# Equilibrium Time-Consistent Strategy for Corporate International Investment Problem with Mean-Variance Criterion

Authors: Jun Long; Sanyun Zeng
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

We analyze a continuous-time model for corporate international investment problem (CIIP) with mean-variance criterion. Based on Nash subgame perfect equilibrium theory, we define an infinitesimal operator and directly derive an extended Hamilton-Jacobi-Bellman (HJB) equation. Besides, we also obtain the equilibrium time-consistent strategy for CIIP. In addition, we discuss two cases of risk aversion coefficient; one is constant and the other is state dependent. Finally, the simulation results are given to illustrate our conclusions and the influence of some parameters on the optimal solution.


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## Body

## 1. Introduction

Generally, a continuous-time stochastic optimal control problem will lead to time inconsistency if the functional has the following form:(1)Jt,w,π=Et,w∫tTCw,Wsπ;πWsπds+Fw,WTπ+Gw,Et,wWTπ,where Ws is a controlled Markov process and Wt=w. The term π is a control applied at time s. The local utility function C, the terminal utility function F, and the nonlinear function G are given functions. The reasons are as follows:(1)
For a fixed initial point(t,Wt), we can determine the optimal control law π^ through maximizing the functional J(t,Wt,π). But the control law π^ at a later point (s,Ws) will no longer be optimal for the functional J(s,Ws,π). This will lead to a time-inconsistent problem.(2)
If the termEt,w[F(w,WTπ)] depends on the wealth Wt=w at time t, this will also lead to a time-inconsistent problem.(3)
Generally, the termG(w,Et,w[WTπ]) is not an expected value of a nonlinear function of terminal wealth but a nonlinear function of the expected value of terminal wealth, and this will also lead to a time-inconsistent problem.WhenC≡0, (1) becomes a mean-variance criterion problem. In this paper, we study corporate international investment problem with mean-variance criterion which is time inconsistent.Because time-inconsistent problem does not satisfy Bellman’s optimality principle, the standard dynamic programming approach fails. In order to deal with time inconsistency, there are three common ways to handle this problem:(1)
We dismiss the entire problem as being silly.(2)
We fix one initial point, for example,(0,W0), and then try to find the control law π^ which maximizes J(0,W0,π). We simply consider that, at a later time point such as (s,Ws), the control law π^ is still optimal for the functional J(s,Ws,π). This is well known as precommitment in the economics literature.(3)
Using game theory approach, we can seriously solve the time-inconsistent problem.The game theory approach which used Nash equilibrium points to handle time inconsistency has a long history which started with Strotz [1], in which a deterministic Ramsay problem was studied. Further works in continuous and discrete time along this line are provided in Goldman [2], Krusell and Smith [3], Peleg and Yaari [4], Pollak [5], Vieille and Weibull [6], Björk and Murgoci [7], and so on.More specifically, Björk and Murgoci [7] deduced an extended HJB equation in a discrete time setting and extended it to continuous-time setting by taking formal limits for the discrete problem. Hu et al. [8] studied a general time-inconsistent stochastic linear-quadratic (LQ) control problem and derived a sufficient condition for equilibrium control via a flow of forward-backward stochastic differential equations. Yong [9] studied a time-inconsistent optimal control problem and proved the existence of an equilibrium control via a forward ordinary differential equation coupled with a backward Riccati-Volterra integral equation. Yong [10] considered a general time-inconsistent optimal control problem for stochastic differential equations with deterministic coefficients. Under suitable conditions, they derived a Hamilton-Jacobi-Bellman equation for the equilibrium value function of the problem. Czichowsky [11] developed a time-consistent formulation of mean-variance portfolio problem and proved the convergence based on a global description of the locally optimal strategy in terms of the structure condition and the Föllmer-Schweizer decomposition of the mean-variance tradeoff. Yong [12] considered a linear-quadratic optimal control problem for mean-field stochastic differential equations with deterministic coefficients and presented open-loop and closed-loop equilibrium solutions for such kind of problems. Björk and Murgoci [13] studied the problem with state-dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time-consistent control was linear in wealth. Bensoussan et al. [14] studied the time-consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous-time settings. Using backward induction, the equilibrium control was shown to be linear in wealth. Wang and Wu [15] studied a partially observed recursive optimization problem. They established Kalman-Bucy filtering equations for a family of parameterized forward and backward stochastic differential equations and obtained the equilibrium control by means of backward separation technique. Cui et al. [16] investigated the effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets and derived the semianalytical expressions for the precommitted efficient mean-variance policy and the minimum-variance signed supermartingale measure (VSSM) and revealed their close relationship. Cui et al. [17] considered a continuous-time mean-variance portfolio optimization problem and derived an explicit time-consistent investment policy.In addition, there are many literatures concerning time consistency in other areas, for example, reinsurance (Zeng and Li [18], Li et al. [19], Zeng et al. [20], Y. Li and Z. Li [21], Zhao et al. [22], Zhao et al. [23], Li et al. [24], Li et al. [25], Liang and Song [26], and Li et al. [27]), asset-liability management (Wei et al. [28]), pricing (Pirvu and Zhang [29]), dividend strategies (Chen et al. [30]), and jumps (Zeng et al. [20] and Dang and Forsyth [31]).In recent years, there are some literatures on corporate international investment problem (CIIP). To be more specific, Choi [32] considered the CIIP in which the quantity of output is assumed to be unity, and the dynamic processes of price are assumed to follow Bachelier model. Based on Choi [32], Bellalah and Wu [33] assumed that the quantity of output is a function of output price and the dynamic processes follow geometric Brownian motion model. Their model accounted for the effects of exchange rates, taxes, and information costs on corporate international investment and assumed that firm maximized the present value of manager’s expected utility of net cash flows. Finally, they made a conclusion that information played a central role in foreign investments and explained the “home bias” in international finance. Besides, Wu and Zhang [34], D. Zhang and T. Zhang [35], and Huang and Zhang [36] also studied corporate international investment problem.However, these prior literatures did not take into account time inconsistency. So, our study tries to explore this problem. In other words, we study the time-inconsistent problem of corporate international investment and deduce a time-consistent strategy for it.This study contributes to the literature as follows. In the first place, based on Wu and Zhang [34] which considered firms only invested in a real project (home market) and money account and Bellalah and Wu [33] which considered firms only invested in home and foreign market, we investigate those who invest not only in home and foreign market, but also in money account. There is one more point; based on Nash subgame perfect equilibrium theory, we define an infinitesimal operator and directly derive an extended HJB system for the general optimization problem from the prospective of continuous time. Last but not least, combining Björk and Murgoci [7] with Bellalah and Wu [33], we obtain equilibrium time-consistent strategy for corporate international investment problem. In addition, inspired by Björk et al. [37] and Y. Li and Z. Li [21], we discuss not only constant risk aversion coefficient but also state-dependent risk aversion coefficient.This paper is organized as follows. In Section2, we introduce the general optimization problem with mean-variance criterion in continuous time and derive an extended HJB equation directly. In Section 3, we apply this method to corporate international investment problem (CIIP) and deduce the equilibrium time-consistent strategy for CIIP. Some examples illustrating the main features of model are explicitly solved in Section 4. Section 5 contains the main conclusions and some extensions. In the end, some technical details are provided in Appendix.


## 2. The General Optimization Problem with Mean-Variance in Continuous-Time Setting

Björk and Murgoci [7] discussed the time consistency in a discrete time setting and extended it to a continuous-time setting via limit theory, but their process is complicated. In this section, we focus on the continuous-time problem directly. Based on Nash subgame perfect equilibrium theory, we, applying the dynamic programming approach, define an infinitesimal operator and derive an extended HJB system which is consistent with Björk and Murgoci [7] in continuous time. Compared to Björk and Murgoci [7], our approach avoids the complicated transformation process from discrete time to continuous time. Therefore, our process is more simple and straightforward. To some extent, this study is a good supplement to Björk and Murgoci [7] in continuous time.In this section, we consider a controlled Markov process in continuous time on the time interval[0,T]. Define the measurable state space W and the measurable control space Π. The dynamic state process is given by (2)dWt=μt,Wt,πtdt+σt,Wt,πtdBt, where Wt∈W and πt∈Π.


### 2.1. Basic Problem Formulation

For initial point(t,w)∈[0,T]×W, with a fixed control law π, we consider the simplified functional(3)Jt,w,π=Et,wFw,WTπ+Gw,Et,wWTπ.Firstly, we introduce some definitions as follows.Definition 1.
For eachπ∈Π and efficient small number ε>0, an operator Pπ, operating on real valued functions of the form h(t,w), is defined by(4)Pπht,w=Eht+ε,Wt+επ∣Wt=w.A corresponding infinitesimal operator Aπ is defined by(5)Aπ=Pπ-I,where I is the identity operator.Therefore, the infinitesimal operatorAπ can be expressed as(6)Aπht,w=Et,wht+ε,Wt+επ-ht,w.The way that controls are influencing the dynamics of process is formalized by specifying the controlled infinitesimal generator ofWt.Definition 2.
Consider a control lawπ^ (informally viewed as a candidate equilibrium law). For any efficient small number ε>0, one chooses a fixed π∈Π and also fixes an arbitrarily chosen initial point (t,w). The control law πε is defined by (7)πεs,w=π,ifs∈t,t+ε,π^,ifs∈t+ε,T.
If(8)liminfε→0⁡Jt,Wt,π^-Jt,Wt,πεε≥0,for all π∈Π, one says that π^ is an equilibrium control law. The equilibrium value function V is defined by(9)Vt,Wt=Jt,Wt,π^.


### 2.2. Deriving the Extended HJB System

For convenient description, we use the symbol(t,w;WTπ) or (t,w,y;WTπ) to replace (t,w) or (t,w,y), although WTπ may be a determined scalar but not a variable.Denote that(10)ft,w,y;WTπ=Et,wFy,WTπ,gt,w;WTπ=Et,wWTπ,(11)fyt,w;WTπ=ft,w,y;WTπ,ifyis  fixed,Gwgt,y;WTπ=Gw,gt,y;WTπ,ifwis  fixed,(12)G⋄gt,w;ETπ=Gw,gt,w;ETπ,HgπGt,w;ETπ=GwPπgt,w;WTπ-Gwgt,w;WTπ.Then the value function at timet can be expressed as (13)Jt,w,π=ft,w,w;WTπ+Gw,gt,w;WTπ.Naturally, at the next time t+ε with efficient small ε>0, the corresponding value function can be expressed as (14)Jt+ε,wε,π=ft+ε,wε,wε;WTπ+Gwε,gt+ε,wε;WTπ.Taking expectations gives us(15)Et,wJt+ε,wε,π=Et,wft+ε,wε,wε;WTπ+Et,wGwε,gt+ε,wε;WTπ.By (6), we can construct the corresponding infinitesimal operators with respect to J, f, G, and g.Notice that(16)ft,w,w;WTπ=Et,wFw,WTπ=Et,wEt+ε,wεFw,WTπ=Et,wft+ε,wε,w;WTπ,gt,w;WTπ=Et,wWTπ=Et,wEt+ε,wεWTπ=Et,wgt+ε,wε;WTπ.Then, the value function at the time t can be expressed as(17)Jt,w,π=Et,wft+ε,wε,w;WTπ+Gw,Et,wgt+ε,wε;WTπ.Next, we construct the infinitesimal operatorAπJ(t,w,π). By (6), (15), and (17), we have(18)AπJt,w,π=Et,wJt+ε,wε,π-Jt,w,π=Et,wft+ε,wε,wε;WTπ+Et,wGwε,gt+ε,wε;WTπ-Et,wft+ε,wε,w;WTπ-Gw,Et,wgt+ε,wε;WTπ.Firstly, notice the three termsEt,w[f(t+ε,wε,wε;WTπ)], Et,w[f(t+ε,wε,w;WTπ)], and Et,w[G(wε,g(t+ε,wε;WTπ))] in (18); we could construct three corresponding infinitesimal operators Aπf(t,w,w;WTπ), Aπfw(t,w;WTπ), and AπGw,gt,w;WTπ; that is,(19)Aπft,w,w;WTπ=Et,wft+ε,wε,wε;WTπ-ft,w,w;WTπ,Aπfwt,w;WTπ=Et,wft+ε,wε,w;WTπ-ft,w,w;WTπ,AπGw,gt,w;WTπ=Et,wGwε,gt+ε,wε;WTπ-Gw,gt,w;WTπ.Secondly, notice the rest termG(w,Et,w[g(t+ε,wε;WTπ)]) in (18), which can be expressed as G(w,Et,w[g(t+ε,wε;WTπ)])=Gw(Et,w[g(t+ε,wε;WTπ)]). By (11)-(12), we can construct another operator(20)HgπGt,w;ETπ=GwEt,wgt+ε,wε;WTπ-Gwgt,w;WTπ.Therefore, by (18)–(20), the infinitesimal operator AπJ(t,w,π) can be expressed as(21)AπJt,w,π=Aπft,w,w;WTπ-Aπfwt,w;WTπ+AπG⋄gt,w;WTπ-HgπGt,w;WTπ.Collecting all results and noticing the factAπ^V(t,w)=Aπ^J(t,w,π^) which is given in (9), we arrive at the extended HJB system which is consistent with Björk and Murgoci [7].Proposition 3.
The extended HJB system for the Nash equilibrium problem satisfies(22)supπ∈ΠAπVt,w-Aπft,w,w;WTπ+Aπfwt,w;WTπ-AπG⋄gt,w;WTπ+HgπGt,w;WTπ=0,t∈0,TAπ^fwt,w;WTπ=0,t∈0,TAπ^gt,w;WTπ=0,t∈0,TVT,w;WTπ=Fw,WTπ+Gw,WTπ,fwT,w;WTπ=Fw,WTπ,gT,w;WTπ=w.Here, π^ is the equilibrium control law which realizes the supremum in the first equation of (22), and fy, G⋄g, and HgπG are given in (10)–(12).Remark 4.
( 1 ) In the case when F(w,WTπ) does not depend on w, the terms are(23)Aπft,w,w;WTπ-Aπfwt,w;WTπ=0.
( 2 ) By the infinitesimal operator Aπh(t,w)=Et,w[h(t+ε,wε)]-h(t,w) and the first-order Taylor extension h(w0+Δw)=h(w0)+h′(w0)Δw+o(Δw), it is easy to see(24)GwEt,wgt+ε,wε;WTπ=Gwgt,w;WTπ+Aπgt,w;WTπ+oAπgt,w;WTπ=Gwgt,w;WTπ+Gywgt,w;WTπAπgt,w;WTπ+oAπg,where Gywy=∂Gw/∂yy.
(3) A simple calculation shows that (see Björk and Murgoci [7])(25)-AπGgt,w;WTπ+G′gt,w;WTπAπgt,w;WTπ=-12σ2t,w,πG′′gt,w;WTπgw2t,w;WTπ.
( 4 ) In the extended HJB system, the partial derivatives of f and g should be evaluated at (t,w,w;WTπ) and (t,w;WTπ), respectively.


## 2.1. Basic Problem Formulation

For initial point(t,w)∈[0,T]×W, with a fixed control law π, we consider the simplified functional(3)Jt,w,π=Et,wFw,WTπ+Gw,Et,wWTπ.Firstly, we introduce some definitions as follows.Definition 1.
For eachπ∈Π and efficient small number ε>0, an operator Pπ, operating on real valued functions of the form h(t,w), is defined by(4)Pπht,w=Eht+ε,Wt+επ∣Wt=w.A corresponding infinitesimal operator Aπ is defined by(5)Aπ=Pπ-I,where I is the identity operator.Therefore, the infinitesimal operatorAπ can be expressed as(6)Aπht,w=Et,wht+ε,Wt+επ-ht,w.The way that controls are influencing the dynamics of process is formalized by specifying the controlled infinitesimal generator ofWt.Definition 2.
Consider a control lawπ^ (informally viewed as a candidate equilibrium law). For any efficient small number ε>0, one chooses a fixed π∈Π and also fixes an arbitrarily chosen initial point (t,w). The control law πε is defined by (7)πεs,w=π,ifs∈t,t+ε,π^,ifs∈t+ε,T.
If(8)liminfε→0⁡Jt,Wt,π^-Jt,Wt,πεε≥0,for all π∈Π, one says that π^ is an equilibrium control law. The equilibrium value function V is defined by(9)Vt,Wt=Jt,Wt,π^.


## 2.2. Deriving the Extended HJB System

For convenient description, we use the symbol(t,w;WTπ) or (t,w,y;WTπ) to replace (t,w) or (t,w,y), although WTπ may be a determined scalar but not a variable.Denote that(10)ft,w,y;WTπ=Et,wFy,WTπ,gt,w;WTπ=Et,wWTπ,(11)fyt,w;WTπ=ft,w,y;WTπ,ifyis  fixed,Gwgt,y;WTπ=Gw,gt,y;WTπ,ifwis  fixed,(12)G⋄gt,w;ETπ=Gw,gt,w;ETπ,HgπGt,w;ETπ=GwPπgt,w;WTπ-Gwgt,w;WTπ.Then the value function at timet can be expressed as (13)Jt,w,π=ft,w,w;WTπ+Gw,gt,w;WTπ.Naturally, at the next time t+ε with efficient small ε>0, the corresponding value function can be expressed as (14)Jt+ε,wε,π=ft+ε,wε,wε;WTπ+Gwε,gt+ε,wε;WTπ.Taking expectations gives us(15)Et,wJt+ε,wε,π=Et,wft+ε,wε,wε;WTπ+Et,wGwε,gt+ε,wε;WTπ.By (6), we can construct the corresponding infinitesimal operators with respect to J, f, G, and g.Notice that(16)ft,w,w;WTπ=Et,wFw,WTπ=Et,wEt+ε,wεFw,WTπ=Et,wft+ε,wε,w;WTπ,gt,w;WTπ=Et,wWTπ=Et,wEt+ε,wεWTπ=Et,wgt+ε,wε;WTπ.Then, the value function at the time t can be expressed as(17)Jt,w,π=Et,wft+ε,wε,w;WTπ+Gw,Et,wgt+ε,wε;WTπ.Next, we construct the infinitesimal operatorAπJ(t,w,π). By (6), (15), and (17), we have(18)AπJt,w,π=Et,wJt+ε,wε,π-Jt,w,π=Et,wft+ε,wε,wε;WTπ+Et,wGwε,gt+ε,wε;WTπ-Et,wft+ε,wε,w;WTπ-Gw,Et,wgt+ε,wε;WTπ.Firstly, notice the three termsEt,w[f(t+ε,wε,wε;WTπ)], Et,w[f(t+ε,wε,w;WTπ)], and Et,w[G(wε,g(t+ε,wε;WTπ))] in (18); we could construct three corresponding infinitesimal operators Aπf(t,w,w;WTπ), Aπfw(t,w;WTπ), and AπGw,gt,w;WTπ; that is,(19)Aπft,w,w;WTπ=Et,wft+ε,wε,wε;WTπ-ft,w,w;WTπ,Aπfwt,w;WTπ=Et,wft+ε,wε,w;WTπ-ft,w,w;WTπ,AπGw,gt,w;WTπ=Et,wGwε,gt+ε,wε;WTπ-Gw,gt,w;WTπ.Secondly, notice the rest termG(w,Et,w[g(t+ε,wε;WTπ)]) in (18), which can be expressed as G(w,Et,w[g(t+ε,wε;WTπ)])=Gw(Et,w[g(t+ε,wε;WTπ)]). By (11)-(12), we can construct another operator(20)HgπGt,w;ETπ=GwEt,wgt+ε,wε;WTπ-Gwgt,w;WTπ.Therefore, by (18)–(20), the infinitesimal operator AπJ(t,w,π) can be expressed as(21)AπJt,w,π=Aπft,w,w;WTπ-Aπfwt,w;WTπ+AπG⋄gt,w;WTπ-HgπGt,w;WTπ.Collecting all results and noticing the factAπ^V(t,w)=Aπ^J(t,w,π^) which is given in (9), we arrive at the extended HJB system which is consistent with Björk and Murgoci [7].Proposition 3.
The extended HJB system for the Nash equilibrium problem satisfies(22)supπ∈ΠAπVt,w-Aπft,w,w;WTπ+Aπfwt,w;WTπ-AπG⋄gt,w;WTπ+HgπGt,w;WTπ=0,t∈0,TAπ^fwt,w;WTπ=0,t∈0,TAπ^gt,w;WTπ=0,t∈0,TVT,w;WTπ=Fw,WTπ+Gw,WTπ,fwT,w;WTπ=Fw,WTπ,gT,w;WTπ=w.Here, π^ is the equilibrium control law which realizes the supremum in the first equation of (22), and fy, G⋄g, and HgπG are given in (10)–(12).Remark 4.
( 1 ) In the case when F(w,WTπ) does not depend on w, the terms are(23)Aπft,w,w;WTπ-Aπfwt,w;WTπ=0.
( 2 ) By the infinitesimal operator Aπh(t,w)=Et,w[h(t+ε,wε)]-h(t,w) and the first-order Taylor extension h(w0+Δw)=h(w0)+h′(w0)Δw+o(Δw), it is easy to see(24)GwEt,wgt+ε,wε;WTπ=Gwgt,w;WTπ+Aπgt,w;WTπ+oAπgt,w;WTπ=Gwgt,w;WTπ+Gywgt,w;WTπAπgt,w;WTπ+oAπg,where Gywy=∂Gw/∂yy.
(3) A simple calculation shows that (see Björk and Murgoci [7])(25)-AπGgt,w;WTπ+G′gt,w;WTπAπgt,w;WTπ=-12σ2t,w,πG′′gt,w;WTπgw2t,w;WTπ.
( 4 ) In the extended HJB system, the partial derivatives of f and g should be evaluated at (t,w,w;WTπ) and (t,w;WTπ), respectively.


## 3. Corporate International Investment Problem with Mean-Variance Criterion

### 3.1. The Market Setting

Let(Ω,F,P) be a probability space equipped with a filtration F=(Ft)0≤t≤T satisfying the usual conditions; that is, (Ft)0≤t≤T is right-continuous and P-complete, where T is a positive finite constant representing the time horizon. Suppose that all stochastic processes and random variables are defined on the filtered probability space (Ω,F,F,P).


### 3.2. Corporate International Investment Problem (CIIP)

The purpose of this paper is to study corporate international investment problem (CIIP) that a firm invests in both bank and real project which contains two markets: home market and foreign market. We consider a risk-free asset with dynamic price process(26)dS0t=rS0tdt,where r is constant short rate.Choi [32] supposed a “two-country” firm whose static cash flows for time t from its home and foreign operations could be written in a standard way (27)Rt=Pt-CtQt,Rt∗=etPt∗-Ct∗Qt∗,where Pt(Pt∗) and Ct(Ct∗) were uncertain output and input prices in home (foreign) market, Qt(Qt∗) was the quantity of output which was assumed to be certain, and et is uncertain exchange rate.Contrary to the model of Choi [32], Bellalah and Wu [33] took the attitude that the quantity of output was a decreasing function of the output price, and the cash flows of firm’s activity were taxed at a certain rate. Thus they considered that a “two-country” firm’s static cash flows from its home (Rt) and foreign (Rt∗) operations could be written in the following form:(28)Rt=1-τPt-CtQt,Qt=Ptβ,Rt∗=et1-τ∗Pt∗-Ct∗Qt∗,Qt∗=Pt∗β∗,where the terms β and β∗ are negative constants. τ and τ∗ are tax rates in home and foreign market, respectively.Similar to Bellalah and Wu [33], we assume the following dynamics for output price, input price, and exchange rate:(29)dPtPt=μpdt+σdBt1,dCtCt=μcdt+σdBt1,dPt∗Pt∗=μp∗dt+σ∗dBt2,dCt∗Ct∗=μc∗dt+σ∗dBt2,detet=μe+λedt+σedBt3.The initial values of above variables are P0, C0, P0∗, C0∗, and e0, respectively. The terms μp, μp∗, μc, μc∗, and μe are bounded constants which represent the instantaneous expected rates of these variables, respectively. λe reflects the information cost carried by managers in order to get information about foreign exchange markets. The terms σ, σ∗, and σe are the instantaneous volatility coefficients in home, foreign, and exchange market. The terms Bt1, Bt2, and Bt3 are three one-dimensional mutually dependent Brownian motions which represent the external sources of uncertainty in the markets with correlation coefficients ρ12, ρ13, and ρ23.The changes in the cash flows from home market can be written as (see AppendixA)(30)dRtRt=f^tdt+β+1σdBt1,where(31)f^t=12β+12σ2+β+1μp-12σ2+μp-μcC0eμctP0eμpt-C0eμct.However, managers should pay sunk costs in order to obtain information. Therefore, they require an additional return to get compensation for their resources engaged in the search for information. Information costs have the dimension of incremental rates of return which are added to the drift in the process describing the dynamics for different variables. So, the real changes in the cash flows from home market should be written as(32)dRtRt=f^t+λRdt+β+1σdBt1,R0=1-τP0βP0-C0,where λR is the information costs rate in home market.Similarly, the expression of changes in the cash flows from foreign market before exchanging can be written as(33)dR¯tR¯t=f¯tdt+β∗+1σ∗dBt2,where(34)f¯t=12β∗+12σ∗2+β∗+1μp∗-12σ∗2+μp∗-μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t.Then, fromR∗=etR¯t and (32) and (33), the expression of changes in the cash flows from foreign market after exchanging is given by(35)dRt∗Rt∗=f^∗t+μe+λedt+β∗+1σ∗dBt2+σedBt3,where(36)f^∗t=f¯t+β∗+1σ∗σeρ23.However, managers also should pay sunk costs in order to obtain information in foreign market. Thus, they also require an additional return to get compensation for their resources engaged in the search for information. So the real changes in the cash flows from foreign market can be expressed as(37)dRt∗Rt∗=f^∗t+λR∗+μe+λedt+β∗+1σ∗dBt2+σedBt3,R0∗=e01-τ∗P0β∗P0∗-C0∗.


### 3.3. The Wealth Process

Suppose thatWtπ is the corporate wealth at time t≥0. Let π1(t) represent the money invested in home market and π2(t) represent those invested in foreign market. Then, the rest money Wtπ-π1(t)-π2(t) is invested in home money account.From (32), (37), and bank account, we know that the corporate wealth can be written as(38)dWtπ=π1f^∗t-r+λR∗+μe+λe+π2f^t-r+λR+Wtπrdt+π1β+1σdBt1+π2β∗+1σ∗dBt2+π2σedBt3.We assume that the initial value of corporate wealthW0>0 and π=(π1,π2) is a locally measurable bounded process, almost surely.Definition 5.
A strategyπ=(π1,π2)0≤t≤T is said to be admissible if it satisfies the following conditions:(i)
Stochastic processπ=(π1,π2)0≤t≤T is Ft-progressively measurable(ii)
E[∫0T(π12+π22)dt]<+∞(iii)
The wealth process in (38) has a unique solution Wtπ on [0,T]For convenience, we will writeWtπ as Wt in next section.Next, we consider the CIIP with constant risk aversion coefficient and state-dependent risk aversion coefficient, respectively.


### 3.4. Corporate International Investment Problem (CIIP) with Constant Risk Aversion Coefficient

In this section, we consider the mean-variance model for corporate international investment problem:(39)maxπt∈ΠEt,wWTπ-γ2Vart,wWTπs.t.dWt=π1f^∗t-r+λR∗+μe+λe+π2f^t-r+λR+Wtrdt+π1β+1σdBt1+π2β∗+1σ∗dBt2+π2σedBt3,where Wt is the corporate wealth at time t>0, Et,w is the expected value operator under Wt=w conditions, and Vart,w is the variance operator under Wt=w conditions. The terms π1,π2 are the amounts of corporate capital budget invested in home and foreign market, respectively. And γ is the constant risk aversion coefficient which leads to aggressive or passive corporate investments. The primary motive of an aggressive company, which is usually a leader in the industry and a risk-taker, is to anticipate greater expected return for investing in foreign market. However, the motivation of a passive company, which is a follower and risk-avoider, is to maintain its existing market position rather than to increase it. Generally, this passive company reacts to the leader when the chance of foreign investment has been “proven” (see Choi [32]).Denote the value functional by(40)Jt,w,π=Et,wWTπ-γ2Vart,w⁡WTπ. The object is to maximize expected return with a risk penalty.According to the definition of conditional variance(41)Vart,w⁡WT=Et,wWT2-Et,w2WT, we can rewrite our value functional as (42)Jt,w,π=Et,wWTπ-γ2Vart,w⁡WTπ=Et,wWTπ-γ2Et,wWTπ2-Et,w2WTπ=Et,wWTπ-γ2WTπ2+γ2Et,wWTπ2≔Et,wFWTπ+GEt,wWTπ,where F(x)=x-(γ/2)x2 and G(x)=(γ/2)x2.From Proposition3 and (1)–(3) in Remark 4, the extended HJB system is then given by the following system.Proposition 6.
The extended HJB system for the Nash equilibrium problem satisfies(43)supπ∈Π⁡AπVt,w-12σ2w,πG′′gt,w;WTπgw2t,w;WTπ=0,VT,w=w,Aπ^gt,w;WTπ^=0,gT,w;WTπ^=w,where π^ is the equilibrium control law which realizes the supremum in the first equation of (43).Furthermore, we have the following HJB system for corporate international investment problem (CIIP).Proposition 7.
The extended HJB system for CIIP satisfies(44)maxπ∈ΠVt+VWWΔu1+12VWW-γgw2W2Δd1+πVWWΔu2-Δu1+12VWW-γgw2W2Δd2+12π2VWW-γgw2W2Δd3=0,VT,w=w,Au^gt,w;WTπ^=0,gT,w;WTπ^=w,where(45)Δu1=f^∗t-r+λR∗+μe+λe,Δu2=f^t-r+λR,Δd1=β+12σ2,Δd2=β+1β∗+1σσ∗ρ12+β+1σσeρ13,Δd3=β∗+12σ∗2+σe2+2β∗+1σ∗σeρ23,and π^ is the equilibrium control law which realizes the supremum in the first equation of (44).The termVt describes the growth rate of the manager’s value function. The term VW describes the marginal increase of stochastic return. The term VWW reflects the effects of stochastic return. The term Δd1=(β+1)2σ2 represents the home returns volatility. The term (β+1)(β∗+1)σσ∗ρ12 reflects the home and foreign returns correlation volatility. And the term (β+1)σσeρ13 shows the home returns and exchange rate correlation volatility. Therefore, the term Δd2 represents the sum of these two correlation volatilities. The term Δd3=(β∗+1)2(σ∗)2+σe2+2(β∗+1)σ∗σeρ23 represents the sum of foreign returns volatility, exchange rate, and their correlation volatility.By the optimal condition of (44), we can deduce an optimal strategy. Further, by conjecturing the form of value function and incorporating the optimal strategy into extended HJB system, we can obtain the value function.Theorem 8.
The equilibrium time-consistent strategy for corporate international investment problem (CIIP) is of the following forms:(46)π1∗t=1γerT-tΔm1,π2∗t=1γerT-tΔm2.And the value functions V and g are of the following forms:(47)Vt,w=erT-tw+1γ∫tTΛs-Γsds,gt,w=erT-tw+1γ∫tTΛsds,where(48)Λt=Δm1Δu1+Δm2Δu2,Γt=-12Δm12Δd1-Δm1Δm2Δd2-12Δm22Δd3,the terms Δu1, Δu2, Δd1, Δd2, and Δd3 are given in (45), and(49)Δm1=Δu1Δd3-Δu2Δd2Δd1Δd3-Δd22,Δm2=Δu2Δd1-Δu1Δd2Δd1Δd3-Δd22.The proof is given in AppendixB.If we denote thatM=(1/γ)(1/(Δd1Δd3-(Δd2)2))er(T-t) and by (46), we have (50)∂π1∗∂λR=-MΔd2,∂π1∗∂λR∗=MΔd3,∂π1∗∂μe=MΔd3,∂π1∗∂λe=MΔd3,∂π2∗∂λR=MΔd1,∂π2∗∂λR∗=-MΔd2,∂π2∗∂μe=-MΔd2,∂π2∗∂λe=-MΔd2.Note thatΔdi>0, i=1,2,3, so we can get the following arguments.Remark 9.
IfM≥0, π1∗ increases with respect to foreign market information cost λR∗, exchange market information cost λe, and exchange rate μe and decreases with respect to home market information cost λR. π2∗ decreases with respect to λR∗, λe, and μe and increases with respect to λR and vice versa.Now, we explain the economic meanings of the equilibrium time-consistent strategy. By (49), we have (51)Δm1=1Δd1Δd3-Δd22Δd3λR∗+λe-Δd2λR+μeΔd3-rΔd3-Δd2+Δd3μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd2μpP0eμptP0eμpt-C0eμct+β+Δd2μcC0eμctP0eμpt-C0eμct-Δd3μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t+12Δd3β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd2ββ+1σ2,Δm2=-1Δd1Δd3-Δd22Δd2λR∗+λe-Δd1λR+μeΔd2-rΔd2-Δd1+Δd2μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd1μpP0eμptP0eμpt-C0eμct+β+Δd1μcC0eμctP0eμpt-C0eμct-Δd2μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t+12Δd2β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd1ββ+1σ2. For simplifying, we denote that (52)H11=Δd3λR∗+λe-Δd2λR+μeΔd3-rΔd3-Δd2+Δd3μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd2μpP0eμptP0eμpt-C0eμct+β+Δd2μcC0eμctP0eμpt-C0eμct-Δd3μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t,H12=12Δd3β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd2ββ+1σ2,H21=Δd2λR∗+λe-Δd1λR+μeΔd2-rΔd2-Δd1+Δd2μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd1μpP0eμptP0eμpt-C0eμct+β+Δd1μcC0eμctP0eμpt-C0eμct-Δd2μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t,H22=12Δd2β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd1ββ+1σ2. Then, the equilibrium time-consistent strategy in (46) can be expressed as (53)π1∗t=MH11+H12,π2∗t=-MH21+H22. The terms MH11 and -MH21 are referred to as the “speculative demand” or the aggressive demand. The terms MH12 and -MH22 are called the “hedging demand.” They describe the demand for home and foreign investments on risk-hedged basis, respectively.These expressions of the speculative demand involve some foreign direct investment theories, some of which emphasize the cost advantage of host countries and advance the comparative cost savings offered by foreign production vis-a-vis home production. These are reflected in the termsΔd2μc(C0eμct/P0eμpt-C0eμct)-Δd3μc∗(C0∗eμc∗t/(P0∗eμp∗t-C0∗eμc∗t)) in H11 and Δd1μc(C0eμct/P0eμpt-C0eμct)-Δd2μc∗(C0∗eμc∗t/(P0∗eμp∗t-C0∗eμc∗t)) in H21. Other foreign direct investment theories, however, incorporate the demand side effect. In this case, foreign investment depends on whether output price is higher in foreign market than in home market, which is reflected in the terms Δd3μp∗(P0∗eμp∗t/(P0∗eμp∗t-C0∗eμc∗t)+β∗)-Δd2μp(P0eμpt/(P0eμpt-C0eμct)+β) in H11 and Δd2μp∗(P0∗eμp∗t/(P0∗eμp∗t-C0∗eμc∗t)+β∗)-Δd1μp(P0eμpt/(P0eμpt-C0eμct)+β) in H21. The investment flow is also affected by the exchange rate and the domestic short rate. The effect of exchange rate can be seen in the terms μeΔd3 in H11 and μeΔd2 in H21. The effect of short rate can be seen in the terms -r(Δd3-Δd2) in H11 and -r(Δd2-Δd1) in H21.We also take into account information costs as well as the degree of competition in home and foreign markets. In fact, if managers do not know about the foreign investment opportunity or they do not want to spend much information cost on foreign and exchange market, they will choose to invest in home market which is referred to as the “home bias puzzle.” Therefore, information plays an important role in foreign investment. It can help managers to make decisions about investment in foreign market. These effects can be seen in the termsΔd3(λR∗+λe)-Δd2λR in H11 and Δd2(λR∗+λe)-Δd1λR in H21.The hedging demand describes a portfolio story as in Choi [32] and Bellalah and Wu [33]. If the variability of home returns is higher and the correlation between home and foreign returns is lower, managers will increase the foreign investments. In this case, the investment flow is affected by the cross-covariances and the degree of competition in home and foreign markets. These effects are reflected in the terms H12 and H22.Remark 10.
We find that the optimal investment strategy does not depend on the wealth processWtπ∗.Inspired by Zeng et al. [20], we can derive the relationship between conditional expectation and conditional variance of terminal wealth for corporate international investment problem (CIIP).Theorem 11.
The efficient frontier of problem (39) at initial point (t,Wt) is given by(54)Et,wWTπ∗=erT-tWt-Vart,w⁡WTπ∗·∫tTΛsds2∫tTΓsds,where Λ(t) and Γ(t) are given in (48).The proof is given in AppendixC.Remark 12.
For all(t,Wt)∈[0,T]×W, the relationship between Et,w[WTπ∗] and Vart,w[WTπ∗] is given by (54). It is also called the efficient frontier of problem (39) at initial point (t,Wt) in the modern portfolio theory.


### 3.5. Corporate International Investment Problem with State-Dependent Risk Aversion Coefficient

In this section, we consider corporate international investment problem (CIIP) under mean-variance criterion with state-dependent risk aversion coefficient which is discussed in Björk et al. [37]; that is, the value functional is given by(55)Jt,w,π=Et,wWTπ-γw2Vart,w⁡WTπ,where γ(w)=γ/w.The target of manager is to find the equilibrium time-consistent strategy or optimal solutionπ∗=(π1∗,π2∗) and the equilibrium value function (56)Vt,w=Jt,w,π∗.As indicated in Section 2.2, for these questions, value function (55) can be written as (57)Jt,w,π=Et,wFw,WTπ+Gw,Et,wWTπ,where (58)Fx,y=y-γ2xy2,Gx,y=γ2xy2.Note that (59)Vt,w=Jt,w,π∗=Et,wWTπ∗-γ2wVart,w⁡WTπ∗.We recall the probabilistic interpretations (60)ft,w,y;WTπ∗=Et,wWTπ∗-γ2yEt,wWTπ∗2,gt,w;WTπ∗=Et,wWTπ∗.From these, we have (61)Vt,w=ft,w,w;WTπ∗+γ2wg2t,w;WTπ∗.By Proposition3, we get the following results after a large number of elementary calculations.Proposition 13.
With notations as above, the extended HJB system takes the form(62)ft+γwggt+supπ∈Π⁡fw+γwggwπ1Δu1+π2Δu2+Wtr+12fww+γwggwwπ12Δd1+2π1π2Δd2+π22Δd3=0, and the optimal solution is given by(63)π1∗=fw+γ/wggwfww+γ/wggwwΔu2Δd2-Δu1Δd3Δd1Δd3-Δd22=fw+γ/wggwfww+γ/wggwwΔm1,π2∗=fw+γ/wggwfww+γ/wggwwΔu1Δd2-Δu2Δd1Δd1Δd3-Δd22=fw+γ/wggwfww+γ/wggwwΔm2.The proof is given in AppendixD.Following Y. Li and Z. Li [21], we have two theorems as follows.Theorem 14.
The equilibrium time-consistent strategy for CIIP is given by(64)π1∗=c1tw+k1t,π2∗=c2tw+k2t,where c1(t), c2(t), k1(t), and k2(t) satisfy the following integral equations:(65)c1t=Δm1·1γe∫tTΨ1s-Φ1sds+e∫tT2Ψ1s-Φ1sds-1,c2t=Δm1·1γe∫tTΨ1s-Φ1sds+e∫tT2Ψ1s-Φ1sds-1,k1t=-Δm2·12-e2∫tTΨ1sds∫tTe-∫tsΦ1τdτΦ2sds,k2t=-Δm1·12-e2∫tTΨ1sds∫tTe-∫tsΦ1τdτΦ2sds,
with(66)Ψ1t=c1tΔu1+c2tΔu2+r,Ψ2t=k1tΔu1+k2tΔu2,Φ1t=2c1tΔu1+c2tΔu2+r+c12tβ+12σ2+c22tβ∗+12σ∗2+σe2+2c1tc2tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+2c22tβ∗+1σ∗σeρ23,Φ2t=2k1tΔu1+k2tΔu2+2c1tk1tβ+12σ2+2c2tk2tβ∗+12σ∗2+σe2+2c1tk2t+c2tk1tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+4c2tk2tβ∗+1σ∗σeρ23,Φ3t=k12tβ+12σ2+k22tβ∗+12σ∗2+σe2+k1tk2tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+k22tβ∗+1σ∗σeρ23, and the symbols Δu1,Δu2 are given in (45).The proof is given in AppendixE.Theorem 15.
Integral equations (65) admit a unique solution c1t,c2t,k1t,k2t∈C10,T∪C2[0,T].The proof can be seen in Y. Li and Z. Li [21].


## 3.1. The Market Setting

Let(Ω,F,P) be a probability space equipped with a filtration F=(Ft)0≤t≤T satisfying the usual conditions; that is, (Ft)0≤t≤T is right-continuous and P-complete, where T is a positive finite constant representing the time horizon. Suppose that all stochastic processes and random variables are defined on the filtered probability space (Ω,F,F,P).


## 3.2. Corporate International Investment Problem (CIIP)

The purpose of this paper is to study corporate international investment problem (CIIP) that a firm invests in both bank and real project which contains two markets: home market and foreign market. We consider a risk-free asset with dynamic price process(26)dS0t=rS0tdt,where r is constant short rate.Choi [32] supposed a “two-country” firm whose static cash flows for time t from its home and foreign operations could be written in a standard way (27)Rt=Pt-CtQt,Rt∗=etPt∗-Ct∗Qt∗,where Pt(Pt∗) and Ct(Ct∗) were uncertain output and input prices in home (foreign) market, Qt(Qt∗) was the quantity of output which was assumed to be certain, and et is uncertain exchange rate.Contrary to the model of Choi [32], Bellalah and Wu [33] took the attitude that the quantity of output was a decreasing function of the output price, and the cash flows of firm’s activity were taxed at a certain rate. Thus they considered that a “two-country” firm’s static cash flows from its home (Rt) and foreign (Rt∗) operations could be written in the following form:(28)Rt=1-τPt-CtQt,Qt=Ptβ,Rt∗=et1-τ∗Pt∗-Ct∗Qt∗,Qt∗=Pt∗β∗,where the terms β and β∗ are negative constants. τ and τ∗ are tax rates in home and foreign market, respectively.Similar to Bellalah and Wu [33], we assume the following dynamics for output price, input price, and exchange rate:(29)dPtPt=μpdt+σdBt1,dCtCt=μcdt+σdBt1,dPt∗Pt∗=μp∗dt+σ∗dBt2,dCt∗Ct∗=μc∗dt+σ∗dBt2,detet=μe+λedt+σedBt3.The initial values of above variables are P0, C0, P0∗, C0∗, and e0, respectively. The terms μp, μp∗, μc, μc∗, and μe are bounded constants which represent the instantaneous expected rates of these variables, respectively. λe reflects the information cost carried by managers in order to get information about foreign exchange markets. The terms σ, σ∗, and σe are the instantaneous volatility coefficients in home, foreign, and exchange market. The terms Bt1, Bt2, and Bt3 are three one-dimensional mutually dependent Brownian motions which represent the external sources of uncertainty in the markets with correlation coefficients ρ12, ρ13, and ρ23.The changes in the cash flows from home market can be written as (see AppendixA)(30)dRtRt=f^tdt+β+1σdBt1,where(31)f^t=12β+12σ2+β+1μp-12σ2+μp-μcC0eμctP0eμpt-C0eμct.However, managers should pay sunk costs in order to obtain information. Therefore, they require an additional return to get compensation for their resources engaged in the search for information. Information costs have the dimension of incremental rates of return which are added to the drift in the process describing the dynamics for different variables. So, the real changes in the cash flows from home market should be written as(32)dRtRt=f^t+λRdt+β+1σdBt1,R0=1-τP0βP0-C0,where λR is the information costs rate in home market.Similarly, the expression of changes in the cash flows from foreign market before exchanging can be written as(33)dR¯tR¯t=f¯tdt+β∗+1σ∗dBt2,where(34)f¯t=12β∗+12σ∗2+β∗+1μp∗-12σ∗2+μp∗-μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t.Then, fromR∗=etR¯t and (32) and (33), the expression of changes in the cash flows from foreign market after exchanging is given by(35)dRt∗Rt∗=f^∗t+μe+λedt+β∗+1σ∗dBt2+σedBt3,where(36)f^∗t=f¯t+β∗+1σ∗σeρ23.However, managers also should pay sunk costs in order to obtain information in foreign market. Thus, they also require an additional return to get compensation for their resources engaged in the search for information. So the real changes in the cash flows from foreign market can be expressed as(37)dRt∗Rt∗=f^∗t+λR∗+μe+λedt+β∗+1σ∗dBt2+σedBt3,R0∗=e01-τ∗P0β∗P0∗-C0∗.


## 3.3. The Wealth Process

Suppose thatWtπ is the corporate wealth at time t≥0. Let π1(t) represent the money invested in home market and π2(t) represent those invested in foreign market. Then, the rest money Wtπ-π1(t)-π2(t) is invested in home money account.From (32), (37), and bank account, we know that the corporate wealth can be written as(38)dWtπ=π1f^∗t-r+λR∗+μe+λe+π2f^t-r+λR+Wtπrdt+π1β+1σdBt1+π2β∗+1σ∗dBt2+π2σedBt3.We assume that the initial value of corporate wealthW0>0 and π=(π1,π2) is a locally measurable bounded process, almost surely.Definition 5.
A strategyπ=(π1,π2)0≤t≤T is said to be admissible if it satisfies the following conditions:(i)
Stochastic processπ=(π1,π2)0≤t≤T is Ft-progressively measurable(ii)
E[∫0T(π12+π22)dt]<+∞(iii)
The wealth process in (38) has a unique solution Wtπ on [0,T]For convenience, we will writeWtπ as Wt in next section.Next, we consider the CIIP with constant risk aversion coefficient and state-dependent risk aversion coefficient, respectively.


## 3.4. Corporate International Investment Problem (CIIP) with Constant Risk Aversion Coefficient

In this section, we consider the mean-variance model for corporate international investment problem:(39)maxπt∈ΠEt,wWTπ-γ2Vart,wWTπs.t.dWt=π1f^∗t-r+λR∗+μe+λe+π2f^t-r+λR+Wtrdt+π1β+1σdBt1+π2β∗+1σ∗dBt2+π2σedBt3,where Wt is the corporate wealth at time t>0, Et,w is the expected value operator under Wt=w conditions, and Vart,w is the variance operator under Wt=w conditions. The terms π1,π2 are the amounts of corporate capital budget invested in home and foreign market, respectively. And γ is the constant risk aversion coefficient which leads to aggressive or passive corporate investments. The primary motive of an aggressive company, which is usually a leader in the industry and a risk-taker, is to anticipate greater expected return for investing in foreign market. However, the motivation of a passive company, which is a follower and risk-avoider, is to maintain its existing market position rather than to increase it. Generally, this passive company reacts to the leader when the chance of foreign investment has been “proven” (see Choi [32]).Denote the value functional by(40)Jt,w,π=Et,wWTπ-γ2Vart,w⁡WTπ. The object is to maximize expected return with a risk penalty.According to the definition of conditional variance(41)Vart,w⁡WT=Et,wWT2-Et,w2WT, we can rewrite our value functional as (42)Jt,w,π=Et,wWTπ-γ2Vart,w⁡WTπ=Et,wWTπ-γ2Et,wWTπ2-Et,w2WTπ=Et,wWTπ-γ2WTπ2+γ2Et,wWTπ2≔Et,wFWTπ+GEt,wWTπ,where F(x)=x-(γ/2)x2 and G(x)=(γ/2)x2.From Proposition3 and (1)–(3) in Remark 4, the extended HJB system is then given by the following system.Proposition 6.
The extended HJB system for the Nash equilibrium problem satisfies(43)supπ∈Π⁡AπVt,w-12σ2w,πG′′gt,w;WTπgw2t,w;WTπ=0,VT,w=w,Aπ^gt,w;WTπ^=0,gT,w;WTπ^=w,where π^ is the equilibrium control law which realizes the supremum in the first equation of (43).Furthermore, we have the following HJB system for corporate international investment problem (CIIP).Proposition 7.
The extended HJB system for CIIP satisfies(44)maxπ∈ΠVt+VWWΔu1+12VWW-γgw2W2Δd1+πVWWΔu2-Δu1+12VWW-γgw2W2Δd2+12π2VWW-γgw2W2Δd3=0,VT,w=w,Au^gt,w;WTπ^=0,gT,w;WTπ^=w,where(45)Δu1=f^∗t-r+λR∗+μe+λe,Δu2=f^t-r+λR,Δd1=β+12σ2,Δd2=β+1β∗+1σσ∗ρ12+β+1σσeρ13,Δd3=β∗+12σ∗2+σe2+2β∗+1σ∗σeρ23,and π^ is the equilibrium control law which realizes the supremum in the first equation of (44).The termVt describes the growth rate of the manager’s value function. The term VW describes the marginal increase of stochastic return. The term VWW reflects the effects of stochastic return. The term Δd1=(β+1)2σ2 represents the home returns volatility. The term (β+1)(β∗+1)σσ∗ρ12 reflects the home and foreign returns correlation volatility. And the term (β+1)σσeρ13 shows the home returns and exchange rate correlation volatility. Therefore, the term Δd2 represents the sum of these two correlation volatilities. The term Δd3=(β∗+1)2(σ∗)2+σe2+2(β∗+1)σ∗σeρ23 represents the sum of foreign returns volatility, exchange rate, and their correlation volatility.By the optimal condition of (44), we can deduce an optimal strategy. Further, by conjecturing the form of value function and incorporating the optimal strategy into extended HJB system, we can obtain the value function.Theorem 8.
The equilibrium time-consistent strategy for corporate international investment problem (CIIP) is of the following forms:(46)π1∗t=1γerT-tΔm1,π2∗t=1γerT-tΔm2.And the value functions V and g are of the following forms:(47)Vt,w=erT-tw+1γ∫tTΛs-Γsds,gt,w=erT-tw+1γ∫tTΛsds,where(48)Λt=Δm1Δu1+Δm2Δu2,Γt=-12Δm12Δd1-Δm1Δm2Δd2-12Δm22Δd3,the terms Δu1, Δu2, Δd1, Δd2, and Δd3 are given in (45), and(49)Δm1=Δu1Δd3-Δu2Δd2Δd1Δd3-Δd22,Δm2=Δu2Δd1-Δu1Δd2Δd1Δd3-Δd22.The proof is given in AppendixB.If we denote thatM=(1/γ)(1/(Δd1Δd3-(Δd2)2))er(T-t) and by (46), we have (50)∂π1∗∂λR=-MΔd2,∂π1∗∂λR∗=MΔd3,∂π1∗∂μe=MΔd3,∂π1∗∂λe=MΔd3,∂π2∗∂λR=MΔd1,∂π2∗∂λR∗=-MΔd2,∂π2∗∂μe=-MΔd2,∂π2∗∂λe=-MΔd2.Note thatΔdi>0, i=1,2,3, so we can get the following arguments.Remark 9.
IfM≥0, π1∗ increases with respect to foreign market information cost λR∗, exchange market information cost λe, and exchange rate μe and decreases with respect to home market information cost λR. π2∗ decreases with respect to λR∗, λe, and μe and increases with respect to λR and vice versa.Now, we explain the economic meanings of the equilibrium time-consistent strategy. By (49), we have (51)Δm1=1Δd1Δd3-Δd22Δd3λR∗+λe-Δd2λR+μeΔd3-rΔd3-Δd2+Δd3μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd2μpP0eμptP0eμpt-C0eμct+β+Δd2μcC0eμctP0eμpt-C0eμct-Δd3μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t+12Δd3β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd2ββ+1σ2,Δm2=-1Δd1Δd3-Δd22Δd2λR∗+λe-Δd1λR+μeΔd2-rΔd2-Δd1+Δd2μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd1μpP0eμptP0eμpt-C0eμct+β+Δd1μcC0eμctP0eμpt-C0eμct-Δd2μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t+12Δd2β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd1ββ+1σ2. For simplifying, we denote that (52)H11=Δd3λR∗+λe-Δd2λR+μeΔd3-rΔd3-Δd2+Δd3μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd2μpP0eμptP0eμpt-C0eμct+β+Δd2μcC0eμctP0eμpt-C0eμct-Δd3μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t,H12=12Δd3β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd2ββ+1σ2,H21=Δd2λR∗+λe-Δd1λR+μeΔd2-rΔd2-Δd1+Δd2μp∗P0∗eμp∗tP0∗eμp∗t-C0∗eμc∗t+β∗-Δd1μpP0eμptP0eμpt-C0eμct+β+Δd1μcC0eμctP0eμpt-C0eμct-Δd2μc∗C0∗eμc∗tP0∗eμp∗t-C0∗eμc∗t,H22=12Δd2β∗β∗+1σ∗2+2β∗+1σ∗σeρ23-12Δd1ββ+1σ2. Then, the equilibrium time-consistent strategy in (46) can be expressed as (53)π1∗t=MH11+H12,π2∗t=-MH21+H22. The terms MH11 and -MH21 are referred to as the “speculative demand” or the aggressive demand. The terms MH12 and -MH22 are called the “hedging demand.” They describe the demand for home and foreign investments on risk-hedged basis, respectively.These expressions of the speculative demand involve some foreign direct investment theories, some of which emphasize the cost advantage of host countries and advance the comparative cost savings offered by foreign production vis-a-vis home production. These are reflected in the termsΔd2μc(C0eμct/P0eμpt-C0eμct)-Δd3μc∗(C0∗eμc∗t/(P0∗eμp∗t-C0∗eμc∗t)) in H11 and Δd1μc(C0eμct/P0eμpt-C0eμct)-Δd2μc∗(C0∗eμc∗t/(P0∗eμp∗t-C0∗eμc∗t)) in H21. Other foreign direct investment theories, however, incorporate the demand side effect. In this case, foreign investment depends on whether output price is higher in foreign market than in home market, which is reflected in the terms Δd3μp∗(P0∗eμp∗t/(P0∗eμp∗t-C0∗eμc∗t)+β∗)-Δd2μp(P0eμpt/(P0eμpt-C0eμct)+β) in H11 and Δd2μp∗(P0∗eμp∗t/(P0∗eμp∗t-C0∗eμc∗t)+β∗)-Δd1μp(P0eμpt/(P0eμpt-C0eμct)+β) in H21. The investment flow is also affected by the exchange rate and the domestic short rate. The effect of exchange rate can be seen in the terms μeΔd3 in H11 and μeΔd2 in H21. The effect of short rate can be seen in the terms -r(Δd3-Δd2) in H11 and -r(Δd2-Δd1) in H21.We also take into account information costs as well as the degree of competition in home and foreign markets. In fact, if managers do not know about the foreign investment opportunity or they do not want to spend much information cost on foreign and exchange market, they will choose to invest in home market which is referred to as the “home bias puzzle.” Therefore, information plays an important role in foreign investment. It can help managers to make decisions about investment in foreign market. These effects can be seen in the termsΔd3(λR∗+λe)-Δd2λR in H11 and Δd2(λR∗+λe)-Δd1λR in H21.The hedging demand describes a portfolio story as in Choi [32] and Bellalah and Wu [33]. If the variability of home returns is higher and the correlation between home and foreign returns is lower, managers will increase the foreign investments. In this case, the investment flow is affected by the cross-covariances and the degree of competition in home and foreign markets. These effects are reflected in the terms H12 and H22.Remark 10.
We find that the optimal investment strategy does not depend on the wealth processWtπ∗.Inspired by Zeng et al. [20], we can derive the relationship between conditional expectation and conditional variance of terminal wealth for corporate international investment problem (CIIP).Theorem 11.
The efficient frontier of problem (39) at initial point (t,Wt) is given by(54)Et,wWTπ∗=erT-tWt-Vart,w⁡WTπ∗·∫tTΛsds2∫tTΓsds,where Λ(t) and Γ(t) are given in (48).The proof is given in AppendixC.Remark 12.
For all(t,Wt)∈[0,T]×W, the relationship between Et,w[WTπ∗] and Vart,w[WTπ∗] is given by (54). It is also called the efficient frontier of problem (39) at initial point (t,Wt) in the modern portfolio theory.


## 3.5. Corporate International Investment Problem with State-Dependent Risk Aversion Coefficient

In this section, we consider corporate international investment problem (CIIP) under mean-variance criterion with state-dependent risk aversion coefficient which is discussed in Björk et al. [37]; that is, the value functional is given by(55)Jt,w,π=Et,wWTπ-γw2Vart,w⁡WTπ,where γ(w)=γ/w.The target of manager is to find the equilibrium time-consistent strategy or optimal solutionπ∗=(π1∗,π2∗) and the equilibrium value function (56)Vt,w=Jt,w,π∗.As indicated in Section 2.2, for these questions, value function (55) can be written as (57)Jt,w,π=Et,wFw,WTπ+Gw,Et,wWTπ,where (58)Fx,y=y-γ2xy2,Gx,y=γ2xy2.Note that (59)Vt,w=Jt,w,π∗=Et,wWTπ∗-γ2wVart,w⁡WTπ∗.We recall the probabilistic interpretations (60)ft,w,y;WTπ∗=Et,wWTπ∗-γ2yEt,wWTπ∗2,gt,w;WTπ∗=Et,wWTπ∗.From these, we have (61)Vt,w=ft,w,w;WTπ∗+γ2wg2t,w;WTπ∗.By Proposition3, we get the following results after a large number of elementary calculations.Proposition 13.
With notations as above, the extended HJB system takes the form(62)ft+γwggt+supπ∈Π⁡fw+γwggwπ1Δu1+π2Δu2+Wtr+12fww+γwggwwπ12Δd1+2π1π2Δd2+π22Δd3=0, and the optimal solution is given by(63)π1∗=fw+γ/wggwfww+γ/wggwwΔu2Δd2-Δu1Δd3Δd1Δd3-Δd22=fw+γ/wggwfww+γ/wggwwΔm1,π2∗=fw+γ/wggwfww+γ/wggwwΔu1Δd2-Δu2Δd1Δd1Δd3-Δd22=fw+γ/wggwfww+γ/wggwwΔm2.The proof is given in AppendixD.Following Y. Li and Z. Li [21], we have two theorems as follows.Theorem 14.
The equilibrium time-consistent strategy for CIIP is given by(64)π1∗=c1tw+k1t,π2∗=c2tw+k2t,where c1(t), c2(t), k1(t), and k2(t) satisfy the following integral equations:(65)c1t=Δm1·1γe∫tTΨ1s-Φ1sds+e∫tT2Ψ1s-Φ1sds-1,c2t=Δm1·1γe∫tTΨ1s-Φ1sds+e∫tT2Ψ1s-Φ1sds-1,k1t=-Δm2·12-e2∫tTΨ1sds∫tTe-∫tsΦ1τdτΦ2sds,k2t=-Δm1·12-e2∫tTΨ1sds∫tTe-∫tsΦ1τdτΦ2sds,
with(66)Ψ1t=c1tΔu1+c2tΔu2+r,Ψ2t=k1tΔu1+k2tΔu2,Φ1t=2c1tΔu1+c2tΔu2+r+c12tβ+12σ2+c22tβ∗+12σ∗2+σe2+2c1tc2tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+2c22tβ∗+1σ∗σeρ23,Φ2t=2k1tΔu1+k2tΔu2+2c1tk1tβ+12σ2+2c2tk2tβ∗+12σ∗2+σe2+2c1tk2t+c2tk1tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+4c2tk2tβ∗+1σ∗σeρ23,Φ3t=k12tβ+12σ2+k22tβ∗+12σ∗2+σe2+k1tk2tβ+1β∗+1σσ∗ρ12+β+1σσeρ13+k22tβ∗+1σ∗σeρ23, and the symbols Δu1,Δu2 are given in (45).The proof is given in AppendixE.Theorem 15.
Integral equations (65) admit a unique solution c1t,c2t,k1t,k2t∈C10,T∪C2[0,T].The proof can be seen in Y. Li and Z. Li [21].


## 4. Simulating Results

In this section, we give some simulating results of the optimal investment strategy for CIIP. Suppose that the time unit in the model is one year and letβ=-0.02, β∗=-0.02, μp=0.08, μp∗=0.06, μc=0.02, μc∗=0.03, σ=0.2, σ∗=0.15, σe=0.05, ρ12=0.8, ρ13=0.6, ρ23=0.4, μe=0.02, λR=0.005, λR∗=0.01, λe=0.01, P0=6=P0∗, and C0=2=C0∗. Besides, let the interest rate r=0.003, and risk aversion coefficient γ=5. For simplifying, we set π1=π1∗/Wt, π2=π2∗/Wt, and then π0=1-π1∗/Wt-π2∗/Wt.Example 16.
In this example, we present that the optimal investment strategyπ=(π1,π2) changes with time t.
As can be seen from Figure1, the variable π1 increases with respect to time t, which shows that the money invested in home market is growing over time. The variable π2 decreases with respect to time t, which shows that the money invested in foreign market is declining over time. And the trend of variable π0=1-π1-π2, representing the money invested in risk-free asset, is similar to π1.Figure 1
Portfolio changes with timet.Example 17.
In this example, we present that the optimal investment strategyπ=(π1,π2) changes with parameters λR∗, λR, λe, and μe.
To find the relationship between the portfolio and a single parameter, we set other parameters equal to zero. Besides, letT=30 and t=5, so we get four figures for different parameters.
In Figure2(a), we get the curve demonstrating that the portfolio changes with home market information cost λR. When λR=-0.035, we can get the optimal investment strategy π1=0.2671, π2=0.1714, and π0=0.5615, which shows that 26.71% of the wealth should be invested in home market, 17.14% of the wealth should be invested in foreign market, and 56.15% of that should be invested in risk-free asset.
In Figure2(b), we get the curve demonstrating that the portfolio changes with foreign market information cost λR∗. When λR∗=0.03, we can get the optimal investment strategy π1=0.1899, π2=0.4975, and π0=0.3127, which shows that 18.99% of the wealth should be invested in home market, 49.75% of the wealth should be invested in foreign market, and 31.27% of that should be invested in risk-free asset.
In Figure2(c), we get the curve demonstrating that the portfolio changes with exchange market information cost λe. When λe=0.045, we can get the optimal investment strategy π1=0.4976, π2=0.2006, and π0=0.3018, which shows that 49.76% of the wealth should be invested in home market, 20.06% of the wealth should be invested in foreign market, and 30.18% of that should be invested in risk-free asset.
In Figure2(d), we get the curve demonstrating that the portfolio changes with exchange rate μe. When μe=0.025, we can get the optimal investment strategy π1=0.0873, π2=0.5964, and π0=0.3163, which shows that 8.73% of the wealth should be invested in home market, 59.64% of the wealth should be invested in foreign market, and 31.63% of that should be invested in risk-free asset.Figure 2
(a) Portfolio changes with home market information costλR. (b) Portfolio changes with foreign market information cost λR∗. (c) Portfolio changes with exchange market information cost λe. (d) Portfolio changes with exchange rate μe.
(a) (b) (c) (d)


## 5. Conclusion

In this paper, we analyze a continuous-time model for corporate international investment problem (CIIP) with mean-variance criterion. Inspired by Björk and Murgoci [7], based on Nash subgame perfect equilibrium theory, we define an infinitesimal operator and deduce an extended HJB equation in continuous time. Besides, combining Björk and Murgoci [7] with Bellalah and Wu [33], we obtain the equilibrium time-consistent strategy for corporate international investment problem. In addition, following Björk et al. [37] and Y. Li and Z. Li [21], we discuss two cases of risk aversion coefficient: one is constant and the other is state dependent. Finally, the simulating results are given to illustrate our conclusion and the influence of some parameters on the optimal solution.Several possible extensions of this study may be interesting. Firstly, our study could be extended to stochastic interest rate and inflation. Secondly, jump-diffusion could be taken into account in our model. Thirdly, we could also consider that the prices of input and output hold for CEV model or mean-reverse model.



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