Portfolio Selection with Jumps under Regime Switching

Abstract

We investigate a continuous-time version of the mean-variance portfolio selection model with jumps under regime switching. The portfolio selection is proposed and analyzed for a market consisting of one bank account and multiple stocks. The random regime switching is assumed to be independent of the underlying Brownian motion and jump processes. A Markov chain modulated diffusion formulation is employed to model the problem.

1. Introduction

The jump diffusion process has come to play an important role in many branches of science and industry. In their book [1], Øksendal and Sulem have studied the optimal control, optimal stopping, and impulse control for jump diffusion processes. In mathematical finance theory, many researchers have developed option pricing theory, for example, Merton [2] was the first to use the jump processes to describe the stock dynamics, and Bardhan and Chao [3] were amongst the first authors to consider market completeness in a discontinuous model. The jump diffusion models have been discussed by Chan [4], Föllmer and Schweizer [5], El Karoui and Quenez [6], Henderson and Hobson [7], and Merculio and Runggaldier [8], to name a few.

On the other hand, regime-switching models have been widely used for price processes of risky assets. For example, in [9] the optimal stopping problem for the perpetual American put has been considered, and the finite expiry American put and barrier options have been priced. The asset allocation has been discussed in [10], and Elliott et al. [11] have investigated volatility problems. Regime-switching models with a Markov-modulated asset have already been applied to option pricing in [12–14] and references therein. Moreover, Markowitz's mean-variance portfolio selection with regime switching has been studied by Yin and Zhou [15], Zhou and Yin [16], and Zhou and Li [17].

Portfolio selection is an important topic in finance; multiperiod mean-variance portfolio selection has been studied by, for example, Samuelson [18], Hakansson [19], and Pliska [20] among others. Continuous-time mean-variance hedging problems were attacked by Duffie and Richardson [21] and Schweizer [22] where optimal dynamic strategies were derived, based on the projection theorem, to hedge contingent claims in incomplete markets.

In this paper, we will extend the results of Yin and Zhou [15] to SDEs with jumps under regime switching. After dealing with the difficulty from the jump processes, we obtain similar results to those of Yin and Zhou [15].

2. SDEs under Regime Switching with Jumps

Throughout this paper, let (Ω,ℱ,𝑃) be a fixed complete probability space on which it is defined a standard 𝑑-dimensional Brownian motion 𝑊(𝑡)≡(𝑊₁(𝑡),…,𝑊_𝑑(𝑡))^′ and a continuous-time stationary Markov chain 𝛼(𝑡) taking value in a finite state space 𝕊={1,2,…,𝑙}. Let 𝑁(𝑡,𝑧) be as 𝑛-dimensional Poisson process and denote the compensated Poisson process by𝑁(𝑑𝑡,𝑑𝑧)=(𝑁₁(𝑑𝑡,𝑑𝑧₁),…,𝑁_𝑛(𝑑𝑡,𝑑𝑧_𝑛))^′=(𝑁₁(𝑑𝑡,𝑑𝑧₁)−𝜈₁(𝑑𝑧₁)𝑑𝑡,…,𝑁_𝑛(𝑑𝑡,𝑑𝑧_𝑛)−𝜈(𝑑𝑧_𝑛)𝑑𝑡)^′, (2.1) where 𝑁_𝑗, 𝑗=1,…,𝑛, are independent 1-dimensional Poisson random measures with characteristic measure 𝜈_𝑗,𝑗=1,…,𝑛, coming from 𝑛 independent 1-dimensional Poisson point processes. We assume that 𝑊(𝑡), 𝛼(𝑡), and 𝑁(𝑑𝑡,𝑑𝑧) are independent. The Markov chain 𝛼(𝑡) has a generator 𝑄=(𝑞_𝑖𝑗)_𝑙×𝑙 given by𝑞𝑃{𝛼(𝑡+Δ)=𝑗∣𝛼(𝑡)=𝑖}=^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝑞_𝑖𝑖Δ+𝑜(Δ),if𝑖=𝑗, (2.2) where Δ>0. Here 𝑞_𝑖j≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while𝑞_𝑖𝑖=−_𝑗≠𝑖𝑞_𝑖𝑗, (2.3) and stationary transition probabilities 𝑝_𝑖𝑗(𝑡)=𝑃(𝛼(𝑡)=𝑗∣𝛼(0)=𝑖),𝑡≥0,𝑖,𝑗=1,2,…,𝑙. (2.4) Define ℱ_𝑡=𝜎{𝑊(𝑠),𝛼(𝑠),𝑁(𝑠,⋅)∶0≤𝑠≤𝑡}. Let |⋅| denote the Euclidean norm as well as the matrix trace norm and 𝑀^′ the transpose of any vector or matrix 𝑀. We denote by 𝐿²_{ℱ(0,𝑇;ℝ}^𝑚₎ the set of all ℝ^𝑚-valued, measurable stochastic processes 𝑓(𝑡) adapted to {ℱ_𝑡}_𝑡≥0, such that 𝔼∫^𝑇₀|𝑓(𝑡)|²𝑑𝑡<+∞.

Consider a market in which 𝑑+1 assets are traded continuously. One of the assets is a bank account whose price 𝑃₀(𝑡) is subject to the following stochastic ordinary differential equation:𝑑𝑃₀(𝑡)=𝑟(𝑡,𝛼(𝑡))𝑃₀𝑃(𝑡)𝑑𝑡,𝑡∈[0,𝑇],₀(0)=𝑝₀>0, (2.5) where 𝑟(𝑡,𝑖)≥0,𝑖=1,2,…,𝑙, are given as the interest rate process corresponding to different market modes. The other 𝑑 assets are stocks whose price processes 𝑃_𝑚(𝑡), 𝑚=1,2,…,𝑑, satisfy the following system of stochastic differential equations (SDEs):𝑑𝑃_𝑚(𝑡)=𝑃_𝑚(𝑡){𝑏_𝑚(𝑡,𝛼(𝑡))𝑑𝑡+^𝑑_𝑛=1𝜎_𝑚𝑛(𝑡,𝛼(𝑡))𝑑𝑊_𝑛(𝑡)+^𝑛_𝑗=1ℝ𝜌_𝑚𝑗(𝑡,𝛼(𝑡),𝑧_𝑗)𝑁_𝑗(𝑑𝑡,𝑑𝑧_𝑗𝑃)},𝑡∈[0,𝑇],_𝑚(0)=𝑝_𝑚>0, (2.6) where for each 𝑖=1,2,…,𝑙, 𝑏∶[0,𝑇]×𝕊→ℝ^𝑑×1, 𝜎∶[0,𝑇]×𝕊→ℝ^𝑑×𝑑, 𝜌∶[0,𝑇]×𝕊×ℝ^𝑛→ℝ^𝑑×𝑛 is the appreciation rate process, and 𝜎_𝑚(𝑡,𝑖)∶=(𝜎_𝑚1(𝑡,𝑖),…,𝜎_𝑚𝑑(𝑡,𝑖)) are adapted processes such that the integrals exist. And each column 𝜌^(𝑘) of the 𝑑×𝑛 matrix 𝜌=[𝜌_𝑖𝑗] depends on 𝑧 only through the 𝑘th coordinate 𝑧_𝑘, that is, 𝜌^(𝑘)(𝑡,𝑖,𝑧)=𝜌^(𝑘)(𝑡,𝑖,𝑧_𝑘),𝑧=(𝑧₁,…,𝑧_𝑛)∈ℝ^𝑛. (2.7)

Remark 2.1. Generally speaking, one uses noncompensated Poisson processes in a jump diffusion model (see Kushner [23]). However, we use compensated Poisson processes in (2.6) instead of using noncompensated Poisson processes, this is because firstly, using relationship (2.1) we can easily transform a jump diffusion model driven by noncompensated Poisson processes into a jump diffusion model driven by compensated Poisson processes; secondly, using compensated Poisson processes we can keep the Riccati Equation (4.2) similar to that of a diffusion model without jump processes, and then 𝐻(𝑡,𝑖) in (4.3) has a financial interpretation.

Define the volatility matrix, for each 𝑖=1,…,𝑙,𝜎𝜎(𝑡,𝑖)∶=¹⋮𝜎(𝑡,𝑖)_𝑑(𝑡,𝑖)≡(𝜎_𝑚𝑛(𝑡,𝑖))_𝑑×𝑑,𝑏𝑏(𝑡,𝑖)=¹⋮𝑏(𝑡,𝑖)_𝑑(𝑡,𝑖)∈ℝ^𝑑×1,𝜌𝜌(𝑡,𝑖,𝑧)=¹⋮𝜌(𝑡,𝑖,𝑧)_𝑑(𝑡,𝑖,𝑧)∈ℝ^𝑑×𝑛, (2.8) where 𝜌_𝑚(𝑡,𝑖,𝑧)=(𝜌_𝑚1(𝑡,𝑖,𝑧),…,𝜌_𝑚𝑛(𝑡,𝑖,𝑧)). (2.9)

We assume throughout this paper that the following nondegeneracy condition:𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′≥𝛿𝐼,∀𝑡∈[0,𝑇],𝑖=1,2,…,𝑙, (2.10) is satisfied for some 𝛿>0. We also assume that all the functions 𝑟(𝑡,𝑖), 𝑏_𝑚(𝑡,𝑖), and 𝜎_𝑚𝑛(𝑡,𝑖), 𝜌_𝑚𝑛(𝑡,𝑖,𝑧) are measurable and uniformly bounded in 𝑡.

Suppose that the initial market mode 𝛼(0)=𝑖₀. Consider an agent with an initial wealth 𝑥₀>0. These initial conditions are fixed throughout the paper. Denote by 𝑥(𝑡) the total wealth of the agent at time 𝑡≥0. Assume that the trading of shares takes place continuously and that transaction cost and consumptions are not considered. Suppose the right portfolio (𝜋₀(𝑡),𝜋₁(𝑡),…,𝜋_𝑑(𝑡)) exists, where 𝜋₀(𝑡) is the money invested in the bond, and 𝜋_𝑖(𝑡) is the money invested in the 𝑖th stock. Then 𝑥(𝑡)=^𝑑_𝑖=0𝜋_𝑖(𝑡)=^𝑑_𝑖=0𝜂_𝑖(𝑡)𝑃_𝑖(𝑡),𝑥(0)=𝑥₀, (2.11) where 𝜂₀(𝑡) is the number of bond units bought by the investor, and 𝜂_𝑖(𝑡) is the amount of units for the 𝑖th stock. We call 𝑥(𝑡) the wealth process for this investor in the market. Now let us derive intuitively the stochastic differential equation (SDE) for the wealth process as follows. Suppose the portfolio is self-financed, that is, in a short time 𝑑𝑡 the investor does not put in or withdraw any money from the market. Let the money 𝑥(𝑡) change in the market due to the market own performance, that is, self-finance produces 𝑑𝑥(𝑡)=𝜂₀(𝑡)𝑑𝑃₀(𝑡)+^𝑑_𝑖=1𝜂_𝑖(𝑡)𝑑𝑃_𝑖(𝑡). (2.12) Now substituting (2.5) and (2.6) into the above equation, after a simple calculation we arrive at𝑑𝑥(𝑡)=𝑟(𝑡,𝛼(𝑡))𝑥(𝑡)𝑑𝑡+^d_𝑚=1𝜋_𝑚(𝑡)(𝑏_𝑚+(𝑡,𝛼(𝑡))−𝑟(𝑡,𝛼(𝑡)))𝑑𝑡_{𝑑𝑑𝑚=1𝑛=1}𝜋_𝑚(𝑡)𝜎_𝑚𝑛(𝑡,𝛼(𝑡))𝑑𝑊_𝑛+(𝑡)_{𝑑𝑛𝑚=1𝑗=1ℝ}𝜋_𝑚(𝑡)𝜌_𝑚𝑗(𝑡,𝛼(𝑡),𝑧_𝑗)𝑁_𝑗(𝑑𝑡,𝑑𝑧_𝑗),𝑥(0)=𝑥₀>0,𝛼(0)=𝑖₀, (2.13) where 𝜋(𝑡)=(𝜋₁(𝑡),…,𝜋_𝑑(𝑡))^′ which we call a portfolio of the agent. And 𝜋_𝑚(𝑡) is the total market value of the agent's wealth in the 𝑚th asset, 𝑚=0,1,…,𝑑, at time 𝑡.

Setting𝐵(𝑡,𝑖)∶=(𝑏₁(𝑡,𝑖)−𝑟(𝑡,𝑖),…,𝑏_𝑑(𝑡,𝑖)−𝑟(𝑡,𝑖)),𝑖=1,2,…,𝑙, (2.14) we can rewrite the wealth equation (2.13) as𝑑𝑥(𝑡)=𝑟(𝑡,𝛼(𝑡))𝑥(𝑡)𝑑𝑡+𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝑑𝑡+𝜋^′+(𝑡)𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_ℝ𝑛𝜋^′(𝑡)𝜌(𝑡,𝛼(𝑡),𝑧)𝑁(𝑑𝑡,𝑑𝑧),𝑥(0)=𝑥₀>0,𝛼(0)=𝑖₀. (2.15)

Definition 2.2. A portfolio 𝜋(⋅) is said to be admissible if 𝜋(⋅)∈𝐿²_ℱ(0,𝑇;ℝ^𝑑) and the SDE (2.15) has a unique solution 𝑥(⋅) corresponding to 𝜋(⋅). In this case, we refer to (𝑥(⋅),𝜋(⋅)) as an admissible (wealth, portfolio) pair.

Remark 2.3. Most works in the literature define a portfolio, say 𝜋(⋅), as the fractions of wealth allocated to different stocks. That is, 𝑢(𝑡)=𝜋(𝑡)𝑥(𝑡),𝑡∈[0,𝑇]. (2.16)

With this definition, (2.15) can be rewritten as𝑑𝑥(𝑡)=𝑥(𝑡)[𝑟(𝑡,𝛼(𝑡))+𝐵(𝑡,𝛼(𝑡))𝑢(𝑡)]𝑑𝑡+𝑥(𝑡)𝑢(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_ℝ𝑛𝑥(𝑡)𝑢(𝑡)^′𝜌(𝑡,𝛼(𝑡),𝑧)𝑁(𝑑𝑡,𝑑𝑧),𝑥(0)=𝑥₀>0,𝛼(0)=𝑖₀. (2.17) It is well known that this equation has a unique solution (see [1, page 10, Theorem 1.19]. We can use the same method in [18, Example  1.15, page  8] to show positivity of the solution of (2.17) if the initial wealth 𝑥₀ is positive and 𝑢(𝑡)^′𝜌(𝑡,𝑖,𝑧)>−1. A wealth process with possible zero or negative values is sensible at least for some circumstances. The nonnegativity of wealth process is better imposed as an additional constraint, rather than as a built-in feature. In our formulation, a portfolio is well defined even if the wealth is zero or negative, and the nonnegativity of the wealth could be a constraint.

The agent's objective is to find an admissible portfolio 𝜋(⋅) among all the admissible portfolios whose expected terminal wealth is 𝔼𝑥(𝑇)=𝜁 for some given 𝜁∈ℝ¹, so that the risk measured by the variance of the terminal wealthVar𝑥(𝑇)≡𝔼[𝑥(𝑇)−𝔼𝑥(𝑇)]²=𝔼[𝑥(𝑇)−𝜁]² (2.18) is minimized. Finding such a portfolio 𝜋(⋅) is referred to as the mean-variance portfolio selection problem. Specifically, we have the following formulation.

Definition 2.4. The mean-variance portfolio selection is a constrained stochastic optimization problem, parameterized by 𝜁∈ℝ¹: minimize𝐽_𝑀𝑉(𝑥₀,𝑖₀,𝜋(⋅))∶=𝔼[𝑥(𝑇)−𝜁]²,subjectto𝔼𝑥(𝑇)=𝜁,(𝑥(⋅),𝜋(⋅))admissible. (2.19)

Moreover, the problem is called feasible if there is at least one portfolio satisfying all the constraints. The problem is called finite if it is feasible and the infimum of 𝐽_𝑀𝑉(𝑥₀,𝑖₀,𝜋(⋅)) is finite. Finally, an optimal portfolio to the above problem, if it ever exists, is called an efficient portfolio corresponding to 𝜁; the corresponding (Var𝑥(𝑇),𝜁)∈ℝ² and (𝜎_𝑥(𝑇),𝜁)∈ℝ² are interchangeably called an efficient point, where 𝜎_𝑥(𝑇) denotes the standard deviation of 𝑥(𝑇). The set of all the efficient points is called the efficient frontier.

For more details of mean-variance portfolio selection see [15, 16], We need more notations; let Δ_𝑖𝑗 be consecutive, left closed, right open intervals of the real line each having length 𝛾_𝑖𝑗 such that Δ₁₂=[0,𝑞₁₂Δ),₁₃=[𝑞₁₂,𝑞₁₂+𝑞₁₃⋮Δ),_1𝑙=[^𝑙−1_𝑗=2𝑞_1𝑗,^𝑙_𝑗=2𝑞_1𝑗Δ),₂₁=[^𝑙_𝑗=2𝑞_1𝑗,^𝑙_𝑗=2𝑞_1𝑗+𝑞₂₁Δ),₂₃=[^𝑙_𝑗=2𝑞_1𝑗+𝑞₂₁,^𝑙_𝑗=2𝑞_1𝑗+𝛾₂₁+𝑞₂₃⋮Δ),_2𝑙=[^𝑙_𝑗=2𝑞_1𝑗+^𝑙−1_{𝑗=1,𝑗≠2}𝑞_2𝑗,^𝑙_𝑗=2𝑞_1𝑗+^𝑙_{𝑗=1,𝑗≠2}𝑞_2𝑗). (2.20) For future use, we cite the generalized Itô lemma (see [1, 24, 25]) as the following lemma.

Lemma 2.5. Given a 𝑑-dimensional process 𝑦(⋅) satisfying 𝑑𝑦(t)=𝑓(𝑡,𝑦(𝑡),𝛼(𝑡))𝑑𝑡+𝑔(𝑡,𝑦(𝑡),𝛼(𝑡))𝑑𝑊(𝑡)+_ℝ𝑛𝛾(𝑡,𝑦(𝑡),𝛼(𝑡),𝑧)𝑁(𝑑𝑡,𝑑𝑧), (2.21) where 𝑓,𝑔, and 𝛾 satisfy Lipschitz condition with appropriate dimensions, each column 𝛾^(𝑘) of the matrix 𝛾=[𝛾_𝑖𝑗] depends on 𝑧 only through the 𝑘𝑡ℎ coordinate 𝑧_𝑘. Let 𝜑(𝑡,𝑥,𝑖)∈𝐶^1,2([0,𝑇]×ℝ^𝑛×𝑆;ℝ), one then has 𝑑𝜑(𝑡,𝑦(𝑡),𝛼(𝑡))=Γ𝜑(𝑡,𝑦(𝑡),𝛼(𝑡))𝑑𝑡+𝜑_𝑥(𝑡,𝑦(𝑡),𝛼(𝑡))^′+𝑔(𝑡,𝑦(𝑡),𝛼(𝑡))𝑑𝑊(𝑡)_{𝑛𝑘=1ℝ}{𝜑(𝑡,𝑦(𝑡)+𝛾^(𝑘)(𝑡,𝛼(𝑡),𝑧_𝑘),𝛼(𝑡))−𝜑(𝑡,𝑦(𝑡),𝛼(𝑡))−𝜑_𝑥(𝑡,𝑦(𝑡),𝛼(𝑡))^′𝛾^(𝑘)(𝑡,𝛼(𝑡),𝑧)}𝜈(𝑑𝑧_𝑘+)𝑑𝑡_{𝑛𝑘=1ℝ}{𝜑(𝑡,𝑦(𝑡)+𝛾^(𝑘)(𝑡,𝛼(𝑡),𝑧),𝛼(𝑡))−𝜑(𝑡,𝑦(𝑡),𝛼(𝑡))}𝑁_𝑘(𝑑𝑡,𝑑𝑧_𝑘)+_ℝ(𝜑(𝑡,𝑦(𝑡),𝛼(0)+ℎ(𝛼(𝑡),𝑙))−𝜑(𝑡,𝑦(𝑡),𝛼(𝑡)))𝜇(𝑑𝑡,𝑑𝑙), (2.22) where Γ𝜑(𝑡,𝑥,𝑖)∶=𝜑_𝑡(𝑡,𝑥,𝑖)+𝜑_𝑥(𝑡,𝑥,𝑖)^′+1𝑓(𝑡,𝑥,𝑖)2trace[𝑔(𝑡,𝑥,𝑖)^′𝜑_𝑥𝑥(𝑡,𝑥,𝑖)𝑔(𝑡,𝑥,𝑖)]+^𝑙_𝑗=1𝑞_𝑖𝑗+𝜑(𝑡,𝑥,𝑗)_{𝑛𝑘=1ℝ}{𝜑(𝑡,𝑦(𝑡)+𝛾^(𝑘)(𝑡,𝛼(𝑡),𝑧_𝑘),𝛼(𝑡))−𝜑(𝑡,𝑦(𝑡),𝛼(𝑡))−𝜑_𝑥(𝑡,𝑦(𝑡),𝛼(𝑡))^′𝛾^(𝑘)(𝑡,𝛼(𝑡),𝑧)}𝜈(𝑑𝑧_𝑘), (2.23) where 𝜇 is a martingale measure, ℎ(𝑖,𝑦)=𝑗−𝑖,if𝑦∈Δ_𝑖𝑗,0,otherwise, (2.24) and 𝜇(𝑑𝑡,𝑑𝑙)=𝛾(𝑑𝑡,𝑑𝑙)−𝜇(𝑑𝑙)𝑑𝑡 is a martingale measure. And 𝛾(𝑑𝑡,𝑑𝑦) is a Poisson random measure with intensity 𝑑𝑡×𝜇(𝑑𝑦), in which 𝜇 is the Lebesgue measure on ℝ.

3. Feasibility

Since the problem (2.19) involves a terminal constraint 𝔼𝑥(𝑇)=𝜁, in this section, we derive conditions under which the problem is at least feasible. First of all, the following generalized Itô lemma [25] for Markov-modulated processes is useful.

The associated wealth process 𝑥⁰(⋅) satisfies𝑑𝑥⁰(𝑡)=𝑟(𝑡,𝛼(𝑡))𝑥⁰𝑥(𝑡)𝑑𝑡,₀(0)=𝑥₀>0,𝛼(0)=𝑖₀, (3.1) with its expected terminal wealth𝜁⁰∶=𝔼𝑥⁰(𝑇)=𝔼𝑒^{∫𝑇0𝑟(𝑠,𝛼(𝑠))𝑑𝑠}𝑥₀. (3.2)

Lemma 3.1. Let 𝜓(⋅,𝑖), 𝑖=1,2,…,𝑙, be the solutions to the following system of linear ordinary differential equations (ODEs): 𝜓̇(𝑡,𝑖)=−𝑟(𝑡,𝑖)𝜓(𝑡,𝑖)−^𝑙_𝑗=1𝑞_𝑖𝑗𝜓(𝑡,𝑗),𝜓(𝑇,𝑖)=1,𝑖=1,2,…,𝑙. (3.3) Then the mean-variance problem (2.19) is feasible for every 𝜁∈ℝ¹ if and only if 𝜚∶=𝔼^𝑇₀|𝜓(𝑡,𝛼(𝑡))𝐵(𝑡,𝛼(𝑡))|²𝑑𝑡>0. (3.4)

Proof. To prove the “if" part, construct a family of admissible portfolios 𝜋^𝛽(⋅)=𝛽𝜋(⋅) for 𝛽∈ℝ¹ where 𝜋(𝑡)=𝐵(𝑡,𝛼(𝑡))^′𝜓(𝑡,𝛼(𝑡)). (3.5) Assume that 𝑥^𝛽(𝑡) is the solution of (2.15). Let 𝑥^𝛽(𝑡)=𝑥⁰(𝑡)+𝛽𝑦(𝑡), where 𝑥⁰(⋅) satisfies (3.1), and 𝑦(⋅) is the solution to the following equation: 𝑑𝑦(𝑡)=[𝑟(𝑡,𝛼(𝑡))𝑦(𝑡)+𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)]𝑑𝑡+𝜋(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_ℝ𝑛𝜋(𝑡)^′𝜌(𝑡,𝛼(𝑡),𝑧)𝑁(𝑑𝑡,𝑑𝑧),𝑦(0)=0,𝛼(0)=𝑖₀. (3.6) Therefore, problem (2.19) is feasible for every 𝜁∈ℝ¹ if there exists 𝛽∈ℝ such that 𝜁=𝔼𝑥^𝛽(𝑇)≡𝔼𝑥⁰(𝑇)+𝛽𝔼𝑦(𝑇). Equivalently, (2.19) is feasible for every 𝜁∈ℝ if 𝔼𝑦(𝑇)≠0. Applying the generalized Itô formula (Lemma 2.5) to 𝜑(𝑡,𝑥,𝑖)=𝜓(𝑡,𝑖)𝑥, we have =̇+𝑑[𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)]𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)𝑑𝑡+𝜓(𝑡,𝛼(𝑡))[𝑟(𝑡,𝛼(𝑡))𝑦(𝑡)+𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)]𝑑t_𝑙𝑗=1𝑞_{𝛼(𝑡)𝑗}𝜓(𝑡,𝑗)𝑦(𝑡)𝑑𝑡+𝜋(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_{𝑛𝑘=1ℝ}{𝜓(𝑡,𝛼(𝑡))(𝑦(𝑡)+𝜋(𝑡)^′𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧))−𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)−𝜓(𝑡,𝛼(𝑡))𝜋(𝑡)^′𝜌^(𝑘)+(𝑡,𝛼(𝑡),𝑧)}𝜈(𝑑𝑧)𝑑𝑡_{𝑛𝑘=1ℝ}{𝜓(𝑡,𝛼(𝑡))(𝑦(𝑡)+𝜋(𝑡)^′𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧))−𝜓(𝑡,𝛼(𝑡))𝜋(𝑡)^′𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)}𝑁_𝑘(𝑑𝑡,𝑑𝑧_𝑘)+_ℝ{𝜓(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))𝑦(𝑡)−𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)}𝜇(𝑑𝑡,𝑑𝑙)=−𝑟(𝑡,𝛼(𝑡))𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)𝑑𝑡−^𝑙_𝑗=1𝑞_{𝛼(𝑡)𝑗}+𝜓(𝑡,𝑗)𝑦(𝑡)𝑑𝑡+𝑟(𝑡,𝛼(𝑡))𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)𝑑𝑡+𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝜓(𝑡,𝛼(𝑡))𝑑𝑡_𝑙𝑗=1𝑞_{𝛼(𝑡)𝑗}𝜓(𝑡,𝑗)𝑦(𝑡)𝑑𝑡+𝜋(𝑡)^′𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)+^𝑛_𝑘=1ℝ{𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)}𝑁_𝑘(𝑑𝑡,𝑑𝑧_𝑘)+_ℝ{𝜓(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))𝑦(𝑡)−𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)}𝜇(𝑑𝑡,𝑑𝑙)=𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝜓(𝑡,𝛼(𝑡))𝑑𝑡+𝜋(𝑡)^′𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)+^𝑛_𝑘=1ℝ{𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)}𝑁_𝑘(𝑑𝑡,𝑑𝑧_𝑘)+_ℝ{𝜓(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))𝑦(𝑡)−𝜓(𝑡,𝛼(𝑡))𝑦(𝑡)}𝜇(𝑑𝑡,𝑑𝑙). (3.7) Integrating from 0 to 𝑇, taking expectation, and using (3.5), we obtain 𝔼𝑦(𝑇)=𝔼^𝑇₀𝜓(𝑡,𝛼(𝑡))𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝑑𝑡=𝔼^𝑇₀|𝜓(𝑡,𝛼(𝑡))𝐵(𝑡,𝛼(𝑡))|²𝑑𝑡. (3.8) Consequently, 𝔼𝑦(𝑇)≠0 if (3.4) holds.
Conversely, suppose that problem (2.19) is feasible for every 𝜁∈ℝ¹. Then for each 𝜁∈ℝ, there is an admissible portfolio 𝜋(⋅) so that 𝔼𝑥(𝑇)=𝜁. However, we can always decompose 𝑥(𝑡)=𝑥⁰(𝑡)+𝑦(𝑡) where 𝑦(⋅) satisfies (3.6). This leads to 𝔼𝑥⁰(𝑇)+𝔼𝑦(𝑇)=𝜁. However, 𝔼𝑥⁰(𝑇)≡𝜁⁰ is independent of 𝜋(⋅); thus it is necessary that there is a 𝜋(⋅) with 𝔼𝑦(𝑇)≠0. It follows then from (3.8) that (3.4) is valid.

Theorem 3.2. The mean-variance problem (2.19) is feasible for every 𝜁∈ℝ if and only if 𝔼^𝑇₀|𝐵(𝑡,𝛼(𝑡))|²𝑑𝑡>0. (3.9)

Proof. By virtue of Lemma (3.1), it suffices to prove that 𝜓(𝑡,𝑖)>0∀𝑡∈[0,𝑇],𝑖=1,2,…,𝑙. To this end, note that (3.3) can be rewritten as 𝜓̇(𝑡,𝑖)=[−𝑟(𝑡,𝑖)−𝑞_𝑖𝑖]𝜓(𝑡,𝑖)−^𝑙_𝑗≠𝑖𝑞_𝑖𝑗𝜓(𝑡,𝑗),𝜓(𝑇,𝑖)=1,𝑖=1,2,…,𝑙. (3.10) Treating this as a system of terminal-valued ODEs, a variation-of-constant formula yields 𝜓(𝑡,𝑖)=𝑒^{−∫𝑇𝑡[−𝑟(𝑠,𝑖)−𝑞𝑖𝑖]𝑑𝑠}+^𝑇_𝑡𝑒^{−∫𝑠𝑡[−𝑟(𝜏,𝑖)−𝑞𝑖𝑖𝑙]𝑑𝜏}_𝑗≠𝑖𝑞_𝑖𝑗𝜓(𝑠,𝑗)𝑑𝑠,𝑖=1,2,…,𝑙. (3.11) Construct a sequence 𝜓^(𝑘)(⋅,𝑖) (known as the Picard sequence) as follows: 𝜓⁽⁰⁾𝜓(𝑡,𝑖)=1,𝑡∈[0,𝑇],𝑖=1,2,…,𝑙,_(𝑘+1)(𝑡,𝑖)=𝑒^{−∫𝑇𝑡[−𝑟(𝑠,𝑖)−𝑞𝑖𝑖]𝑑𝑠}+^𝑇_𝑡𝑒^{−∫𝑠𝑡[−𝑟(𝜏,𝑖)−𝑞𝑖𝑖𝑙]𝑑𝜏}_𝑗≠𝑖𝑞_𝑖𝑗𝜓^(𝑘)(𝑠,𝑗)𝑑𝑠,𝑡∈[0,𝑇],𝑖=1,2,…,𝑙,𝑘=0,1,…. (3.12) Noting that 𝑞_𝑖𝑗≥0 for all 𝑗≠𝑖, we have 𝜓^(𝑘)(𝑡,𝑖)≥𝑒^{−∫𝑇𝑡[−𝑟(𝑠,𝑖)−𝑞𝑖𝑖]𝑑𝑠}>0,𝑘=0,1,…. (3.13) On the other hand, it is well known that 𝜓(𝑡,𝑖) is the limit of the Picard sequence 𝜓^(𝑘)(𝑡,𝑖) as 𝑘→∞. Thus 𝜓(𝑡,𝑖)>0. This proves the desired result.

Corollary 3.3. If (3.9) holds, then for any 𝜁∈ℝ, an admissible portfolio that satisfies 𝔼𝑥(𝑇)=𝜁 is given by 𝜋(𝑡)=𝜁−𝜁⁰𝜚𝐵(𝑡,𝛼(𝑡))^′𝜓(𝑡,𝛼(𝑡)), (3.14) where 𝑥⁰ and 𝜚 are given by (3.2) and (3.4), respectively.

Proof. This is immediate from the proof of the “if" part of Lemma (3.1) 𝔼𝑥(𝑇)=𝜁=𝑥⁰(𝑇)+𝔼𝑦(𝑇),𝜁−𝜁⁰=𝔼𝑦(𝑇)=𝔼^𝑇₀𝜓(𝑡,𝛼(𝑡))𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝑑𝑡. (3.15) Then one has 𝜋(𝑡)=𝜁−𝜁⁰𝜚𝐵(𝑡,𝛼(𝑡))^′𝜓(𝑡,𝛼(𝑡)). (3.16)

Corollary 3.4. If 𝔼∫^𝑇₀|𝐵(𝑡,𝛼(𝑡))|²𝑑𝑡=0, then any admissible portfolio 𝜋(⋅) results in 𝔼𝑥(𝑇)=𝜁⁰.

Proof. This is seen from the proof of the “only if" part of Lemma (3.1) 𝔼𝑥(𝑇)=𝔼𝑥⁰(𝑇)+𝔼𝑦(𝑇)=𝜁⁰+𝜓(𝑡,𝛼(𝑡))𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)𝑑𝑡=𝜁⁰ (3.17) since 𝔼∫^𝑇₀|𝐵(𝑡,𝛼(𝑡))|²𝑑𝑡=0.

Remark 3.5. Condition (3.9) is very mild. For example, (3.9) holds as long as there is one stock whose appreciation-rate process is different from the interest-rate process at any market mode, which is obviously a practically reasonable assumption. On the other hand, if (3.9) fails, then Corollary (3.4) implies that the mean-variance problem (2.19) is feasible only if 𝜁=𝜁⁰. This is pathological and trivial case that does not warrant further consideration. Therefore, from this point on we will assume that (3.9) holds or, equivalently, the mean-variance problem (2.19) is feasible for any 𝜁.

Having addressed the issue of feasibility, we proceed with the study of optimality. The mean-variance problem (2.19) under consideration is a dynamic optimization problem with a constraint 𝔼𝑥(𝑇)=𝜁. To handle this constraint, we apply the Lagrange multiplier technique. Define𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)∶=𝔼{|𝑥(𝑇)−𝜁|²+2𝜆[𝑥(𝑇)−𝜁]}=𝔼[𝑥(𝑇)+𝜆−𝜁]²−𝜆²,𝜆∈ℝ. (3.18)

Our first goal is to solve the following unconstrained problem parameterized by the Lagrange multiplier 𝜆:minimize𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)=𝔼[𝑥(𝑇)+𝜆−𝜁]²−𝜆²,subjectto(𝑥(⋅),𝜋(⋅))admissible. (3.19) This turns out to be a Markov-modulated stochastic linear-quadratic optimal control problem, which will be solved in the next section.

4. Solution to the Unconstrained Problem

In this section we solve the unconstrained problem (3.19). Firstly define𝛾(𝑡,𝑖)∶=𝐵(𝑡,𝑖)[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝑖)^′,𝑖=1,2,…,𝑙. (4.1)

Consider the following two systems of ODEs:𝑃̇(𝑡,𝑖)=[𝛾(𝑡,𝑖)−2𝑟(𝑡,𝑖)]𝑃(𝑡,𝑖)−^𝑙_𝑗=1𝑞_𝑖𝑗̇1𝑃(𝑡,𝑗),0≤𝑡≤𝑇,𝑃(𝑇,𝑖)=1,𝑖=1,2,…,𝑙,𝐻(𝑡,𝑖)=𝑟(𝑡,𝑖)𝐻(𝑡,𝑖)−𝑃(𝑡,𝑖)^𝑙_𝑗=1𝑞_𝑖𝑗𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)],0≤𝑡≤𝑇,𝐻(𝑇,𝑖)=1,𝑖=1,2,…,𝑙. (4.2)(4.3) The existence and uniqueness of solutions to the above two systems of equations are evident as both are linear with uniformly bounded coefficients.

Proposition 4.1. The solutions of (4.2) and (4.3) must satisfy 𝑃(𝑡,𝑖)>0 and 0<𝐻(𝑡,𝑖)≤1, ∀𝑡∈[0,𝑇], 𝑖=1,2,…,𝑙. Moreover, if for a fixed 𝑖, 𝑟(𝑡,𝑖)>0, a.e., 𝑡∈[0,𝑇], then 𝐻(𝑡,𝑖)<1, ∀𝑡∈[0,𝑇).

Proof. The assertion 𝑃(𝑡,𝑖)>0 can be proved in exactly the same way as that of 𝜓(𝑡,𝑖)>0; see the proof of Theorem 3.2. Having proved the positivity of 𝑃(𝑡,𝑖), one can then show that 𝐻(𝑡,𝑖)>0 using the same argument because now 𝑃(𝑡,𝑗)/𝑃(𝑡,𝑖)>0.
To prove that 𝐻(𝑡,𝑖)≤1, first note that the following system of ODEs:𝑑1𝑑𝑡𝐻(𝑡,𝑖)=−𝑃(𝑡,𝑖)^𝑙_𝑗=1𝑞_𝑖𝑗𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)],𝐻(𝑇,𝑖)=1,𝑖=1,2,…,𝑙, (4.4) has the only solutions 𝐻(𝑡,𝑖)≡1,  𝑖=1,2,…,𝑙, due to the uniqueness of solutions. Set 𝐻(𝑡,𝑖)∶=𝐻(𝑡,𝑖)−𝐻(𝑡,𝑖)≡1−𝐻(𝑡,𝑖), (4.5) which solves the following equations: 𝑑1𝑑𝑡𝐻(𝑡,𝑖)=𝑟(𝑡,𝑖)𝐻(𝑡,𝑖)−𝑟(𝑡,𝑖)−𝑃(𝑡,𝑖)^𝑙_𝑗=11𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)]=[𝑟(𝑡,𝑖)+𝑃(𝑡,𝑖)^𝑙_𝑗≠𝑖1𝑃(𝑡,𝑗)]𝐻(𝑡,𝑖)−𝑟(𝑡,𝑖)−𝑃(𝑡,𝑖)^𝑙_𝑗=1𝑃(𝑡,𝑗)𝐻(𝑡,𝑗),𝐻(𝑇,𝑖)=0,𝑖=1,2,…,𝑙. (4.6) A variation-of-constant formula leads to 𝐻(𝑡,𝑖)=∫^𝑇_𝑡𝑒^{−∫𝑠𝑡[𝑟(𝜏,𝑖)+(1/𝑃(𝜏,𝑖))∑𝑙𝑗≠𝑖𝑃(𝜏,𝑗)]𝑑𝜏}1[𝑟(𝑠,𝑖)+𝑃(𝑠,𝑖)^𝑙_𝑗∑₌₁𝑃(𝑠,𝑗)𝐻(𝑠,𝑗)]𝑑𝑠. (4.7) A similar trick using the construction of Picard's sequence yields that 𝐻(𝑡,𝑖)≥0. In addition, 𝐻(𝑡,𝑖)>0, ∀𝑡∈[0,𝑇), if 𝑟(𝑡,𝑖)>0, a.e., 𝑡∈[0,𝑇]. The desired result then follows from the fact that 𝐻(𝑡,𝑖)=1−𝐻(𝑡,𝑖).

Remark 4.2. Equation (4.2) is a Riccati type equation that arises naturally in studying the stochastic LQ control problem (3.19) whereas (4.3) is used to handle the nonhomogeneous terms involved in (3.19); see the proof of Theorem 4.3. On the other hand, 𝐻(𝑡,𝑖) has a financial interpretation: for fixed (𝑡,𝑖), 𝐻(𝑡,𝑖) is a deterministic quantity representing the risk-adjusted discount factor at time 𝑡 when the market mode is 𝑖 (note that the interest rate itself is random).

Theorem 4.3. Problem (3.19) has an optimal feedback control 𝜋^∗(𝑡,𝑥,𝑖)=−[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝑖)^′[𝑥+(𝜆−𝜁)𝐻(𝑡,𝑖)]. (4.8) Moreover, the corresponding optimal value is inf_{𝜋(⋅)admissible}𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)=[𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1](𝜆−𝜁)²+2[𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑥₀−𝜁](𝜆−𝜁)+𝑃(0,𝑖₀)𝑥²₀−𝜁², (4.9) where 𝜃∶=𝔼^𝑇₀^𝑙_𝑗=1𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝛼(𝑡))]²=𝑑𝑡_{𝑙𝑙𝑖=1𝑗=1𝑇0}𝑃(𝑡,𝑗)𝑝_𝑖0𝑖(𝑡)𝑞_𝑖𝑗[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)]²𝑑𝑡≥0, (4.10) with the transition probabilities 𝑝_𝑖0𝑖(𝑡) given by (2.4).

Proof. Let 𝜋(⋅) be any admissible control and 𝑥(⋅)the corresponding state trajectory of (2.15). Applying the generalized Itô formula (Lemma 2.5) to 𝜑(𝑡,𝑥,𝑖)=𝑃(𝑡,𝑖)[𝑥+(𝜆−𝜁)𝐻(𝑡,𝑖)]², (4.11) we obtain 𝑑{𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]²=̇}𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]²̇+𝑑𝑡+2𝑃(𝑡,𝛼(𝑡))(𝜆−𝜁)[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]𝐻(𝑡,𝛼(𝑡))𝑑𝑡+2{𝑟(𝑡,𝛼(𝑡))𝑥(𝑡)+𝐵(𝑡,𝛼(𝑡))𝜋(𝑡)}×𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]𝑑𝑡_𝑙𝑗=1𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝑗)]²+1𝑑𝑡22𝑃(𝑡,𝛼(𝑡))𝜋(𝑡)^′[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′]𝜋(𝑡)𝑑𝑡+𝑃(𝑡,𝛼(𝑡))𝜋(𝑡)^′{_ℝ𝑛𝜌(𝑡,𝛼(𝑡),𝑧)𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)}𝜋(𝑡)𝑑𝑡+2𝑃(𝑡,𝛼(𝑡))𝑥(𝑡)²𝜋(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_{𝑛𝑘=1ℝ}𝑃(𝑡,𝛼(𝑡)){2[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)+𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)²}𝑑𝑁(𝑑𝑡,𝑑𝑧)+_ℝ{𝑃(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(0)+ℎ(𝛼(𝑡),(𝑙)))]²−𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]²}𝜇(𝑑𝑡,𝑑𝑙)=𝑃(𝑡,𝛼(𝑡)){𝜋(𝑡)^′[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝛼(𝑡),𝑧)𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)]𝜋(𝑡)+2𝜋(𝑡)^′𝐵(𝑡,𝛼(𝑡))^′[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]+𝛾(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]}𝑑𝑡+(𝜆−𝜁)^2𝑙_𝑗=1𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)]²𝑑𝑡+2𝑃(𝑡,𝛼(𝑡))𝑥(𝑡)²𝜋(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_{𝑛𝑘=1ℝ}𝑃(𝑡,𝛼(𝑡)){2[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)+𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)²+}𝑑𝑁(𝑑𝑡,𝑑𝑧)_ℝ{𝑃(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(0)+ℎ(𝛼(𝑡),(𝑙)))]²−𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]²}𝜇(𝑑𝑡,𝑑𝑙)=𝑃(𝑡,𝛼(𝑡))[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]^′[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝛼(𝑡))𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)]×[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]𝑑𝑡+(𝜆−𝜁)^2𝑙_𝑗=1𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝑖)]²𝑑𝑡+2𝑃(𝑡,𝛼(𝑡))𝑥(𝑡)²𝜋(𝑡)^′+𝜎(𝑡,𝛼(𝑡))𝑑𝑊(𝑡)_{𝑛𝑘=1ℝ}𝑃(𝑡,𝛼(𝑡)){2[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)+𝜌^(𝑘)(𝑡,𝛼(𝑡),𝑧)²+}𝑑𝑁(𝑑𝑡,𝑑𝑧)_ℝ{𝑃(𝑡,𝛼(0)+ℎ(𝛼(𝑡),𝑙))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(0)+ℎ(𝛼(𝑡),(𝑙)))]²−𝑃(𝑡,𝛼(𝑡))[𝑥(𝑡)+(𝜆−𝜁)𝐻(𝑡,𝛼(𝑡))]²}𝜇(𝑑𝑡,𝑑𝑙), (4.12)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) where 𝜋^∗(𝑡,𝑥,𝑖) is defined as the right-hand side of (4.8). Integrating the above from 0 to 𝑇 and taking expectations, we obtain 𝔼[𝑥(𝑇)+𝜆−𝜁]²=𝑃(0,𝑖₀)[𝑥₀+(𝜆−𝜁)𝐻(0,𝑖₀)]²+𝜃(𝜆−𝜁)²+𝔼^𝑇₀𝑃(𝑡,𝛼(𝑡))[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]^′×[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝛼(𝑡),𝑧)𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)]×[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]𝑑𝑡. (4.13) Consequently, 𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)=𝔼[𝑥(𝑇)+𝜆−𝜁]²−𝜆²=[𝑃(0,𝑖₀)𝐻(0,𝑖₀)+𝜃−1](𝜆−𝜁)²+2[𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑥₀−𝜁](𝜆−𝜁)+𝑃(0,𝑖₀)𝑥²₀−𝜁²+𝔼^𝑇₀𝑃(𝑡,𝛼(𝑡))[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]^′×[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝛼(𝑡),𝑧)𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)]×[𝜋(𝑡)−𝜋^∗(𝑡,𝑥(𝑡),𝛼(𝑡))]𝑑𝑡. (4.14) Since 𝑃(𝑡,𝛼(𝑡))>0 by Proposition (4.1), it follows immediately that the optimal feedback control is given by (4.8) and the optimal value is given by (4.9), provided that the corresponding equation (2.15) under the feedback control (4.8) has a solution. But under (4.8), the system (2.15) is a nonhomogeneous linear SDE with coefficients modulated by 𝛼(𝑡). Since all the coefficients of this linear equation are uniformly bounded and 𝛼(𝑡) is independent of 𝑊(𝑡), the existence and uniqueness of the solution to the equation are straightforward based on a standard successive approximation scheme.
Finally, since𝜃∶=𝔼^𝑇₀^𝑙_𝑗≠𝑖𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝛼(𝑡))]²𝑑𝑡 (4.15) and 𝑞_𝑖𝑗≥0 for all  𝑖≠𝑗, we must have 𝜃≥0. This completes the proof.

5. Efficient Frontier

In this section we proceed to derive the efficient frontier for the original mean-variance problem (2.19).

Theorem 5.1 (efficient portfolios and efficient frontier). Assume that (3.9) holds. Then one has 𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1<0. (5.1) Moreover, the efficient portfolio corresponding to 𝑧, as a function of the time 𝑡, the wealth level 𝑥, and the market mode 𝑖, is 𝜋^∗(𝑡,𝑥,𝑖)=−[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝑖)^′[𝑥+(𝜆^∗−𝜁)𝐻(𝑡,𝑖)], (5.2) where 𝜆^∗=𝜁−𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑥₀𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1+𝜁. (5.3) Furthermore, the optimal value of Var𝑥(𝑇), among all the wealth processes 𝑥(⋅) satisfying 𝔼𝑥(𝑇)=𝜁, is Var𝑥^∗(𝑇)=𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃1−𝜃−𝑃(0,𝑖₀)𝐻(0,𝑖₀)²[𝜁−𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑃(0,𝑖₀)𝐻(0,𝑖₀)²𝑥+𝜃⁰]²+𝑃(0,𝑖₀)𝜃𝑃(0,𝑖₀)𝐻(0,𝑖₀)²𝑥+𝜃²⁰. (5.4)

Proof. By assumption (3.9) and Theorem 3.2, the mean-variance problem (2.19) is feasible for any 𝜁∈ℝ¹. Moreover, using exactly the same approach in the proof of Theorem 4.3, one can show that problem (2.19) without the constraint 𝔼𝑥(𝑇)=𝜁 must have a finite optimal value, hence so does the problem (2.19). Therefore, (2.19) is finite for any 𝜁∈ℝ¹. Now we need to prove that 𝐽_𝑀𝑉(𝑥₀,𝑖₀,𝜋(⋅)) is strictly convex in 𝜋(⋅). We can easily get 𝔼(2𝑥₁𝑥₂)≤𝔼(𝑥²₁+𝑥²₂),𝔼(2𝜅(1−𝜅)𝑥₁𝑥₂)≤𝔼(𝜅(1−𝜅)𝑥²₁+𝜅(1−𝜅)𝑥²₂),𝔼(𝜅²𝑥²₁+(1−𝜅)²𝑥²₂+2𝜅(1−𝜅)𝑥₁𝑥₂)≤𝔼(𝜅𝑥²₁+(1−𝜅)𝑥²₂),𝔼(𝜅𝑥₁+(1−𝜅)𝑥₂−𝜁)²≤𝔼(𝜅(𝑥₁−𝜁)²)+𝔼((1−𝜅)(𝑥₂−𝜁)²), (5.5) where 𝜅∈[0,1]. So, we obtain 𝔼(𝜅𝑥₁−𝜅𝜁+(1−𝜅)𝑥₂−(1−𝜅)𝜁)²≤𝔼(𝜅(𝑥₁−𝜁)²)+𝔼((1−𝜅)(𝑥₂−𝜁)²), (5.6) which proves 𝐽_𝑀𝑉(𝑥₀,𝑖₀,𝜋(⋅)) is strictly convex in 𝜋(⋅). that Affine space means the complement of points at infinity. It can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one. Since 𝐽_𝑀𝑉(𝑥₀,𝑖₀,𝜋(⋅)) is strictly convex in 𝜋(⋅) and the constraint function 𝔼𝑥(𝑇)−𝜁 is affine in 𝜋(⋅), we can apply the well-known duality theorem (see [26, page 224, Theorem 1]) to conclude that for any 𝜁∈ℝ¹, the optimal value of (2.19) is sup_𝜆∈ℝ1inf_{𝜋(⋅)admissible}𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)=max_𝜁∈ℝ1inf_{𝜋(⋅)admissible}(𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)+⟨𝜁,𝜁^∗⟩)>−∞. (5.7) By Theorem 4.3, inf_{𝜋(⋅)admissible}𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆) is a quadratic function (4.9) in 𝜆−𝜁. It follows from the finiteness of the supremum value of this quadratic function that 𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1≤0. (5.8) Now if 𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1=0, (5.9) then again by Theorem 4.3 and (5.7) we must have 𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑥₀−𝜁=0, (5.10) for every 𝜁∈ℝ¹, which is a contradiction. This proves (5.1). On the other hand, in view of (5.7), we maximize the quadratic function (4.9) over 𝜆−𝜁 and conclude that the maximizer is given by (5.3) whereas the maximum value is given by the right-hand side of (5.4). Finally, the optimal control (5.2) is obtained by (4.8) with 𝜆=𝜆^∗.The efficient frontier (5.4) reveals explicitly the tradeoff between the mean (return) and variance (risk) at the terminal. Quite contrary to the case without Markovian jumps [17], the efficient frontier in the present case is no longer a perfect square (or, equivalently, the efficient frontier in the mean-standard deviation diagram is no more a straight line). As a consequence, one is not able to achieve a risk-free investment. This, certainly, is expected since now the interest rate process is modulated by the Markov chain, and the interest rate risk cannot be perfectly hedged by any portfolio consisting of the bank account and stocks [27], because the Markov chain is independent of the Brownian motion. Nevertheless, expression (5.4) does disclose the minimum variance, namely, the minimum possible terminal variance achievable by an admissible portfolio, along with the portfolio that attains this minimum variance.

Theorem 5.2 (minimum variance). The minimum terminal variance is Var𝑥^∗_min(𝑇)=𝑃(0,𝑖₀)𝜃𝑃(0,𝑖₀)𝐻(0,𝑖₀)²𝑥+𝜃²⁰≥0 (5.11) with the corresponding expected terminal wealth 𝜁_min∶=𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑃(0,𝑖₀)𝐻(0,𝑖₀)²𝑥+𝜃⁰ (5.12) and the corresponding Lagrange multiplier 𝜆^∗_min=0. Moreover, the portfolio that achieves the above minimum variance, as a function of the time 𝑡, the wealth level 𝑥, and the market mode 𝑖, is 𝜋^∗_min(𝑡,𝑥,𝑖)=−[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝑖)^′[𝑥−𝜁_min𝐻(𝑡,𝑖)]. (5.13)

Proof. The conclusions regarding (5.11) and (5.12) are evident in view of the efficient frontier (5.4). The assertion 𝜆^∗_min=0 can be verified via (5.3) and (5.12). Finally, (5.13) follows from (5.2).

Remark 5.3. As a consequence of the above theorem, the parameter 𝑠 can be restricted to 𝜁≥𝜁_min when one defines the efficient frontier for the mean-variance problem (2.19).

Theorem 5.4 (mutual fund theorem). Suppose that an efficient portfolio 𝜋^∗₁(⋅) is given by (5.2) corresponding to 𝜁=𝜁₁>𝜁_min. Then a portfolio 𝜋^∗(⋅) is efficient if and only if there is a 𝜇≥0 such that 𝜋^∗(𝑡)=(1−𝜇)𝜋^∗_min(𝑡)+𝜇𝜋^∗₁(𝑡),𝑡∈[0,𝑇], (5.14) where 𝜋^∗_min(⋅) is the minimum variance portfolio defined in Theorem 5.2.

Proof. We first prove the “if" part. Since both 𝜋^∗_min(⋅) and 𝜋^∗₁(⋅) are efficient, by the explicit expression of any efficient portfolio given by (5.2), 𝜋^∗(𝑡)=(1−𝜇)𝜋^∗₀(⋅)+𝜇𝜋^∗₁(𝑡) must be in the form of (5.2) corresponding to 𝜁=(1−𝜇)𝜁_min+𝜇𝜁₁ (also noting that 𝑥^∗(⋅) is linear in 𝜋^∗(⋅)). Hence 𝜋^∗(𝑡) must be efficient.
Conversely, suppose that 𝜋^∗(⋅) is efficient corresponding to a certain 𝜁≥𝜁_min. Write 𝜁=(1−𝜇)𝜁_min+𝜇𝜁₁ with some 𝜇≥0. Multiplying𝜋^∗_min(𝑡)=−[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝛼(𝑡))^′[𝑥^∗_min(𝑡)−𝜁_min𝐻(𝑡,𝛼(𝑡))] (5.15) by (1−𝜇), multiplying 𝜋^∗₁(𝑡)=−[𝜎(𝑡,𝛼(𝑡))𝜎(𝑡,𝛼(𝑡))^′+_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧)]⁻¹𝐵(𝑡,𝛼(𝑡))^′[𝑥^∗₁(𝑡)+(𝜆^∗₁−𝜁₁)𝐻(𝑡,𝛼(𝑡))] (5.16) by 𝜇, and summing them up, we obtain that (1−𝜇)𝜋^∗_min(𝑡)+𝜇𝜋^∗₁(𝑡) is represented by (5.2) with 𝑥^∗(𝑡)=(1−𝜇)𝑥^∗_min(𝑡)+𝜇𝑥^∗₁(𝑡) and 𝜁=(1−𝜇)𝜁_min+𝜇𝜁₁. This leads to (5.14).

Remark 5.5. The above mutual fund theorem implies that any investor needs only to invest in the minimum variance portfolio and another prespecified efficient portfolio in order to achieve the efficiency. Note that in the case where all the market parameters are deterministic [17], the corresponding mutual fund theorem becomes the one-fund theorem, which yields that any efficient portfolio is a combination of the bank account and a given efficient risky portfolio (known as the tangent fund). This is equivalent to the fact that the fractions of wealth among the stocks are the same among all efficient portfolios. However, in the present Markov-modulated case this feature is no longer available.

Since the wealth processes 𝑥(⋅) are with jumps, it is more complicated when we solve the unconstrained problem (3.19). Firstly, we aim to derive conditions of feasibility. It is not hard to prove feasibility of the constrained stochastic optimization problem (2.19), which we get the unconstrained problem (3.19) from. Then we solve the unconstrained problem (3.19). If we assume that 𝛾(𝑡,𝑖)∶=𝐵(𝑡,𝑖)[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′]⁻¹𝐵(𝑡,𝑖)^′𝜋,𝑖=1,2,…,𝑙,_∗(𝑡,𝑥,𝑖)∶=−[𝜎(𝑡,𝑖)𝜎(𝑡,𝑖)^′]⁻¹𝐵(𝑡,𝑖)^′[𝑥+(𝜆−𝜁)𝐻(𝑡,𝑖)], (5.17) we have inf_{𝜋(⋅)admissible}𝐽(𝑥₀,𝑖₀,𝜋(⋅),𝜆)=[𝑃(0,𝑖₀)𝐻(0,𝑖₀)²+𝜃−1](𝜆−𝜁)²+2[𝑃(0,𝑖₀)𝐻(0,𝑖₀)𝑥₀−𝜁](𝜆−𝜁)+𝑃(0,𝑖₀)𝑥²₀−𝜁², (5.18) where 𝜃∶=𝔼{^𝑇₀^𝑙_𝑗=1𝑞_{𝛼(𝑡)𝑗}𝑃(𝑡,𝑗)[𝐻(𝑡,𝑗)−𝐻(𝑡,𝛼(𝑡))]²1𝑑𝑡(𝜆−𝜁)²+𝑃(𝑡,𝛼(𝑡))𝜋(𝑡)^′{_ℝ𝑛𝜌(𝑡,𝛼(𝑡),𝑧)𝜌(𝑡,𝛼(𝑡),𝑧)^′𝜈(𝑑𝑧)}𝜋(𝑡)𝑑𝑡}. (5.19) So, we added one item ∫_ℝ𝑛𝜌(𝑡,𝑖,𝑧)𝜌(𝑡,𝑖,𝑧)^′𝜈(𝑑𝑧) in optimal feedback control 𝜋^∗(𝑡,𝑥,𝑖) (see (3.19)) to simplify the calculation.

Acknowledgments

The author would like to thank Dr. C. Yuan for his helpful comments and discussions. He would like to thank the referees for the careful reading of the first version of this paper.