistic attitude toward the market and the return rates of assets may not be positive. Therefore, the regime-switching model, which is pioneered by Neftci [5] , has become one of the most important extensions of the MV portfolio selestion problem, and for details we refer the reader to the works of Zhou and Yin [6] , Yin and Zhou [7] , Çakmak and Özekici [8] , Xie [9] and Elliott et al. [10] .

Since the early regime-switching models only concern the problems with fixed investment period, recently the uncertain time horizon framework has become another important branch of the MV portfolio selection problem. For the works on uncertain time horizon, we refer the reader to Guo and Hu [11] , Yi et al. [12] , Wu and Li [13] and Yao et al. [14] . Besides, the study of Karatzas and Wang [15] , Martellini and UrosÌŇevicÌĄ [16] and Blanchet-Scalliet et al. [17] further regarded the asset price as an uncertain factor which affects the investment time horizon.

From the existing literature on portfolio selection in a regime-switching market, there are two important factors that are always neglected. Firstly, there is very few research on the extreme market state in which a company might proclaim the bankruptcy. In such an extreme state like the financial crisis in 2008, companies in the market are struggling with heavy liability that bankruptcy is the best option for themselves. Investors who invest in such companies can only retrieve part of the funds from the bankrupt company. Because of universality of the company bankruptcy in this market state, investors would rather invest the money to the risk-less asset than risky assets in order to avoid the investment risk. In reality, bankruptcy is not a rare situation. According to the statistical data by www.afsa.gov.au, the number of business related bankruptcy in Australia for just the June quarter 2018 is 1213, which suggests that it is fairly meaningful and realistic to consider the bankruptcy state.

The other factor is the exit probability which relies on the time and the market state. In this context, when the investment background becomes worse or the market state is currently being bearish, it is more likely that investors will quit the market to avoid the investment risk. On the other hand, the exit probability may differ as time goes by, for example, an investor generally quits the market after the investment reaches maturity, which means that the exit probability is higher at the end of the time horizon than any other time points. Based on the above consideration, we cannot predetermine the state-dependent time horizon since the market state is unknown prior to the corresponding time point, which suggests that this assumption is different and much more realistic than those in the existing literature. Because both the bankruptcy state and the state-dependent investment time horizon often occur in the real world, it is significant and necessary to combine and study these two factors with the MV portfolio selection problem in the regime-switching market. Wu and Zeng [18] studied the portfolio selection with bankruptcy state, but do not consider the factor of exit probability. Based on the multi-period mean-variance portfolio selection. Wu et al. [19] assumed that the uncertain time horizon depends on the market state, however, the exit probability is not considered. Because of this novel research angle in this paper, the wealth process becomes fairly different from previous formulations, and we assume that the conditional exiting probability depends on the current market state including the bankruptcy state. Therefore, in order to obtain the optimal strategy and the corresponding efficient frontier, innovative expressions of the wealth formulation and the iteration process are needed, which is the main difficulty and contribution in this paper. Besides, we prove that our work is a general model including several existing models in literature as special cases. By comparing the efficient frontiers in different situations, numerical examples illustrate that both of the two factors affect the investment strategy and the corresponding efficient frontier significantly.

The rest of this paper is organized as follows. Section 2 introduces the mean-variance formulation of the portfolio selection problem under the regime-switching framework, and the auxiliary problem is also given. The closed-form expression of the optimal strategy in the auxiliary problem is established in Section 3. Section 4 is devoted to detailed transformation and derivation of the optimal strategy and efficient frontier of the original problem. Sections 5 and 6 provide extreme cases and numerical analysis respectively. The conclusion of this paper is presented in Section 7.

2. Problem Formulation

This paper assumes that an investor enters the market at time 0 with initial wealth $w_0$. Denote by $S_t$ the market state at time $t\left(t=0,1,2,\cdots ,T-1\right)$ and assume that $S_t\left(S_t=1,2,\cdots ,L;t=0,1,2,\cdots ,T-1\right)$ is a time-homogeneous Markov chain with a transition matrix Q. Let L denote the bankruptcy state in which the investor will get back $\delta$ of the wealth from the bankrupt company, where $\delta$ refers to the retrieval rate which is assumed to be a random variable ranging between $\left[\mathrm{0,1}\right]$. Since in the bankruptcy market state risky assets will have similar economic performance, this paper assumes that there are only one risky asset and one risk-less asset in the market in order to underline the effects of the bankruptcy state on the optimal investment strategy. Let $R_t\left(S_t\right)$ and $r_t\left(S_t\right)$ signify the return rate of the risky asset and the risk-less asset respectively. With regard to the state-dependent exiting probability, suppose that the investor quits the market with the probability which depends on the current market state. Let $\tau$ represent the exit time and define

$P_t\left(i\right)=P\left(\tau =t|S_t=i\right),t=0,1,\cdots ,T,$ (1)

which satisfies ${\displaystyle {\sum }_{t=0}^T}P\left(\tau =t|S_0=i\right)=1,i=1,2,\cdots ,L$.

2.1. Notation

Throughout this paper, some notations need to be made, we summarize them as follows:

N1. It is unnecessary to define $R_t\left(L\right)$, because the company goes bankrupt and the risky asset has no investment value at state L. For $S_t=1,2,\cdots ,L-1$, define $R_t^e\left(S_t\right)=R_t\left(S_t\right)-r_t\left(S_t\right)$, which means the difference between the return rate of the risky asset and risk-less asset at time $t\left(t=0,1,\cdots ,T-1\right)$, and for $S_t=1,2,\cdots ,L-1$, denote by $r_t^e\left(S_t\right)=E\left(R_t^e\left(S_t\right)\right)$ the expectation of $R_t^e\left(S_t\right)$ at time $t\left(t=0,1,\cdots ,T-1\right)$ and assume it to be nonnegative. Besides, we define $e_t\left(S_t\right)={\left(r_t\left(S_t\right),R_t\left(S_t\right)\right)}^{\prime }\left(t=1,2,\cdots ,T-1\right)$.

N2. ${\pi }_t$ represents the investment amount into the risky asset at time $t\left(t=0,1,\cdots ,T-1\right)$ and $\pi$ means a set of strategies ${\pi }_t\left(t=0,1,\cdots ,T-1\right)$ within the investment time horizon.

N3. For any time-dependent matrix $A_t\left(L\times L\right)$ and any vector $a_t\left(L\times 1\right)$, define ${\displaystyle {\sum }_{t=1}^0}{a}_t=0$ where $0$ is the $L\times 1$ zero vector and ${\displaystyle {\prod }_{t=1}^0}{A}_t=1$ where $1$ is the $L\times L$ unit matrix.

N4. $Q{\left(i,j\right)}_{i,j=1,2,\cdots ,L}$ is the entry of transition matrix Q which means one-step transition probability from market state i to state j. $Q^k\left(i\mathrm{,}j\right)$ signifies k-step transition probability which is the entry of $Q^k$, the kth power of transition matrix Q. Besides, define $\hat{Q}\left(i,j\right)=Q{\left(i,j\right)}_{i,j=1,2,\cdots ,L-1}$ with $Q^0$ representing the identity matrix.

N5. For any matrix $A_{L\times L}$ and any vector $a_{L\times 1}$, $A\left(i\right)$ represents the ith row of A and $a\left(i\right)$ means the ith element of a, and we define $A_a{\left(i\mathrm{,}j\right)}_{L\times L}$ to be a matrix in which $A_a\left(i,j\right)=A\left(i,j\right)a\left(j\right)$, and $\bar{A}$ is a column vector which is equivalent to $A\cdot 1_{L\times 1}$ where $1_{L\times 1}$ is a column vector whose elements are all 1.

N6. $A_t^1\left(t=1,2,\cdots ,T-1\right)$ is a column vector whose ith element is

$A_t^1\left(i\right)=E\left(r_t^2\left(i\right)\right)E\left(R_t^e{\left(i\right)}^2\right)-{\left[E\left(r_t\left(i\right)R_t^e\left(i\right)\right)\right]}^2,i\ne L.$

N7. $A_t^2\left(t=1,2,\cdots ,T-1\right)$ is a column vector whose ith element is

$A_t^2\left(i\right)=E\left(r_t\left(i\right)\right)E\left(R_t^e{\left(i\right)}^2\right)-r_t^e\left(i\right)E\left(r_t\left(i\right)R_t^e\left(i\right)\right),i\ne L.$

N8. $H_t\left(t=1,2,\cdots ,T-1\right)$ is a column vector whose ith element is

$\begin{array}{l}{H}_t\left(i\right)={\displaystyle \underset{S_{t+1}=1}{\overset{L-1}{\sum }}}Q\left(i,S_{t+1}\right)\left[P_{t+1}\left(S_{t+1}\right)+\frac{H_{t+1}\left(S_{t+1}\right)A_{t+1}^1\left(S_{t+1}\right)}{E\left(R_{t+1}^e{\left(S_{t+1}\right)}^2\right)}\right]\\ +Q\left(i,L\right)E\left({\delta }^2\right)[P_{t+1}\left(L\right)+{\displaystyle \underset{S_{t+2}=1}{\overset{L}{\sum }}}Q\left(L,S_{t+2}\right)E\left(r{\left(L\right)}^2\right)\\ \times {\displaystyle \underset{k=t+2}{\overset{T}{\sum }}}{Q}^{k-\left(t+2\right)}{P}_k\left(S_k\right){\displaystyle \underset{j=1}{\overset{k-\left(t+2\right)}{\prod }}}E\left(r_{k-j}{\left(S_{k-j}\right)}^2\right)],i\ne L,t=0,1,\cdots ,T-2,\\ H_{T-1}\left(i\right)=Q\left(i,L\right)P_T\left(L\right)E\left({\delta }^2\right)+{\displaystyle \underset{j=1}{\overset{L-1}{\sum }}}Q\left(i,j\right)P_T\left(j\right),i\ne L.\end{array}$

N9. $G_t\left(t=1,2,\cdots ,T-1\right)$ is a column vector whose ith element is

$\begin{array}{l}{G}_t\left(i\right)={\displaystyle \underset{S_{t+1}=1}{\overset{L-1}{\sum }}}Q\left(i,S_{t+1}\right)\left[P_{t+1}\left(S_{t+1}\right)+\frac{H_{t+1}\left(S_{t+1}\right)A_{t+1}^2\left(S_{t+1}\right)}{E\left(R_{t+1}^e{\left(S_{t+1}\right)}^2\right)}\right]\\ +Q\left(i,L\right)E\left(\delta \right)[P_{t+1}\left(L\right)+{\displaystyle \underset{S_{t+2}=1}{\overset{L}{\sum }}}Q\left(L,S_{t+2}\right)E\left(r\left(L\right)\right)\\ \times {\displaystyle \underset{k=t+2}{\overset{T}{\sum }}}{Q}^{k-\left(t+2\right)}{P}_k\left(S_k\right){\displaystyle \underset{j=1}{\overset{k-\left(t+2\right)}{\prod }}}E\left(r_{k-j}\left(S_{k-j}\right)\right)],i\ne L,t=0,1,\cdots ,T-2,\\ G_{T-1}\left(i\right)=Q\left(i,L\right)P_T\left(L\right)E\left(\delta \right)+{\displaystyle \underset{j=1}{\overset{L-1}{\sum }}}Q\left(i,j\right)P_T\left(j\right),i\ne L.\end{array}$

N10. $K_t\left(t=1,2,\cdots ,T-1\right)$ is a column vector whose ith element is

$K_t\left(i\right)=\frac{{\left(G_t\left(i\right)r_t^e\left(i\right)\right)}^2}{H_t\left(i\right)E\left(R_t^e{\left(i\right)}^2\right)},i\ne L,t=0,1,\cdots ,T-1.$

2.2. Wealth Process and Optimization Problem

Now define the dynamics of the wealth with $\pi$ as follows:

$w_{t+1}^{\pi }=(\begin{array}{l}\left(r_t\left(S_t\right)w_t^{\pi }+R_t^e\left(S_t\right){\pi }_t\right)\left(1_{\left\{S_{t+1}\ne L\right\}}+\delta 1_{\left\{S_{t+1}=L\right\}}\right),S_t\ne L,\\ r_t\left(L\right)w_t^{\pi },S_t=L.\end{array}$ (2)

where $1_{\left\{S_{t+1}\ne L\right\}}$ represents the function whose value is 1 when $S_{t+1}\ne L$. Moreover, we generally suppose that the initial market state $S_0\ne L$. Otherwise, the investor needs to save all the wealth into his bank account from the beginning, which is a trivial question.

For each time point, the investor makes rational strategy to optimize terminal wealth. This paper studies this portfolio selection problem under mean-variance criterion in which the investment risk is measured by the terminal wealth variance. Considering the dependence of the exit probability on the market state and the fixed expectation of terminal wealth d, the objective function under a strategy $\pi$ is presented as follows:

$P\left(d\right)(\begin{array}{l}{\min }_{\pi }Va{r}_{i_0,w_0}\left(w_{T\wedge \tau }^{\pi }\right)\\ s.t.E_{i_0,w_0}\left(w_{T\wedge \tau }^{\pi }\right)=d\text{and}\left(2\right),\end{array}$

where $E_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }\right)$ denotes $E\left(w_{T\wedge \tau }|S_0=i,w_0\right)$ and $Va{r}_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }\right)$ stands for $Var\left(w_{T\wedge \tau }|S_0=i,w_0\right)$ respectively.

Let ${\pi }^{\mathrm{<mo>\; *\; </mo>}}$ be the optimal strategy. Then the portfolio is an efficient portfolio if there exists no strategy $\hat{\pi }$ such that $E_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{{\pi }^{\mathrm{<mo>\; *\; </mo>}}}\right)\le E_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{\hat{\pi }}\right)$ and

$Va{r}_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{{\pi }^{\mathrm{<mo>\; *\; </mo>}}}\right)\ge Va{r}_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{\hat{\pi }}\right)$ and at least one inequality holds true.

$\left(E_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{{\pi }^{\mathrm{<mo>\; *\; </mo>}}}\right)\mathrm{,}Va{r}_{i_0\mathrm{,}{w}_0}\left(w_{T\wedge \tau }^{{\pi }^{\mathrm{<mo>\; *\; </mo>}}}\right)\right)$ is called an efficient point and the set of all efficient points is called the efficient frontier.

According to $Var\left(w_{T\wedge \tau }\right)=E{\left(w_{T\wedge \tau }-E\left(w_{T\wedge \tau }\right)\right)}^2=E{\left(w_{T\wedge \tau }-d\right)}^2$, we have

$\bar{P}\left(d\right)(\begin{array}{l}{\min }_{\pi }{E}_{i_0,w_0}{\left(w_{T\wedge \tau }^{\pi }-d\right)}^2\\ s.t.E_{i_0,w_0}\left(w_{T\wedge \tau }^{\pi }\right)=d\text{and}\left(2\right),\end{array}$

by embedding a Lagrange multiplier $\lambda$ into problem $\bar{P}\left(d\right)$ (Luenberger [20] ), the original problem is transformed into the auxiliary problem $\overline{P1}\left(\lambda \mathrm{,}d\right)$ which has no constraint condition:

$\overline{P1}\left(\lambda ,d\right)(\begin{array}{l}{\min }_{\pi }{E}_{i_0,w_0}\left[{\left(w_{T\wedge \tau }^{\pi }-d\right)}^2+2\lambda \left(w_{T\wedge \tau }^{\pi }-d\right)\right]\\ s.t.\left(2\right).\end{array}$

The Lemma below gives the relationship between the problem $\bar{P}\left(d\right)$ and $\overline{P1}\left(\lambda ,d\right)$ :

Lemma 1 Given that the value function of the problem $\overline{P1}\left(\lambda ,d\right)$ at the beginning is $V_0\left(\lambda \mathrm{,}d\mathrm{<mo>\; ;\; </mo>}{i}_0\mathrm{,}{w}_0\right)$, and the corresponding optimal strategy is denoted by $\left\{{\pi }_t^{*}\left(\lambda ,i_t,w_t\right)|t=0,1,\cdots ,T-1\right\}$, then the value function of the problem $\bar{P}\left(d\right)$ at time 0 is ${\sup }_{\lambda }{V}_0\left(\lambda ,d;i_0,w_0\right)$, and its optimal strategy is $\left\{{\pi }_t^{*}\left({\lambda }^{*},i_t,w_t\right)|t=0,1,\cdots ,T-1\right\}$ where ${\lambda }^{\mathrm{<mo>\; *\; </mo>}}$ is the one which satisfies ${\sup }_{\lambda }{V}_0\left(\lambda ,d;i_0,w_0\right)$.

Note that ${\left[w_{T\wedge \tau }-d\right]}^2+2\lambda \left[w_{T\wedge \tau }-d\right]={\left[w_{T\wedge \tau }-\left(d-\lambda \right)\right]}^2-{\lambda }^2$, and because $\lambda$ has no effect on the choice of the optimal strategy, we neglect ${\lambda }^2$ to obtain the problem $\widehat{P1}\left(\lambda \mathrm{,}d\right)$ as follows:

$\begin{array}{l}\widehat{P1}\left(\lambda ,d\right)(\begin{array}{l}{\min }_{\pi }{E}_{i_0,w_0}\left[{\left(w_{T\wedge \tau }-\left(d-\lambda \right)\right)}^2\right]\\ s.t.\left(2\right),\end{array}\hfill \end{array}$

which is equivalent to problem $\overline{P1}\left(\lambda ,d\right)$.

2.3. Important Assumptions

565t = Throughout this paper, we make the following important assumptions:

A1 Assume that for any i, $i=1,2,\cdots ,L$, $P_T\left(i\right)>0$ always holds true. Otherwise, we do not need to consider the problem in Ttime horizon.

A2 The market state $S_t,t=0,1,\cdots ,T$ is independent of the return rates of the risky asset and the risk-less asset, and the return rate of the risky asset and the risk-less asset are also independent of each other, which can be expressed as follows:

$P_t\left(S_{t+1}\mathrm{,}{R}_t\left(S_t\right)\in B\right)=P_t\left(S_{t+1}\right)P_t\left(R_t\left(S_t\right)\in B\right)\mathrm{,}$

$P_t\left(R_t\left(S_t\right)\in B,r_t\left(S_t\right)\in B\right)=P_t\left(R_t\left(S_t\right)\in B\right)P_t\left(r_t\left(S_t\right)\right),$

for all $B\in B\left(ℝ\right),i=1,2,\cdots ,L$ and $t=0,1,\cdots ,T-1$, where $P_t$ is the probability based on information up to time t and $B\left(ℝ\right)$ is the Borel σ-algebra on $ℝ$.

A3 $P\left(\tau =t|S_0,\cdots ,S_t,\cdots ,S_T\right)=P\left(\tau =t|S_t\right)$, which means that the exit probability is only dependent on the current market state $S_t$, and the time before and after the current market state has no effect on the current exit probability.

A4 For $S_t=1,2,\cdots ,L-1$ and $t=0,1,\cdots ,T-1$, assume that

$E\left[e_t\left(S_t\right)e_t{\left(S_t\right)}^{\prime }\right]$ is positive definite.

A5 The investment strategy $\tau$ is self-finance, which means that there is no exogenous infusion or withdrawal of money during the investment time horizon.

A6 During the investment time horizon, short selling, borrowing and lending are not prohibited, and transaction costs are ignored.

3. Solution for Problem $P1\left(\lambda ,d\right)$

In order to derive the value functions and the optimal strategy of problem $\widehat{P1}\left(\lambda \mathrm{,}d\right)$, we need to further transform the objective function.

By using the law of total probability, we have the following formulation:

$\begin{array}{l}{E}_{i_0,w_0}\left[{\left(w_{T\wedge \tau }-\left(d-\lambda \right)\right)}^2\right]\\ =E_{i_0,w_0}\left\{E\left[{\left(w_{T\wedge \tau }-\left(d-\lambda \right)\right)}^2|S_0,S_1,\cdots ,S_T\right]\right\}\\ =E_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}P\left(\tau =t|S_0,S_1,\cdots ,S_T\right)E\left[{\left(w_t-\left(d-\lambda \right)\right)}^2|S_0,S_1,\cdots ,S_T\right]\right\}\\ =E_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}P\left(\tau =t|S_t\right)E\left[{\left(w_t-\left(d-\lambda \right)\right)}^2|S_0,S_1,\cdots ,S_T\right]\right\}\end{array}$

$\begin{array}{l}=E_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}{P}_t\left(S_t\right)E\left[{\left(w_t-\left(d-\lambda \right)\right)}^2|S_0,S_1,\cdots ,S_T\right]\right\}\\ =E_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}E\left[P_t\left(S_t\right){\left(w_t-\left(d-\lambda \right)\right)}^2|S_0,S_1,\cdots ,S_T\right]\right\}\\ =E_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}{P}_t\left(S_t\right){\left(w_t-\left(d-\lambda \right)\right)}^2\right\}.\end{array}$ (3)

Note that the law of total probability is being applied on the first line and the assumption A2 on the third line of Equation (3). Therefore, by using the above formulation, the problem $\widehat{P1}\left(\lambda \mathrm{,}d\right)$ can be rewritten as follows:

$P1\left(\lambda ,d\right)(\begin{array}{l}{\min }_{\pi }{E}_{i_0,w_0}\left\{{\displaystyle \underset{t=0}{\overset{T}{\sum }}}{P}_t\left(S_t\right){\left(w_t-\left(d-\lambda \right)\right)}^2\right\}\\ s.t.\left(2\right).\end{array}$

Now we consider the optimal strategy for the problem $P1\left(\lambda ,d\right)$ by using the dynamic programming approach. Define the value function as follows:

$\begin{array}{l}{V}_t\left(i,w_t\right)={\min }_{{\pi }_t,{\pi }_{t+1},\cdots ,{\pi }_{T-1}}{E}_{i,w_0}\left[{\displaystyle \underset{n=t}{\overset{T}{\sum }}}{P}_n\left(S_n\right){\left(w_n-\left(d-\lambda \right)\right)}^2\right],t=0,1,\cdots ,T-1,\\ V_T\left(i,w_T\right)=w_T-{\left(d-\lambda \right)}^2,i=1,2,\cdots ,L.\end{array}$ (4)

When $S_n\ne L$, we have the following Bellman’s equations:

$\begin{array}{l}{V}_t\left(i,w_t\right)=P_t\left(i\right){\left(w_t-\left(d-\lambda \right)\right)}^2+{\min }_{{\pi }_t}{E}_{i,w_t}\left[V_{t+1}\left(S_{t+1},w_{t+1}\right)\right]\\ =P_t\left(i\right){\left(w_t-\left(d-\lambda \right)\right)}^2+{\displaystyle \underset{j=1}{\overset{L}{\sum }}}Q\left(i,j\right)E[V_{t+1}(j,{(r_t\left(i\right)w_t}^{}\\ +{\pi }_t{R}_t^e\left(i\right))\left(1_{\left\{S_t\ne L\right\}}+\delta 1_{\left\{S_t=L\right\}}\right))],t=0,1,\cdots ,T-1.\end{array}$ (5)

When $S_t=L$, an investor will invest all the money into the bank account and has wealth of $r_t\left(L\right)w_t$ at time $t+1$. Therefore, the investor will possess the

terminal wealth of ${\displaystyle {\prod }_{k=t+1}^{T-1}}{r}_k\left(S_k\right)r_t\left(L\right)w_t$ that only depends on the return of the

risk-less asset after time t. Based on the state-dependent exit probability, the value function then can be shown to be as follows:

$\begin{array}{l}{V}_t\left(L,w_t\right)=P_t\left(L\right){\left(w_t-\left(d-\lambda \right)\right)}^2\\ +E_{L,w_t}\left[{\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{P}_k\left(S_k\right){\left[{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}{r}_{k-j}\left(S_{k-j}\right)r_t\left(L\right)w_t-\left(d-\lambda \right)\right]}^2\right],t=0,1,\cdots ,T-1.\end{array}$ (6)

Lemma 2 For $i=1,2,\cdots ,L-1$ and $t=0,1,\cdots ,T-1$, $A_t^1\left(i\right)>0,H_t\left(i\right)>0$ and $K_t\left(i\right)>0$.

Proof. From A4, we have

$E\left(e_t\left(S_t\right)e_t{\left(S_t\right)}^{\prime }\right)=\left(\begin{array}{c}E\left(r_t^2\left(S_t\right)\right),E\left(r_t\left(S_t\right)R_t\left(S_t\right)\right)\\ E\left(r_t\left(S_t\right)R_t\left(S_t\right)\right),E\left(R_t^2\left(S_t\right)\right)\end{array}\right)>0,$ (7)

from which, we get

$\left(\begin{array}{c}E\left(r_t^2\left(S_t\right)\right),E\left(r_t\left(S_t\right)R_t^e\left(S_t\right)\right)\\ E\left(r_t\left(S_t\right)R_t^e\left(S_t\right)\right),E\left(R_t^e{\left(S_t\right)}^2\right)\end{array}\right)=\left(\begin{array}{c}1,0\\ -1,1\end{array}\right)E{\left(e_t\left(S_t\right)e_t\left(S_t\right)\right)}^{\prime }\left(\begin{array}{c}1,-1\\ 0,1\end{array}\right)>0,$

Then we obtain the following from the above equation:

$E\left(R_t^e{\left(S_t\right)}^2\right)>0,$

and

$E\left(r_t^2\left(S_t\right)\right)-\frac{{\left[E\left(r_t\left(S_t\right)R_t^e\left(S_t\right)\right)\right]}^2}{E\left(R_t^e{\left(S_t\right)}^2\right)}>0,$ (8)

which can be rewritten as follows:

$E\left(r_t^2\left(S_t\right)\right)E\left(R_t^e{\left(S_t\right)}^2\right)-{\left[E\left(r_t\left(S_t\right)R_t^e\left(S_t\right)\right)\right]}^2=A_t^1\left(S_t\right)>0.$ (9)

According to Equation (9) and N8, we get that $H_t\left(S_t\right)>0$. Then we can deduce that $K_t\left(S_t\right)>0$ from N10.

Note that the above lemma is used to guarantee the existence of the optimal strategy for problem $P1\left(\lambda \mathrm{,}d\right)$ which is presented in the following theorem:

Theorem 1 The optimal investment strategy for $P1\left(\lambda \mathrm{,}d\right)$ is of the following form:

$\begin{array}{l}{\pi }_t^{*}\left(i,w_t\right)=\frac{G_t\left(i\right)r_t^e\left(i\right)}{H_t\left(i\right)E\left(R_t^e{\left(i\right)}^2\right)}\left(d-\lambda \right)-\frac{E\left(r_t\left(i\right)R_t^e\left(i\right)\right)}{E\left(R_t^e{\left(i\right)}^2\right)}{w}_t,i\ne L,t=0,1,\cdots ,T-1,\\ {\pi }_t^{*}\left(L,w_t\right)=0,t=0,1,\cdots ,T-1.\end{array}$ (10)

The corresponding optimal value functions are as follows:

$\begin{array}{l}{V}_t^{*}\left(i,w_t\right)=\left[{\displaystyle \underset{k=t}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-t}(i)-\left({\displaystyle \underset{m=t}{\overset{T-1}{\sum }}}{\hat{Q}}^{m-t}{K}_m\right)\left(i\right)\right]{\left(d-\lambda \right)}^2+\left[P_t\left(i\right)+\frac{H_t\left(i\right)A_t^1\left(i\right)}{E\left(R_t^e{\left(i\right)}^2\right)}\right]w_t^2\\ -2\left[P_t\left(i\right)+\frac{G_t\left(i\right)A_t^2\left(i\right)}{E\left(R_t^e{\left(i\right)}^2\right)}\right]\left(d-\lambda \right)w_t,i\ne L,t=0,1,\cdots ,T-1,\end{array}$ (11)

$\begin{array}{l}{V}_t^{*}\left(L,w_t\right)=Q\left(L,S_{t+1}\right)[E\left(r_t{\left(L\right)}^2\right){\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-\left(t+1\right)}\left(S_k\right){\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}E\left(r_{k-j}{\left(S_{k-j}\right)}^2\right)w_t^2\\ +{\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-\left(t+1\right)}\left(S_k\right){\left(d-\lambda \right)}^2-2E\left(r_t\left(L\right)\right){\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-\left(t+1\right)}\left(S_k\right)\\ \times {\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}E\left(r_{k-j}\left(S_{k-j}\right)\right)\left(d-\lambda \right)w_t]\\ +P_t\left(L\right){\left(w_t-\left(d-\lambda \right)\right)}^2,i=L,t=0,1,\cdots ,T-1.\end{array}$ (12)

Proof. (i) First, we prove Equation (12) according to Equation (6), for $i=L,t=0,1,\cdots ,T-1$, we have

$\begin{array}{l}{V}_t\left(L,w_t\right)=P_t\left(L\right){\left(w_t-\left(d-\lambda \right)\right)}^2\\ +E_{L,w_t}\left[{\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{P}_k\left(S_k\right){\left[{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}{r}_{k-j}\left(S_{k-j}\right)r_t\left(L\right)w_t-\left(d-\lambda \right)\right]}^2\right]\\ =P_t\left(L\right){\left(w_t-\left(d-\lambda \right)\right)}^2+E_{L,w_t}[{\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{P}_k\left(S_k\right)[{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}{r}_{k-j}^2\left(S_{k-j}\right)r_t^2\left(L\right)w_t^2\\ +{\left(d-\lambda \right)}^2-2{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}{r}_{k-j}\left(S_{k-j}\right)r_t\left(L\right)\left(d-\lambda \right)w_t]].\end{array}$ (13)

Based on A2, we rewrite Equation (13) as follows:

$\begin{array}{c}{V}_t\left(L,w_t\right)=P_t\left(L\right){\left(w_t-\left(d-\lambda \right)\right)}^2+Q\left(L,S_{t+1}\right){\displaystyle \underset{k=t+1}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-\left(t+1\right)}\left(S_k\right)\\ \times [{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}E\left(r_{k-j}^2\left(S_{k-j}\right)\right)E\left(r_t^2\left(L\right)\right)w_t^2+{\left(d-\lambda \right)}^2\\ -2{\displaystyle \underset{j=1}{\overset{k-\left(t+1\right)}{\prod }}}E\left(r_{k-j}\left(S_{k-j}\right)\right)E\left(r_t\left(L\right)\right)\left(d-\lambda \right)w_t].\end{array}$ (14)

Note that Equation (14) is equivalent to Equation (12), which means that Equation (12) holds true for $t=0,1,\cdots ,T-1$.

Then we prove the expressions(10)-(11) for $i\ne L$ by using mathematical induction. First, we show that the expressions are true for $t=T-1$ in (ii), and then in (iii) prove that if the expressions hold true for $t=n+1$, then they are also true for $t=n$.

(ii) For $t=T-1$, based on Equation (4) and (5) and (12), we have

$\begin{array}{l}{V}_{T-1}\left(i,w_{T-1}\right)\\ =P_{T-1}\left(i\right){\left(w_{T-1}-\left(d-\lambda \right)\right)}^2+{\min }_{{\pi }_T-1}{E}_{i,w_{T-1}}\left[P_T\left(S_T\right){\left(w_T-\left(d-\lambda \right)\right)}^2\right]\\ =P_{T-1}\left(i\right){\left(w_{T-1}-\left(d-\lambda \right)\right)}^2\\ +{\min }_{{\pi }_{T-1}}\{Q\left(i,L\right)E\left[P_T\left(L\right){\left[\delta \left(w_{T-1}{r}_{T-1}\left(i\right)+{\pi }_{T-1}{R}_{T-1}^e\left(i\right)\right)-\left(d-\lambda \right)\right]}^2\right]\\ +\left(1-Q\left(i,L\right)\right)E\left[{\left[\left(w_{T-1}{r}_{T-1}\left(i\right)+{\pi }_{T-1}{R}_{T-1}^e\left(i\right)\right)-\left(d-\lambda \right)\right]}^2{P}_T\left(S_T\right)\right]\}\end{array}$

$\begin{array}{l}=P_{T-1}\left(i\right){\left(w_{T-1}-\left(d-\lambda \right)\right)}^2\\ +{\min }_{{\pi }_{T-1}}\{{\bar{Q}}_{P_T}\left(i\right){\left(d-\lambda \right)}^2-2G_{T-1}\left(i\right)\left[E\left(r_{T-1}\left(i\right)\right)w_{T-1}+r_{T-1}^e\left(i\right){\pi }_{T-1}\right]\left(d-\lambda \right)\\ +H_{T-1}\left(i\right)\left[E\left(r_{T-1}^2\left(i\right)\right)w_{T-1}^2+E\left(R_{T-1}^e{\left(i\right)}^2\right){\pi }_{T-1}^2+2E\left(r_{T-1}\left(i\right)R_{T-1}^e\left(i\right)\right){\pi }_{T-1}{w}_{T-1}\right]\}.\end{array}$

As $H_{T-1}\left(i\right)>0$ in Lemma 2, the optimal solution ${\pi }_{T-1}^{\mathrm{<mo>\; *\; </mo>}}$ for $V_{T-1}$ exists and can be obtained by setting $\frac{\text{d}{V}_{T-1}\left(i,w_{T-1}\right)}{\text{d}{\pi }_{T-1}}=0$ to yield

${\pi }_{T-1}^{\mathrm{<mo>\; *\; </mo>}}\left(\lambda \mathrm{,}i\mathrm{,}{w}_{T-1}\right)=\frac{G_{T-1}\left(i\right)r_{T-1}^e\left(i\right)}{H_{T-1}\left(i\right)E\left(R_{T-1}^e{\left(i\right)}^2\right)}\left(d-\lambda \right)-\frac{E\left(r_{T-1}\left(i\right)R_{T-1}^e\left(i\right)\right)}{E\left(R_{T-1}^e{\left(i\right)}^2\right)}{w}_{T-1}\mathrm{.}$ (15)

Substituting Equation (15) into the expression for $V_{T-1}\left(i\mathrm{,}{w}_{T-1}\right)$, we have

$\begin{array}{c}{V}_{T-1}^{*}\left(i,w_{T-1}\right)=\left[{\bar{Q}}_{P_T}\left(i\right)+P_{T-1}\left(i\right)-K_{T-1}\left(i\right)\right]{\left(d-\lambda \right)}^2\\ +\left[P_{T-1}\left(i\right)+\frac{H_{T-1}\left(i\right)A_{T-1}^1\left(i\right)}{E\left(R_{T-1}^e{\left(i\right)}^2\right)}\right]w_{T-1}^2\\ -2\left[P_{T-1}\left(i\right)+\frac{G_{T-1}\left(i\right)A_{T-1}^2\left(i\right)}{E\left(R_{T-1}^e{\left(i\right)}^2\right)}\right]\left(d-\lambda \right)w_{T-1}.\end{array}$ (16)

Hence, Equations (10) and (11) are true for $t=T-1$.

(iii) Assume that Equation (10), (11) and (12) hold true for $t=n+1$, then when $t=n$, we have

$\begin{array}{l}{V}_n\left(i,w_n\right)=P_n\left(i\right){\left(w_n-\left(d-\lambda \right)\right)}^2+{\min }_{{\pi }_n}{E}_{i,w_n}\left[V_{n+1}\left(S_{n+1},w_{n+1}\right)\right]\\ =P_n\left(i\right){\left(w_n-\left(d-\lambda \right)\right)}^2+{\min }_{{\pi }_n}{\displaystyle \underset{j=1}{\overset{L-1}{\sum }}}Q\left(i,j\right)E\{[{\displaystyle \underset{k=n+1}{\overset{T}{\sum }}}{\bar{Q}}_{P_k}^{k-\left(n+1\right)}\left(j\right)\begin{array}{c}\begin{array}{l}\\ \end{array}\\ \end{array}\\ -\left[{\displaystyle \underset{m=n+1}{\overset{T-1}{\sum }}}{\hat{Q}}^{m-\left(n+1\right)}{K}_m\right]\left(j\right)]{\left(d-\lambda \right)}^2+\left[P_{n+1}\left(j\right)+\frac{H_{n+1}\left(j\right)A_{n+1}^1\left(j\right)}{E\left(R_{n+1}^e{\left(j\right)}^2\right)}\right]w_{n+1}^2\\ -2\left[P_{n+1}\left(j\right)+\frac{G_{n+1}\left(j\right)A_{n+1}^2\left(j\right)}{E\left(R_{n+1}^e{\left(j\right)}^2\right)}\right]\left(d-\lambda \right)w_{n+1}\}\end{array}$