# Portfolio Selection with Jumps under Regime Switching

Authors: Lin Zhao
Journal: International Journal of Stochastic Analysis
 (2010)

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## 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.


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## Body

## 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),…,Wd(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 by(2.1)Ñ(dt,dz)=(Ñ1(dt,dz1),…,Ñn(dt,dzn))'=(N1(dt,dz1)-ν1(dz1)dt,…,Nn(dt,dzn)-ν(dzn)dt)',
where Nj, j=1,…,n, are independent 1-dimensional Poisson random measures with characteristic measure νj,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=(qij)l×l given by(2.2)P{α(t+Δ)=j∣α(t)=i}={qijΔ+o(Δ),ifi≠j,1+qiiΔ+o(Δ),ifi=j,
where Δ>0. Here qij≥0 is the transition rate from i to j if i≠j while(2.3)qii=-∑j≠iqij,
and stationary transition probabilities
(2.4)pij(t)=P(α(t)=j∣α(0)=i),t≥0,i,j=1,2,…,l.
Define ℱt=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 Lℱ(0,T;ℝm)2 the set of all ℝm-valued, measurable stochastic processes f(t) adapted to {ℱt}t≥0, such that 𝔼∫0T|f(t)|2dt<+∞.Consider a market in whichd+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:(2.5)dP0(t)=r(t,α(t))P0(t)dt,t∈[0,T],P0(0)=p0>0,
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 Pm(t), m=1,2,…,d, satisfy the following system of stochastic differential equations (SDEs):(2.6)dPm(t)=Pm(t){bm(t,α(t))dt+∑n=1dσmn(t,α(t))dWn(t)+∑j=1n∫ℝρmj(t,α(t),zj)Ñj(dt,dzj)},t∈[0,T],Pm(0)=pm>0,
where for each i=1,2,…,l, b:[0,T]×𝕊→ℝd×1, σ:[0,T]×𝕊→ℝd×d, ρ:[0,T]×𝕊×ℝn→ℝd×n is the appreciation rate process, and σm(t,i):=(σm1(t,i),…,σmd(t,i)) are adapted processes such that the integrals exist. And each column ρ(k) of the d×n matrix ρ=[ρij] depends on z only through the kth coordinate zk, that is, (2.7)ρ(k)(t,i,z)=ρ(k)(t,i,zk),z=(z1,…,zn)∈ℝn.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 eachi=1,…,l,(2.8)σ(t,i):=(σ1(t,i)⋮σd(t,i))≡(σmn(t,i))d×d,b(t,i)=(b1(t,i)⋮bd(t,i))∈ℝd×1,ρ(t,i,z)=(ρ1(t,i,z)⋮ρd(t,i,z))∈ℝd×n,
where (2.9)ρm(t,i,z)=(ρm1(t,i,z),…,ρmn(t,i,z)).We assume throughout this paper that the following nondegeneracy condition:(2.10)σ(t,i)σ(t,i)′≥δI,∀t∈[0,T],i=1,2,…,l,
is satisfied for some δ>0. We also assume that all the functions r(t,i), bm(t,i), and σmn(t,i), ρmn(t,i,z) are measurable and uniformly bounded in t.Suppose that the initial market modeα(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),…,πd(t)) exists, where π0(t) is the money invested in the bond, and πi(t) is the money invested in the ith stock. Then (2.11)x(t)=∑i=0dπi(t)=∑i=0dηi(t)Pi(t),x(0)=x0,
where η0(t) is the number of bond units bought by the investor, and ηi(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 (2.12)dx(t)=η0(t)dP0(t)+∑i=1dηi(t)dPi(t).
Now substituting (2.5) and (2.6) into the above equation, after a simple calculation we arrive at(2.13)dx(t)=r(t,α(t))x(t)dt+∑m=1dπm(t)(bm(t,α(t))-r(t,α(t)))dt+∑m=1d∑n=1dπm(t)σmn(t,α(t))dWn(t)+∑m=1d∑j=1n∫ℝπm(t)ρmj(t,α(t),zj)Ñj(dt,dzj),x(0)=x0>0,α(0)=i0,
where π(t)=(π1(t),…,πd(t))' which we call a portfolio of the agent. And πm(t) is the total market value of the agent's wealth in the mth asset, m=0,1,…,d, at time t.Setting(2.14)B(t,i):=(b1(t,i)-r(t,i),…,bd(t,i)-r(t,i)),i=1,2,…,l,
we can rewrite the wealth equation (2.13) as(2.15)dx(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)Ñ(dt,dz),x(0)=x0>0,α(0)=i0.Definition 2.2.
A portfolioπ(·) is said to be admissible if π(·)∈Lℱ2(0,T;ℝd) 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, sayπ(·), as the fractions of wealth allocated to different stocks. That is,
(2.16)u(t)=π(t)x(t),t∈[0,T].With this definition, (2.15) can be rewritten as(2.17)dx(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)Ñ(dt,dz),x(0)=x0>0,α(0)=i0.
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 portfolioπ(·) 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 wealth(2.18)Varx(T)≡𝔼[x(T)-𝔼x(T)]2=𝔼[x(T)-ζ]2
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 byζ∈ℝ1:
(2.19)minimizeJMV(x0,i0,π(·)):=𝔼[x(T)-ζ]2,subjectto{𝔼x(T)=ζ,(x(·),π(·))admissible.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 ofJMV(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 (σx(T),ζ)∈ℝ2 are interchangeably called an efficient point, where σx(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 Δij be consecutive, left closed, right open intervals of the real line each having length γij such that (2.20)Δ12=[0,q12),Δ13=[q12,q12+q13),⋮Δ1l=[∑j=2l-1q1j,∑j=2lq1j),Δ21=[∑j=2lq1j,∑j=2lq1j+q21),Δ23=[∑j=2lq1j+q21,∑j=2lq1j+γ21+q23),⋮Δ2l=[∑j=2lq1j+∑j=1,j≠2l-1q2j,∑j=2lq1j+∑j=1,j≠2lq2j).
For future use, we cite the generalized Itô lemma (see [1, 24, 25]) as the following lemma.Lemma 2.5.
Given ad-dimensional process y(·) satisfying
(2.21)dy(t)=f(t,y(t),α(t))dt+g(t,y(t),α(t))dW(t)+∫ℝnγ(t,y(t),α(t),z)Ñ(dt,dz),
where f,g, and γ satisfy Lipschitz condition with appropriate dimensions, each column γ(k) of the matrix γ=[γij] depends on z only through the kth coordinate zk. Let φ(t,x,i)∈C1,2([0,T]×ℝn×S;ℝ), one then has
(2.22)dφ(t,y(t),α(t))=Γφ(t,y(t),α(t))dt+φx(t,y(t),α(t))′g(t,y(t),α(t))dW(t)+∑k=1n∫ℝ{φ(t,y(t)+γ(k)(t,α(t),zk),α(t))-φ(t,y(t),α(t))‍-φx(t,y(t),α(t))'γ(k)(t,α(t),z)}ν(dzk)dt+∑k=1n∫ℝ{φ(t,y(t)+γ(k)(t,α(t),z),α(t))-φ(t,y(t),α(t))}Ñk(dt,dzk)+∫ℝ(φ(t,y(t),α(0)+h(α(t),l¯))-φ(t,y(t),α(t)))μ(dt,dl¯),
where
(2.23)Γφ(t,x,i):=φt(t,x,i)+φx(t,x,i)′f(t,x,i)+12trace[g(t,x,i)′φxx(t,x,i)g(t,x,i)]+∑j=1lqijφ(t,x,j)+∑k=1n∫ℝ{φ(t,y(t)+γ(k)(t,α(t),zk),α(t))-φ(t,y(t),α(t))-φx(t,y(t),α(t))′γ(k)(t,α(t),z)}ν(dzk),
where μ is a martingale measure,
(2.24)h(i,y)={j-i,ify∈Δij,0,otherwise,
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 processx0(·) satisfies(3.1)dx0(t)=r(t,α(t))x0(t)dt,x0(0)=x0>0,α(0)=i0,
with its expected terminal wealth(3.2)ζ0:=𝔼x0(T)=𝔼e∫0Tr(s,α(s))dsx0.Lemma 3.1.
Letψ(·,i), i=1,2,…,l, be the solutions to the following system of linear ordinary differential equations (ODEs):
(3.3)ψ̇(t,i)=-r(t,i)ψ(t,i)-∑j=1lqijψ(t,j),ψ(T,i)=1,i=1,2,…,l.
Then the mean-variance problem (2.19) is feasible for every ζ∈ℝ1 if and only if
(3.4)ϱ:=𝔼∫0T|ψ(t,α(t))B(t,α(t))|2dt>0.Proof.
To prove the “if" part, construct a family of admissible portfoliosπβ(·)=βπ(·) for β∈ℝ1 where
(3.5)π(t)=B(t,α(t))'ψ(t,α(t)).
Assume that xβ(t) is the solution of (2.15). Let xβ(t)=x0(t)+βy(t), where x0(·) satisfies (3.1), and y(·) is the solution to the following equation:
(3.6)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)Ñ(dt,dz),y(0)=0,α(0)=i0.
Therefore, problem (2.19) is feasible for every ζ∈ℝ1 if there exists β∈ℝ such that ζ=𝔼xβ(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
(3.7)d[ψ(t,α(t))y(t)]=ψ̇(t,α(t))y(t)dt+ψ(t,α(t))[r(t,α(t))y(t)+B(t,α(t))π(t)]dt+∑j=1lqα(t)jψ(t,j)y(t)dt+π(t)′σ(t,α(t))dW(t)+∑k=1n∫ℝ{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+∑k=1n∫ℝ{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)}Ñk(dt,dzk)+∫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-∑j=1lqα(t)jψ(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+∑j=1lqα(t)jψ(t,j)y(t)dt+π(t)′σ(t,α(t))dW(t)+∑k=1n∫ℝ{ψ(t,α(t))y(t)}Ñk(dt,dzk)+∫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)+∑k=1n∫ℝ{ψ(t,α(t))y(t)}Ñk(dt,dzk)+∫ℝ{ψ(t,α(0)+h(α(t),l¯))y(t)-ψ(t,α(t))y(t)}μ(dt,dl¯).
Integrating from 0 to T, taking expectation, and using (3.5), we obtain
(3.8)𝔼y(T)=𝔼∫0Tψ(t,α(t))B(t,α(t))π(t)dt=𝔼∫0T|ψ(t,α(t))B(t,α(t))|2dt.
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
(3.9)𝔼∫0T|B(t,α(t))|2dt>0.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
(3.10)ψ̇(t,i)=[-r(t,i)-qii]ψ(t,i)-∑j≠ilqijψ(t,j),ψ(T,i)=1,i=1,2,…,l.
Treating this as a system of terminal-valued ODEs, a variation-of-constant formula yields
(3.11)ψ(t,i)=e-∫tT[-r(s,i)-qii]ds+∫tTe-∫ts[-r(τ,i)-qii]dτ∑j≠ilqijψ(s,j)ds,i=1,2,…,l.
Construct a sequence ψ(k)(·,i) (known as the Picard sequence) as follows:
(3.12)ψ(0)(t,i)=1,t∈[0,T],i=1,2,…,l,ψ(k+1)(t,i)=e-∫tT[-r(s,i)-qii]ds+∫tTe-∫ts[-r(τ,i)-qii]dτ∑j≠ilqijψ(k)(s,j)ds,t∈[0,T],i=1,2,…,l,k=0,1,….
Noting that qij≥0 for all j≠i, we have
(3.13)ψ(k)(t,i)≥e-∫tT[-r(s,i)-qii]ds>0,k=0,1,….
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
(3.14)π(t)=ζ-ζ0ϱB(t,α(t))′ψ(t,α(t)),
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)
(3.15)𝔼x(T)=ζ=x0(T)+𝔼y(T),ζ-ζ0=𝔼y(T)=𝔼∫0Tψ(t,α(t))B(t,α(t))π(t)dt.
Then one has
(3.16)π(t)=ζ-ζ0ϱB(t,α(t))'ψ(t,α(t)).Corollary 3.4.
If𝔼∫0T|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)
(3.17)𝔼x(T)=𝔼x0(T)+𝔼y(T)=ζ0+ψ(t,α(t))B(t,α(t))π(t)dt=ζ0
since 𝔼∫0T|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. Define(3.18)J(x0,i0,π(·),λ):=𝔼{|x(T)-ζ|2+2λ[x(T)-ζ]}=𝔼[x(T)+λ-ζ]2-λ2,λ∈ℝ.Our first goal is to solve the following unconstrained problem parameterized by the Lagrange multiplierλ:(3.19)minimizeJ(x0,i0,π(·),λ)=𝔼[x(T)+λ-ζ]2-λ2,subjectto(x(·),π(·))admissible.
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 define(4.1)γ(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.Consider the following two systems of ODEs:(4.2)Ṗ(t,i)=[γ(t,i)-2r(t,i)]P(t,i)-∑j=1lqijP(t,j),0≤t≤T,P(T,i)=1,i=1,2,…,l,(4.3)Ḣ(t,i)=r(t,i)H(t,i)-1P(t,i)∑j=1lqijP(t,j)[H(t,j)-H(t,i)],0≤t≤T,H(T,i)=1,i=1,2,…,l.
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 assertionP(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 thatH(t,i)≤1, first note that the following system of ODEs:(4.4)ddtH̃(t,i)=-1P(t,i)∑j=1lqijP(t,j)[H̃(t,j)-H̃(t,i)],H̃(T,i)=1,i=1,2,…,l,
has the only solutions H̃(t,i)≡1,  i=1,2,…,l, due to the uniqueness of solutions. Set
(4.5)Ĥ(t,i):=H̃(t,i)-H(t,i)≡1-H(t,i),
which solves the following equations:
(4.6)ddtĤ(t,i)=r(t,i)Ĥ(t,i)-r(t,i)-1P(t,i)∑j=1lP(t,j)[Ĥ(t,j)-Ĥ(t,i)]=[r(t,i)+1P(t,i)∑j≠ilP(t,j)]Ĥ(t,i)-r(t,i)-1P(t,i)∑j=1lP(t,j)Ĥ(t,j),Ĥ(T,i)=0,i=1,2,…,l.
A variation-of-constant formula leads to
(4.7)Ĥ(t,i)=∫tT[r(s,i)+1P(s,i)∑j=1lP(s,j)Ĥ(s,j)]ds.e-∫ts[r(τ,i)+(1/P(τ,i))∑j≠ilP(τ,j)]dτ[r(s,i)+1P(s,i)∑j=1lP(s,j)Ĥ(s,j)]ds.
A similar trick using the construction of Picard's sequence yields that Ĥ(t,i)≥0. In addition, Ĥ(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 Ĥ(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
(4.8)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)].
Moreover, the corresponding optimal value is
(4.9)infπ(·)admissibleJ(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)x02-ζ2,
where
(4.10)θ:=𝔼∫0T∑j=1lqα(t)jP(t,j)[H(t,j)-H(t,α(t))]2dt=∑i=1l∑j=1l∫0TP(t,j)pi0i(t)qij[H(t,j)-H(t,i)]2dt≥0,
with the transition probabilities pi0i(t) given by (2.4).Proof.
Letπ(·) be any admissible control and x(·)the corresponding state trajectory of (2.15). Applying the generalized Itô formula (Lemma 2.5) to
(4.11)φ(t,x,i)=P(t,i)[x+(λ-ζ)H(t,i)]2,
we obtain
(4.12)d{P(t,α(t))[x(t)+(λ-ζ)H(t,α(t))]2}=Ṗ(t,α(t))[x(t)+(λ-ζ)H(t,α(t))]2dt+2P(t,α(t))(λ-ζ)[x(t)+(λ-ζ)H(t,α(t))]Ḣ(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+∑j=1lqα(t)jP(t,j)[x(t)+(λ-ζ)H(t,j)]2dt+122P(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)+∑k=1n∫ℝP(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}dÑ(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(t,α(0)+h(α(t),l¯))}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))][σ(t,α(t))σ(t,α(t))′+∫ℝnρ(t,α(t),z)ρ(t,α(t),z)′ν(dz)]}dt+(λ-ζ)2∑j=1lqα(t)jP(t,j)[H(t,j)-H(t,i)]2dt+2P(t,α(t))x(t)2π(t)'σ(t,α(t))dW(t)+∑k=1n∫ℝP(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}dÑ(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(t,α(0)+h(α(t),l¯))}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)2∑j=1lqα(t)jP(t,j)[H(t,j)-H(t,i)]2dt+2P(t,α(t))x(t)2π(t)'σ(t,α(t))dW(t)+∑k=1n∫ℝP(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}dÑ(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¯),
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
(4.13)𝔼[x(T)+λ-ζ]2=P(0,i0)[x0+(λ-ζ)H(0,i0)]2+θ(λ-ζ)2+𝔼∫0TP(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.
Consequently,
(4.14)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)x02-ζ2+𝔼∫0TP(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.
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, since(4.15)θ:=𝔼∫0T∑j≠ilqα(t)jP(t,j)[H(t,j)-H(t,α(t))]2dt
and qij≥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
(5.1)P(0,i0)H(0,i0)2+θ-1<0.
Moreover, the efficient portfolio corresponding to z, as a function of the time t, the wealth level x, and the market mode i, is
(5.2)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)],
where
(5.3)λ*=ζ-P(0,i0)H(0,i0)x0P(0,i0)H(0,i0)2+θ-1+ζ.
Furthermore, the optimal value of Varx(T), among all the wealth processes x(·) satisfying 𝔼x(T)=ζ, is
(5.4)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)2+θx0]2+P(0,i0)θP(0,i0)H(0,i0)2+θx02.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 JMV(x0,i0,π(·)) is strictly convex in π(·). We can easily get
(5.5)𝔼(2x1x2)≤𝔼(x12+x22),𝔼(2κ(1-κ)x1x2)≤𝔼(κ(1-κ)x12+κ(1-κ)x22),𝔼(κ2x12+(1-κ)2x22+2κ(1-κ)x1x2)≤𝔼(κx12+(1-κ)x22),𝔼(κx1+(1-κ)x2-ζ)2≤𝔼(κ(x1-ζ)2)+𝔼((1-κ)(x2-ζ)2),
where κ∈[0,1]. So, we obtain
(5.6)𝔼(κx1-κζ+(1-κ)x2-(1-κ)ζ)2≤𝔼(κ(x1-ζ)2)+𝔼((1-κ)(x2-ζ)2),
which proves JMV(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 JMV(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
(5.7)supλ∈ℝ1infπ(·)admissibleJ(x0,i0,π(·),λ)=maxζ∈ℝ1infπ(·)admissible(J(x0,i0,π(·),λ)+〈ζ,ζ*〉)>-∞.
By Theorem 4.3, infπ(·)admissibleJ(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
(5.8)P(0,i0)H(0,i0)2+θ-1≤0.
Now if
(5.9)P(0,i0)H(0,i0)2+θ-1=0,
then again by Theorem 4.3 and (5.7) we must have
(5.10)P(0,i0)H(0,i0)x0-ζ=0,
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(5.11)Varxmin*(T)=P(0,i0)θP(0,i0)H(0,i0)2+θx02≥0
with the corresponding expected terminal wealth
(5.12)ζmin:=P(0,i0)H(0,i0)P(0,i0)H(0,i0)2+θx0
and the corresponding Lagrange multiplier λmin*=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
(5.13)πmin*(t,x,i)=-[σ(t,i)σ(t,i)′+∫ℝnρ(t,i,z)ρ(t,i,z)′ν(dz)]-1B(t,i)′[x-ζminH(t,i)].Proof.
The conclusions regarding (5.11) and (5.12) are evident in view of the efficient frontier (5.4). The assertion λmin*=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 parameters can be restricted to ζ≥ζmin when one defines the efficient frontier for the mean-variance problem (2.19).Theorem 5.4 (mutual fund theorem).
Suppose that an efficient portfolioπ1*(·) is given by (5.2) corresponding to ζ=ζ1>ζmin. Then a portfolio π*(·) is efficient if and only if there is a μ≥0 such that
(5.14)π*(t)=(1-μ)πmin*(t)+μπ1*(t),t∈[0,T],
where πmin*(·) is the minimum variance portfolio defined in Theorem 5.2.Proof.
We first prove the “if" part. Since bothπmin*(·) and π1*(·) are efficient, by the explicit expression of any efficient portfolio given by (5.2), π*(t)=(1-μ)π0*(·)+μπ1*(t) must be in the form of (5.2) corresponding to ζ=(1-μ)ζmin+μζ1 (also noting that x*(·) is linear in π*(·)). Hence π*(t) must be efficient.
Conversely, suppose thatπ*(·) is efficient corresponding to a certain ζ≥ζmin. Write ζ=(1-μ)ζmin+μζ1 with some μ≥0. Multiplying(5.15)πmin*(t)=-[σ(t,α(t))σ(t,α(t))′+∫ℝnρ(t,i,z)ρ(t,i,z)′ν(dz)]-1B(t,α(t))′[xmin*(t)-ζminH(t,α(t))]
by (1-μ), multiplying
(5.16)π1*(t)=-[σ(t,α(t))σ(t,α(t))′+∫ℝnρ(t,i,z)ρ(t,i,z)′ν(dz)]-1B(t,α(t))′[x1*(t)+(λ1*-ζ1)H(t,α(t))]
by μ, and summing them up, we obtain that (1-μ)πmin*(t)+μπ1*(t) is represented by (5.2) with x*(t)=(1-μ)xmin*(t)+μx1*(t) and ζ=(1-μ)ζmin+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 processesx(·) 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 (5.17)γ(t,i):=B(t,i)[σ(t,i)σ(t,i)']-1B(t,i)',i=1,2,…,l,π*(t,x,i):=-[σ(t,i)σ(t,i)′]-1B(t,i)′[x+(λ-ζ)H(t,i)],
we have (5.18)infπ(·)admissibleJ(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)x02-ζ2,
where (5.19)θ:=𝔼{∫0T∑j=1lqα(t)jP(t,j)[H(t,j)-H(t,α(t))]2dt1(λ-ζ)2+P(t,α(t))π(t)′{∫ℝnρ(t,α(t),z)ρ(t,α(t),z)′ν(dz)}π(t)dt}.
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.



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*Source: 697257-2010-07-28.xml*