Abstract
In this paper, we consider a stochastic portfolio optimization model for investment on a risky asset with stochastic yields and stochastic volatility. The problem is formulated as a stochastic control problem, and the goal is to choose the optimal investment and consumption controls to maximize the investor’s expected total discounted utility. The Hamilton–Jacobi–Bellman equation is derived by virtue of the dynamic programming principle, which is a second-order nonlinear equation. Using the subsolution–supersolution method, we establish the existence result of the classical solution of the equation. Finally, we verify that the solution is equal to the value function and derive and verify the optimal investment and consumption controls.
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1 Introduction
In the classical Merton portfolio optimization problem, an investor allocates her/his investment between a risky asset (e.g., stock) and a riskless asset (e.g., bond). The goal is to choose the optimal investment (allocation) strategy and the consumption strategy to maximize the total discounted expected utility. Thousands of papers on this and related topics have been published since the seminal paper by Merton [1]. This kind of portfolio optimization problem can usually be formulated as a stochastic control problem, where the controls are the investment and the consumption. For example, at any time t, the investor chooses how much of the wealth to invest into the risky asset and how much to consume to maximize the total expected utility.
A very popular model to describe the price of the risky asset is the classical geometric Brownian motion model, in which the volatility is assumed as a constant. However, it is widely accepted that stock volatilities exhibit random characteristics, which presents limitations in models with a constant stock volatility. One commonly noted example is the discrepancy between observed option prices and the prices predicted by the Black–Scholes formula. Implied volatilities are known to vary with strike price (for a fixed maturity), creating a volatility smile or smirk. Stochastic volatility models can capture this smile/smirk effect. Fouque et al. [2] discuss further benefits of stochastic volatility models, including the generalization of more realistic return distributions with fatter tails. They also discuss more difficulties presented by such models and ways to address the challenges. Recently, Lorig and Sircar [3] consider a finite-time horizon portfolio optimization model in a general local-stochastic volatility setting, and they derive the approximations for both the value function and the optimal investment strategy. In Fatone et al. [4, 5], some multi-scale stochastic volatility models and the market calibration problems are considered. Some other results with stochastic factor or stochastic volatility models can be found in Zariphopoulou [6], Fleming and Hernández-Hernández [7], Fouque and Han [8], Fouque et al. [9], and the references therein.
Moreover, unlike the constant interest rate assumed in the classical Merton model, the interest rate on the riskless asset may fluctuate from time to time. Portfolio optimization problems with stochastic interest rate models are considered in Fleming and Pang [10] and Pang [11, 12]. In those papers, the interest rate for the riskless asset is assumed to follow a stochastic process with a mean-reverting feature. Goel and Kumar [13] consider a class of risk-sensitive portfolio optimization problems, where a fixed income security with stochastic interest rate is included. They prove the existence results under certain conditions. Moreover, in Fleming and Hernández-Hernández [14], Nagai [15], and Noh and Kim [16], stochastic interest rates and stochastic volatility are both incorporated. Hata and Sheu [17, 18] consider an optimal investment problem in which they allow both the drift and volatility of price to be stochastic. In [19], Kaise and Sheu prove existence and uniqueness of ergodic-type Bellman equations.
The classical Merton model assumes that the risky asset does not pay any dividend (or equivalently, there is no productivity yield with the risky asset), and the investor only makes profits from the asset price changes. However, in the real world, many stocks do pay dividends, and there are some derivatives based on dividends (see Tunaru [20]). Moreover, the dividend yield tends not to be constant due to the possibility of bankruptcy (see Geske [21]), and the dividend yield rate and/or the dividend amount usually changes from time to time. In [21], Geske considers the stochastic dividend in the classical Black–Scholes–Merton option pricing formula, and a new formula is derived in discrete time, under the assumption of a lognormal distribution for dividend yield. Lioui [22] proposes a mean-reverting stochastic process to model the stochastic dividend yield in continuous time, under the complete market case. In Pang and Varga [23], a portfolio optimization problem with stochastic dividend is considered. Chevalier et al. [24] consider an optimal dividend and investment control problem with debt constraints. We want to point out that the (stochastic) dividend model can be used to model any risky asset with productivity yield, such as foreign currency, gold (when lease yield is considered), farm land. In Fleming and Pang [25], a portfolio optimization model with stochastic productivity is considered and the model can be applied to an optimal investment problem on a stock with stochastic dividends.
In this paper, we consider a portfolio optimization problem in which the risky asset price is modeled by a stochastic differential equation with stochastic volatility and stochastic yields (dividends). In particular, we assume that the stochastic volatility process is driven by a mean-reverting Ornstein–Uhlenbeck factor process, and the stochastic dividend yield rate is modeled by a white noise-type stochastic process. Investment and consumption controls are chosen to maximize the expected discounted utility of consumption. Our model can be used to describe an economic unit with productive capital and liabilities in the form of debt.
This paper is an extension of Pang and Varga [23] by including stochastic volatility in the model to make it more realistic. The introduction of stochastic volatility brings more mathematical challenges, and a new method is needed to establish the results. Similar to [23], we assume that the stochastic dividend yield rate is modeled by a white noise-type stochastic process given by (4). There are a couple of reasons we use this model for the dividend. First, the stock price can be treated as the total present value of the future dividends, so the stock price can be written as an integral of discounted future dividends. If we use the popular geometric Brownian motion to describe the stock price, the dividend yield is more like the derivative of the geometric Brownian motion, i.e., a white noise-type stochastic process. Second, for technical reasons, the assumption of (4) for the dividend yield \(b_t\) makes the model mathematically more tractable.
The problem is formulated as a stochastic control problem, and we derive the associated Hamilton–Jacobi–Bellman (HJB) equation using the dynamic programming principle. The HJB equation is a second-order nonlinear partial differential equation for which the current PDE existence results do not apply. By virtue of the subsolution–supersolution method, which is proposed by Fleming and Pang [10] and is later extended by Hata and Sheu [17, 18], we establish the existence results of the HJB equation. Further, we derive the optimal investment and consumption control policies and establish the verification results.
The rest of the paper is organized as follows. In Sect. 2, the problem is introduced and formulated as a stochastic control problem. The HJB equation for the value function is derived by the dynamic programming principle, and some preliminary results are given. In Sect. 3, we establish the existence result of the solution for the HJB equation by virtue of the subsolution–supersolution method. The verification results are given in Sect. 4, and the optimal investment and consumption strategies are derived in this section as well. We conclude the paper in Sect. 5.
2 Problem Formulation
We consider a portfolio optimization problem of Merton’s type. In particular, we consider an investor who, at time t, owns \(N_t\) shares of stock at price per share \(P_t\). The total worth of investments is given by \(K_t = N_tP_t\). Then, we can write
where \(I_t=\frac{d N_t}{N_t}\) is the investment rate at time t. Here, we assume that the number of shares \(N_t\) is of finite variation, so the term \(dK_t\cdot dN_t\) vanishes here.
The investor’s debt, \(L_t,\) increases with interest payments, investment, and consumption and decreases with income. Thus, the equation for the change in debt is given by
where \(r\ge 0\) is a constant interest rate, \(C_t\) is the consumption rate, and \(D_t\) is the rate of income from the risky asset yield. For example, \(D_t\) can be treated as the total earned dividends per unit time at time t. The total dividend is equal to the total number of shares times the productivity of capital, or dividend rate, \(b_t\):
The investor’s net worth is given by \(X_t = K_t - L_t\), and we require \(X_t > 0\).
It is worth noting that the model applies to any economic unit with productive capital and liabilities. For another example, consider a farm on which products are grown and then sold for profits. \(N_t\) may represent the number of acres of the farm, with \(P_t\) being the property value per acre. Debt increases with property taxes, the purchase of new land, and consumption and decreases with income from selling the produce. From here on, we continue our explanations with the investor example.
We assume that the dividend rate has a constant average growth rate of b with a white noise. In particular, we assume that the dividend rate \(b_t\) is governed by the following equation:
where \(b, \sigma _1 >0\) are constants and \(B_{1,t}\) is a standard Brownian motion. We assume that the stock price follows a geometric Brownian motion with a stochastic volatility:
where \(\mu >0\) is a constant, and the function \(\sigma _2(z) \in C^1(\mathbb {R})\) satisfies
for some constants \(\tilde{\sigma }_2, \hat{\sigma }_2\), and \(\hat{\sigma }_2'\), and \(B_{2,t}\) is a standard Brownian motion. We further assume that the volatility is driven by a mean-reverting Ornstein–Uhlenbeck process given by
where \(a, \sigma _3, \bar{z} \) are positive constants and \(B_{3,t}\) is a standard Brownian motion. The mean-reverting feature captures the tendency of the stochastic volatility to revert back to its invariant, or long-run, distribution.
Remark 2.1
In Eq. (5), instead of assuming \(\mu \) is a constant, we can assume that \(\mu \) is a positive, bounded, and smooth function of \(Z_t\). All the results still hold, and all the arguments are very similar. In this paper, we just consider the constant \(\mu \) case for notation convenience.
In the above equations, we introduce three one-dimensional standard Brownian motions, \(B_{1,t}, B_{2,t},\) and \(B_{3,t}\). We allow \(B_{1,t}\) and \(B_{2,t}\) to be correlated with a correlation constant \(\rho \in [\rho _0,1]\) for some constant \(-1<\rho _0 <0,\) and we suppose \(B_{3,t}\) is uncorrelated with \(B_{1,t}\) and \(B_{2,t}\). That is,
It is reasonable to assume that the dividend process and the stock price process are not perfectly negatively correlated. So here we restrict \(\rho _0 \ne -1\).
By virtue of (1) and (2), we can get the equation for the investor’s net worth \(X_t=K_t-L_t\) as
Define \(k_t \equiv \displaystyle \frac{K_t}{X_t}\) and \( c_t \equiv \displaystyle \frac{C_t}{X_t}\) as the control variables. Noting that
and using (3), (4), and (5), we can get
Let \(Y_t \equiv \log P_t\). Then, the above equation can be written as
The \(Y_t\) process follows
where \({\tilde{\mu }}(Z_t) = \mu - \frac{1}{2}\sigma _2^2(Z_t)\). Note that \({\tilde{\mu }}(z)\) is bounded:
We define the admissible control space \(\varPi \) as the following:
Definition 2.1
(Admissible Control Space) The pair \((k_t, c_t)\) is said to be in the admissible control space\(\varPi \) if \((k_t,c_t)\) is an \(\mathbb {R}^2\)-process which is progressively measurable with respect to a \((B_{1,t}, B_{2,t}, B_{3,t})\)-adapted family of \(\sigma _1\)-algebras \(\{ \mathcal {F}_t, t \ge 0\}\). Moreover, we require that \(k_t, c_t \ge 0,\) and
We consider the hyperbolic absolute risk aversion (HARA) utility function
\(\gamma =0\) is the value corresponding to the log utility case \(U(C)=\log C\). In this paper, we only consider \(0<\gamma <1\). The method to solve the problem for the \(\gamma <0\) case is the same, so we omit it in this paper. The goal is to maximize the expected total discounted HARA utility of consumption subject to the constraints \((k_t,c_t) \in \varPi \) and \(X_t > 0\). The objective function is
and the corresponding value function is given by
where the discount factor \(\beta >0\) is a constant, and x, y, and z are the initial values of the state variables \(X_t, Y_t\), and \(Z_t,\) respectively.
The state variables \(X_t, Y_t\), and \(Z_t\) are given by (9), (10), and (7), respectively. Using the dynamic programming principle (refer to Fleming and Soner [26] for details), we get the following HJB equation for V(x, y, z) :
where
We look for a solution of the following form:
Substituting this form into Eq. (14), we can get the equation for W by canceling \(x^\gamma \):
Define a function \(Q(y, z) = \log W (y,z)\). Then, we can get the equation for Q:
where the function G(y, z, p) is defined by
The candidates for the optimal controls are
Substitute the \((k^*, c^*)\) given by (20) and (21) into (18), and we can rewrite the equation for Q as
where
Equation (22) is the reduced HJB equation for Q(y, z), and we want to show that \(V(x,y,z) = \frac{1}{\gamma }x^\gamma e^{Q(y,z)}\) is equal to the value function given by (13). Next, we establish the existence of the solution Q(y, z) to (23) in Sect. 3. Then, in Sect. 4, we verify that \(V(x,y,z) = \frac{1}{\gamma }x^\gamma e^{Q(y,z)}\) is the value function, and the optimal investment and consumption strategies are given by (20) and (21), respectively.
We present some useful results before we move to the next section. First, we have the following lemma about the q(y, z) function defined by (15).
Lemma 2.1
For \(\rho \in [\rho _0, 1]\), where \(-1<\rho _0 < 0\), we have
where \(q_0 \equiv \tilde{\sigma }_2^2(1-\rho ^2_0)\).
The proof can be found in “Appendix A:”. \(\square \)
Note that if \(k^* = 0\), then \(G = 0\). If \(k^* > 0,\) then
So we have \(G(y,z,p) \ge 0\). Define
Then, it is easy to verify that \(G(y, z, 0) = \varPsi (y,z)\). We have the following lemma about the function \(\varPsi \).
Lemma 2.2
\(\varPsi (y,z)\) is bounded.
The proof can be found in “Appendix B:”. \(\square \)
3 Existence Results
In this section, we prove the existence of a classical solution to (22). In particular, we use the subsolution–supersolution method to establish the existence results. The method is first introduced by Fleming and Pang [10] to solve the HJB equations that arise in stochastic control problems, and it is later extended in Hata and Sheu [17, 18]. The problem we consider here is not covered by the problems considered in [17, 18]. Due to the particular structure of (22), the results of [17, 18] cannot be applied here and there are some technical difficulties that we have to overcome.
3.1 Subsolution and Supersolution
We first define subsolutions and supersolutions of (22).
Definition 3.1
Q(y, z) is a subsolution (supersolution) of (22), if
In addition, if \(\tilde{Q}\) is a subsolution, \(\hat{Q}\) is a supersolution, and \(\tilde{Q} \le \hat{Q},\) then, \(\langle \tilde{Q}, \hat{Q}\rangle \) is an ordered pair of subsolution–supersolution.
Next, we show that there exists a pair of ordered subsolution and supersolutions.
Lemma 3.1
Suppose \(0< \gamma < 1\) and
where \({\bar{\varPsi }}\) is defined by
In addition, define
Then, \(\langle K_1,K_2\rangle \) is an ordered pair of subsolution–supersolution to (22).
The proof is straightforward, and we omit it here. The main existence result is as follows:
Theorem 3.1
Suppose \(\sigma _2^2(z) \in C^{1,\alpha }({\bar{B}}_R)\). Define \(\tilde{Q} \equiv K_1\) and \(\hat{Q} \equiv K_2\), where \(K_1\) and \(K_2\) are given by (29). Then, there exists a solution \( Q \in C^{2,\beta }(\mathbb {R}^2)\) to (22) such that \(\tilde{Q} \le Q(y,z) \le \hat{Q}\) for all \((y,z) \in \mathbb {R}^2\).
The proof of Theorem 3.1 is given in Sect. 3.4. We first prove that, on the closed ball \({\bar{B}}_R \equiv \{(y, z) \in \mathbb {R}^2 : y^2 + z^2 \le R^2\}\), there exists a classical solution to the following boundary value problem
for a particular choice of \(\psi \). Once we have a solution to (30) on \({\bar{B}}_R\) for each R, we take the limit as \(R \rightarrow \infty \) to show the existence of solution to (22). The details are provided in the proof of Theorem 3.1 in Sect. 3.4.
3.2 Existence Results of Boundary Value Problem (30)
Following the approach taken by Hata and Sheu [17], we start by introducing the parameter \(\tau \in [0,1]\) into our equation:
where \(H(y,z, u, p, s, \tau )\) is equal to H(y, z, u, p, s) with \(\gamma \) replaced by \(\tau \gamma :\)
and
For \(0<\tau \le 1\), (31) is the reduced HJB equation corresponding to the value function
by taking \(V^\tau (x,y,z)=\frac{1}{\tau \gamma } x^{\tau \gamma } e^{Q^\tau (y,z)}\). We consider \(\tau =0\) to be a limiting case of \(0<\tau \le 1\). This corresponds to the consumption problem for log utility, for which (31) has a unique solution [please refer to (49)].
The boundary value problem for (31) is
The above boundary value problem is used to obtain the existence result over the whole space.
The following theorem states the sufficient conditions for the existence of solution to (30).
Theorem 3.2
Let \(0<\alpha <1 \) and \(R>0\) be fixed. We assume the following conditions:
-
(a)
\(H(y,z,u,p,s,1) = H(y,z,u,p,s)\).
-
(b)
\(\sigma _2^2(z) \in C^{1,\alpha }({\bar{B}}_R); \, H(\cdot ,\cdot ,\cdot ,\cdot ,\cdot , \tau ) \in C^\alpha ({\bar{B}}_R \times \mathbb {R}^3)\) for \(\tau \in [0,1],\) and the function \(H(y,z,u,p,s,\tau )\) is continuous when considered as a mapping from [0, 1] into \(C^\alpha ({\bar{B}}_R \times \mathbb {R}\times \mathbb {R}^2)\).
-
(c)
\(\psi \in C^{2,\alpha }({\bar{B}}_R)\).
-
(d)
There exists a constant M such that every \(C^{2,\alpha }({\bar{B}}_R)\)-classical solution \(Q^\tau \) of (34) satisfies
\(|Q^\tau (y,z)| < M, \quad (y,z) \in {\bar{B}}_R\), where M is independent of \(Q^\tau \) and \(\tau \).
-
(e)
There are \({\bar{k}}>0, \underline{c}, {\bar{c}},\) such that the following inequalities hold for \((y,z) \in {\bar{B}}_R, |u| \le M, \eta \in \mathbb {R}^2, \tau \in [0,1],\) and arbitrary (p, s):
$$\begin{aligned}&\underline{c} \sum _{i=1}^2 \eta _i^2 \le \sigma _2^2(z)\eta _1^2 + \sigma _3^2\eta _2^2 \le {\bar{c}} \sum _{i=1}^2 \eta _i^2,\nonumber \\&\quad |H(y,z,u,p,s,\tau )| + \left| \frac{d \sigma _2^2(z)}{d z} \right| \le {\bar{c}} (1 + p^2 + s^2)^{\frac{{\bar{k}}}{2}}. \end{aligned}$$(35) -
(f)
There is an \(M_1 > 0\) such that for any \(\tau \in [0,1]\), \(|Q_\tau ^0(y,z)| < M_1\) and \( (y,z) \in {\bar{B}}_R\), where \(Q^0_\tau (\cdot )\) is an arbitrary solution of
$$\begin{aligned} \begin{array}{ll} \frac{\sigma _2^2(z)}{2} Q_{yy} + \frac{\sigma _3^2}{2} Q_{zz} - \tau H(y,z, Q, Q_y,Q_z,0) = 0 &{}\text {on} \quad B_R, \\ Q = 0 &{}\text {on} \quad \partial B_R. \end{array} \end{aligned}$$
Then, boundary value problem (30) is solvable in \(C^{2,\alpha }({\bar{B}}_R)\).
See Theorem 3.4 and the proof in [17] for more details. \(\square \)
The next step is to prove that the conditions in Theorem 3.2 hold for the problem considered in this paper. Then we can get the existence of solution to boundary value problem (30). Several results are needed before that can be done. We begin with the following definition.
Definition 3.2
Let \(\tilde{Q}, \hat{Q}\) be continuous second-order differentiable functions defined on \({\bar{B}}_R\). \(\tilde{Q}\) (\(\hat{Q}\)) is called a subsolution (supersolution) of (30) if it satisfies
In addition, \(\langle \tilde{Q}, \hat{Q}\rangle \) is called an ordered pair of subsolution–supersolution if they also satisfy
We can define an ordered pair of subsolution–supersolution to (34) in a similar manner. The following lemma is used later.
Lemma 3.2
For G given by (19), there exist constants \({\tilde{C}}_1, {\tilde{C}}_2 > 0\) such that
The proof can be found in “Appendix C:”. \(\square \)
Next, we establish a comparison result.
Lemma 3.3
Let \(0<\tau \le 1\). Assume \(\tilde{Q}, \hat{Q}\) are second-order continuous differentiable functions on \({\bar{B}}_R\) and satisfy
Then, \(\tilde{Q} \le \hat{Q}\) holds in \({\bar{B}}_R\).
The proof can be found in “Appendix D:”. From Lemma 3.3, we can get the following result:
Corollary 3.1
For \(0<\tau \le 1\), the solution \(Q^\tau \in C^2({\bar{B}}_R)\) of (34) is unique.
The proof can be found in “Appendix E:”. \(\square \)
To establish the existence result for (30) by virtue of Theorem 3.2, we need to verify that the conditions \((a){-}(f)\) are satisfied.
The coefficients of (22), \(\sigma _2^2(z), {\tilde{\mu }}(z), g_0(y,z), g_1(y,z),\) and \(g_2(y,z),\) are Lipschitz continuous for all \((y,z) \in \mathbb {R}\). If \(Q(y,z) \in C^1({\bar{B}}_R)\), it follows that \(H(y,z,Q,Q_y, Q_z) \in C^\alpha ({\bar{B}}_R\times \mathbb {R} \times \mathbb {R}^2)\). This helps to verify condition (b).
The next two theorems provide us with bounds on \(Q^\tau \), which are useful when we verify the conditions (d) and (f).
Theorem 3.3
Suppose \(\sigma _2^2(z) \in C^{1,\alpha }({\bar{B}}_R), \psi \) is continuous, and let \(\hat{Q}\) be a supersolution of (34) for \(\tau = 1\). Suppose that \(Q^\tau \) is a solution of (34) with \(0<\tau \le 1\). Then,
where f(y, z) satisfies
and is given by
where \( t_R = \inf \left\{ t \ge 0 ; \sqrt{Y_t^2 + Z_t^2} = R \right\} . \) Moreover, for \(0< \gamma < 1,\)
Theorem 3.4
Let \(0<\tau \le 1\). Suppose \(\sigma _2^2(z) \in C^{1,\alpha }(\bar{B}_R),\) and that \(Q_\tau ^0\) is a solution of
Then,
where \( {\bar{t}}_R = \inf \left\{ t\ge 0; \sqrt{\hat{Y}_t^2 + \hat{Z}_t^2} = R \right\} , \) and \(\hat{Y}_t, \hat{Z}_t\) are defined by
The proofs of Theorems 3.3 and 3.4 can be found in “Appendix G: and H:,” respectively.
The following theorem gives us the existence of solution to boundary value problem (30) with \(\psi = \tilde{Q},\) on the closed ball given by
Theorem 3.5
Assume \(\sigma _2^2(z) \in C^{1,\alpha }({\bar{B}}_R), \) (22) has an ordered pair of subsolution–supersolution \(\langle \tilde{Q}, \hat{Q}\rangle \), and \(\tilde{Q} \in C^{2,\beta }({\bar{B}}_R)\) for some \(0<\beta \le 1\). Further assume that \(\hat{Q}\) is a supersolution of (34) for \(\tau =1\), and \(Q^\tau \) is a solution of (34) with \(0<\tau \le 1\). Then, the boundary value problem
has a unique solution in \(C^{2,\beta }({\bar{B}}_R)\).
Proof
Notice that (44) is equivalent to (30) with \(\psi = \tilde{Q}\). We use Theorem 3.2 to prove the existence of solution. Note that conditions (a), (b), and (c) are automatically satisfied. By Theorem 3.3, we have inequality (38):
Since \(0<\tau \le 1\) and \(f > 0\), we can write
where \(\bar{M}\equiv \max _{\bar{B}_R} \log \big [e^{\hat{Q}(y,z)} + f(y,z)\big ]\). Also by Theorem 3.3, we have the following inequality:
Note that \(\underline{M}\) and \({\bar{M}}\) are independent of both \(\tau \) and \(Q^\tau \). Here, we take
Then, we have the bound
where M is independent of \(\tau \) and \(Q^\tau \). Hence condition (d) of Theorem 3.2 is satisfied.
For condition (e), take \(\underline{c} \equiv \min \{\tilde{\sigma }_2^2, \sigma _3^2\}\) and \({\bar{c}} \equiv \max \{\hat{\sigma }_2^2, \sigma _3^2\}\). Then, for any \((\eta _1,\eta _2) \in \mathbb {R}^2,\)
For the second part of condition (e), we have that for \(|z| \le M,\)
Thus, condition (e) is satisfied, with \({\bar{k}}=2\).
By Theorem 3.4, we have estimate (43). Then, it is easy to get that \( |Q_0^\tau (y,z)| < M_1\), where \(M_1 = \max \big \{\beta \mathbf {E}[{\bar{t}}_R], \log (1 + \mathbf {E}[{\bar{t}}_R]) \big \} + C_1,\) for some constant \(C_1 > 0\). Thus, condition (f) of Theorem 3.2 is satisfied.
In (34), take \(\psi \equiv \tilde{Q}\). For the \(\tau = 0\) case, (34) has the unique solution
where f is given by (40). Indeed, setting \(\tau = 0\) in (34) and substituting (49) into the equation satisfied on \(B_R\), we obtain
by (39). On the boundary \(\partial B_R, \tau _R = 0\). Then, by (40) we see that \(f=1\), and so \(Q_0^\tau = 0\) on \(\partial B_R\). Therefore, \(Q_0^\tau = \log f\) is a solution to (34) for \(\tau = 0\). Uniqueness can be proven using a method similar to that in Corollary 3.1.
The solution for \(\tau =0\) corresponds to the solution of the log utility problem, which is the limit case of the HARA utility case when parameter \(\gamma \) goes to 0. We do not discuss the log utility case in detail in this paper as our main focus is on the nonlog HARA utility with \(0<\gamma <1\).
By Theorem 3.2, (44) has a solution in \(C^{2,\alpha }({\bar{B}}_R)\). For \(\tau = 1\), (34) is equivalent to (30), which is equivalent to (44) when \(\psi = \tilde{Q}\). Therefore, by Corollary 3.1, the solution to (44) is unique. \(\square \)
3.3 A Uniform Bound for \(\sup _{B_R} |DQ|^2\)
Before proving existence of a classical solution to (22), we must prove the existence of a uniform bound for \(\sup _{B_R} |DQ|^2\) for any R.
Theorem 3.6
Let \(Q_{{\tilde{R}}}\) be a smooth function satisfying HJB equation (22) in \(B_{{\tilde{R}}}\). For each \(R>0\) and \({{\tilde{R}}}>2R\), we have
where C is a nonnegative constant independent of R and \({\tilde{R}}\), and \(C_R\) is a constant depending only on R.
To prove Theorem 3.6, we need the following lemma:
Lemma 3.4
Let \(Q(x) \in C^2(\mathbb {R}^N)\), and \(a^{i,j}(x) \in C^2(\mathbb {R}^N)\) for all i, j ranging from 1 to N. Then,
where \(\epsilon >0\) is a small constant.
The proof can be found in “Appendix F:”. \(\square \)
Now, we can give the proof of Theorem 3.6.
Proof of Theorem 3.6
We write G as defined in (19) in its quadratic form,
where \(g_2, g_1,\) and \(g_0\) are defined by
Then, \(Q_{{\tilde{R}}}\) satisfies
on \(B_{{\tilde{R}}}\). Note that this is the case of \(be^{-y} + \mu - r \ge 0\), which is true if we assume \(\mu >r\). The more simple case of \(G = 0\) (if \(be^{-y} + \mu - r < 0\)) can be proved using the same steps as in this proof, so we do not prove this case explicitly.
For simplicity, we drop the subscript \({{\tilde{R}}}\) and set \(Q \equiv Q_{{\tilde{R}}}\). Differentiating (52) with respect to y and z, we obtain
and
respectively. Next, we take the sum of Eq. (53) multiplied by \(Q_y\) and Eq. (54) multiplied by \(Q_z\). Rearranging the result (to a form that is useful in a later step), we get
Define
Then, we have
By (55), we can get
By Lemma 3.4, we have that
Then, we can get the following inequality of the right-hand side (RHS) of (57):
where the last step is based on the fact that
for any constant \(\epsilon \) such that \(0 < \epsilon \le \min \{\tilde{\sigma }_2^2,\sigma _3^2\},\) and \(\tilde{\sigma }_2\) given by (6).
Consider the matrix inequality \((tr(AB))^2 \le N \nu _2 (tr(AB^2))\), where A and B are \(N \times N\) symmetric matrices, A is positive semidefinite, and \(\nu _2\) is the maximum eigenvalue of A. We use this inequality with \(A=\begin{bmatrix}\sigma _2^2(z)&0\\ 0&\sigma _3^2\end{bmatrix}\) and \(B = \begin{bmatrix} Q_{yy}&Q_{yz} \\ Q_{yz}&Q_{zz} \end{bmatrix}\) to get
Then, we have
We use \(C_R\) to represent an arbitrary constant depending only on R, and we use C for a nonnegative constant independent of R and \({{\tilde{R}}}\).
Fix arbitrary \(\xi \in B_R\), and let \(B_R(\xi )\) be an open ball with radius R and center \(\xi \). Let \(\phi \in C^\infty _0(\mathbb {R}^2)\) be a cutoff function such that
Suppose the maximum of \(\phi \varPhi \) in \({\bar{B}}_R(\xi )\) is attained at \((y_0, z_0)\). By the maximum principle, at \((y_0,z_0)\) we have
where the second condition means that the Hessian of \((\phi \varPhi )(y_0,z_0)\) is negative semidefinite. Then, at \((y_0,z_0)\), we have
where the last two steps are based on (58) and (52).
Note that there exist \(0< \mu _1 < \mu _2\) such that
In fact, we can take \(\mu _1 = \min \{L^2,\sigma _3^2\}\) and \(\mu _2 = \max \{U^2 + \gamma {\tilde{C}}_2\}\), where \({\tilde{C}}_2\) is the bound of the coefficient to \(p^2\) in (36). Then, we have
where \(\kappa >0\) is a constant that depends only on \(\mu _1\), and the last step follows from the identity \(|DQ| \le \frac{1}{2}(|DQ|^2 + 1)\).
Now we split the problem up into cases. First, consider the case
Then, it is easy to see that
Noting that \((y_0,z_0)\) is the maximizer of \(\phi \varPhi \) in \(\bar{B}_R(\xi )\), we can get that
Therefore,
So in this case, we have
Next, consider the case \( -\kappa |DQ|^2 + C_R - \gamma g_0 + \beta - \gamma r - (1-\gamma )e^{\frac{Q}{\gamma -1}} \le 0\) at \((y_0,z_0)\). By virtue of (60), we obtain
at \((y_0,z_0)\). If \(\kappa \varPhi \le C_R - \gamma g_0 + \beta - \gamma r - (1-\gamma )e^{\frac{Q}{\gamma -1}} = C_R + \beta - \gamma r,\) then,
so that
If \(C_R \ge \varPhi (y_0,z_0)\), we have a similar result:
for some nonnegative constant C.
Finally, suppose both \(\kappa \varPhi \ge C_R - \gamma g_0 + \beta - \gamma r - (1-\gamma )e^{\frac{Q}{\gamma -1}} = C_R + \beta - \gamma r\) and \(C_R \le \varPhi (y_0,z_0)\). Then,
where \(C_1, C_2\), and \(C_3\) are positive constants independent of \(R, {{\tilde{R}}},\) and \((\beta - \gamma r)\). In above inequality (66), we used the inequality
and the fact that \(\phi \le 1\).
Let \(Y \equiv ( \phi (y_0,z_0)\varPhi (y_0,z_0))^{\frac{1}{2}}\). Then, from (66) we have
where \({\tilde{C}}_3 = C_3C_R\). The quadratic on the right-hand side is concave down. Since the quadratic is greater than or equal to zero, Y is bounded by the zeros of the quadratic. This, along with the fact that \(Y \ge 0\), implies
By virtue of \(|DQ|^2(\xi ) \le \phi (y_0,z_0)|DQ|^2(y_0,z_0)\), we obtain the bound
In each case, we have a bound for \(|DQ|^2\) in \(B_{{\tilde{R}}}\) that can be written in the form \(C_R + C(\beta - \gamma r)\), where \(C_R\) is a constant depending only on R, and C is a positive constant. \(\square \)
3.4 Existence of Solution to HJB Equation (22)
Now we are ready to prove the existence result.
Proof of Theorem 3.1
By Theorem 3.5, there is a unique solution \(Q^l\) to
for \(l = 1, 2, 3, \ldots \) Since \(\tilde{Q}\) and \(Q^{l+1}\) satisfy the following,
we see that they satisfy (37) with \(\tau =1\). Thus, by Lemma 3.3,
In particular,
On \(\partial B_l\), we also have \(Q^l = \tilde{Q}\). Therefore,
Now we see that \(Q^l\) and \(Q^{l+1}\) satisfy
By Lemma 3.3 again,
Thus, for any k such that \(|(y, z)| \le k, Q^l(y,z)\) is nondecreasing in l for \(l > k,\) and is bounded above by \(\hat{Q}\). So by taking \(l \rightarrow \infty \), we can get that \(\{Q^l(y,z)\)} converges pointwise to some function \({\tilde{Q}}(y,z)\).
By Theorem 3.6, we have a uniform bound for \(|DQ^l|^2\) on \({\bar{B}}_R\) for any \(R > 0\). By the Arzela–Ascoli theorem, \(\{Q^l\}\) contains a subsequence that converges to a function \( Q \in C^{2,\beta }({\bar{B}}_R)\) as \(l \rightarrow \infty \). Since \(\{Q^l\}\) also converges pointwise to \({\tilde{Q}}\), we must have \({\tilde{Q}} \equiv Q\). By (48) and (50), it follows that \({\tilde{Q}}\) is a solution to (22) on \({\bar{B}}_R\) for any R. Take R to infinity, and we can get that \({\tilde{Q}}\) is a solution to (22) on \(\mathbb {R}^2\). \(\square \)
4 The Verification Theorem
In Sect. 3, we prove existence of a classical solution \({\tilde{Q}}(y,z)\) to HJB equation (22). In this section, we prove that \({\tilde{V}} = \frac{1}{\gamma }x^\gamma e^{{\tilde{Q}}}\) is equal to the value function given by (13). In effect, we maximize the expected discounted utility of an investor whose net worth depends on stochastic dividends and stochastic volatility of stock price.
We begin by stating a useful result.
Lemma 4.1
Let \({\tilde{Q}}(y,z)\) be a classical solution to (22) such that
where \(K_1\) and \(K_2\) are defined by (29). Define \(k^*(y,z)\) and \( c^*(y,z)\) by (20) and (21), respectively. Then, under the control policies \((k^*, c^*)\), we have
for all fixed \(T\ge 0\) and \(m > 0\).
The proof can be found in “Appendix I:”. \(\square \)
We now state and prove the verification theorem.
Theorem 4.1
(Verification Theorem) Suppose \(0< \gamma < 1\) and (28) holds. Let \(\tilde{Q}(y,z)\) denote a classical solution of (22) which satisfies \(K_1 \le {\tilde{Q}} \le K_2\), where \(K_1, K_2\) are defined by (29). Denote
Then, we have
where V(x, y, z) is the value function defined by (13). Moreover, the optimal control policy is
Before we give the proof of the above theorem, we like to make the following remark:
Remark 4.1
From (22), we can get that its solution \(\tilde{Q}\) not only depends on the average dividend rate b, but also depends on the stochastic dividend volatility parameter \(\sigma _1\), and it is true for the value function V, too. In addition, the optimal investment control \(k^*\), given by (73), and the optimal consumption control \(c^*\), given by (74), both depend on b and \(\sigma _1\) explicitly or implicitly through the dependence on \(\tilde{Q}\). Therefore, under the stochastic dividend case, both the average dividend rate (in terms of b) and the volatility of the dividend rate (in terms of \(\sigma _1\)) have some effects on the value function as well as the optimal investment and consumption controls.
Proof of Theorem 4.1
Since \({\tilde{Q}}\) is a classical solution of (22), it is not hard to show that \({\tilde{V}}\), given by (72), is a classical solution of (14). For any admissible control \((k_t,c_t) \in \varPi \), using Ito’s rule, we can get
Further, by Ito’s rule, we have
Then, using the fact that \(\tilde{V}\) is a solution of (14), we can get
where
Then, we can get
It is easy to show that \(m_{1, T}, m_{1, T} \), and \(m_{3, T} \) are local martingales. Define
Then, we have
Replacing T with \(T \wedge \tau _R\) in (77) and taking expectations, we arrive at the following:
The second inequality is true because \({\tilde{V}} > 0\). Now let R approach infinity, and by Fatou’s lemma, we can get that
Now, letting \(T \rightarrow \infty \) and using the monotone convergence theorem, we have
Since (78) holds for all arbitrary values of \((k_t,c_t) \in \varPi \), we have
Next we show the reverse inequality, \({\tilde{V}}(x,y,z) \le V(x,y,z)\). For \((k^*, c^*)\) given by (73) and (74), it is easy to check that \((k_t^*,c_t^*) \in \varPi \). Using \(k_t^*\) and \(c_t^*\) instead of arbitrary \(k_t, c_t > 0\), we have equality in (77):
where \(m^*_{1,T}, m^*_{2,T},\) and \( m^*_{3,T}\) are equal to the expressions for \(m_{1, T}, m_{2,T},\) and \(m_{3, T}\), respectively, with arbitrary \(k_t\) and \(c_t\) replaced with \(k^*_t\) and \(c^*_t\). It is not hard to verify that \(m^*_{1,T}, m^*_{2,T},\) and \( m^*_{3,T}\) are martingales (see the proof of Lemma 4.1). Therefore, we can get
From (78), we can see that
This implies that
which is easily seen with a proof by contradiction. Note that \(c_t^*\) is bounded below by a positive constant:
Then, we have
which implies
Using this along with the fact that \(e^{{\tilde{Q}}} \le e^{K_2}\), we have that
Now taking the lim inf of (81) as T approaches infinity, we get
The second equality is true due to the monotone convergence theorem. Finally, by (86) and the definition of V, we have
Combining (79) and (87), we have \({\tilde{V}}(x,y,z) = V(x,y,z)\). \(\square \)
5 Conclusions
We consider a portfolio optimization problem with stochastic dividend and stochastic volatility in this paper. The problem is formulated as a stochastic control problem, and the HJB equation is derived. We then establish the existence results of the HJB equation by virtue of the subsolution–supersolution method and some PDE techniques. It is verified that the solution of the HJB equation is the same as the value function, and the optimal investment and consumption strategies are derived. It turns out that both the average dividend rate and the volatility of the dividend play some roles in the value function, the optimal investment strategy, and the consumption strategy.
There are some extensions that can be done beyond the work presented in this paper. For example, one could consider other utility functions such as exponential utility. Another problem is to consider the optimization problem over a finite-time horizon. Moreover, some other realistic features, such as delay effects and transaction costs, can be added to the model. Those can be topics for future research.
References
Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51(3), 247–257 (1969)
Fouque, J.-P., Papanicolaou, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)
Lorig, M., Sircar, R.: Portfolio optimization under local-stochastic volatility: coefficient Taylor series approximations and implied sharpe ratio. SIAM. J. Financ. Math. 7, 418–447 (2016)
Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: Calibration of a multi-scale stochastic volatility model using European option prices. Math. Methods Econ. Finance 3(1), 49–61 (2008)
Fatone, L., Mariani, F., Recchioni, M.C., Zirilli, F.: An explicitly solvable multi-scale stochastic volatility model: option pricing and calibration problems. J. Futures Mark 29(9), 862–893 (2009)
Zariphopoulou, T.: A solution approach to valuation with un-hedgeable risks. Finance Stoch. 5(1), 61–82 (2001)
Fleming, W.H., Hernández-Hernández, D.: An optimal consumption model with stochastic volatility. Finance Stoch. 7, 245–262 (2003)
Fouque, J.-P., Han, C.-H.: Pricing Asian options with stochastic volatility. Quant. Finance 3(5), 352–362 (2003)
Fouque, J.-P., Sircar, R., Zariphopoulou, T.: Portfolio optimization and stochastic volatility asymptotics. Math. Finance 27(3), 704–745 (2017)
Fleming, W.H., Pang, T.: An application of stochastic control theory to financial economics. SIAM J. Control Optim. 43(2), 502–531 (2004)
Pang, T.: Portfolio optimization models on infinite time horizon. J. Optim. Theory Appl. 122(3), 573–597 (2004)
Pang, T.: Stochastic portfolio optimization with log utility. Int. J. Theor. Appl. Finance 9(6), 869–887 (2006)
Goel, M., Kumar, K.S.: Risk-sensitive portfolio optimization problems with fixed income securities. J. Optim. Theory Appl. 142(1), 67–84 (2009)
Fleming, W.H., Hernández-Hernández, D.: The tradeoff between consumption and investment in incomplete financial markets. Appl. Math. Optim. 52(2), 219–235 (2005)
Nagai, H.: H–J–B equations of optimal consumption-investment and verification theorems. Appl. Math. Optim. 71(2), 279–311 (2015)
Noh, E.J., Kim, J.H.: An optimal portfolio model with stochastic volatility and stochastic interest rate. J. Math. Anal. Appl. 375(2), 510–522 (2011)
Hata, H., Sheu, S.J.: On the Hamilton–Jacobi–Bellman equation for an optimal consumption problem: I. Existence of solution. SIAM J. Control Optim 50(4), 2373–2400 (2012)
Hata, H., Sheu, S.J.: On the Hamilton–Jacobi–Bellman equation for an optimal consumption problem: II. Verification theorem. SIAM J. Control Optim. 50(4), 2401–2430 (2012)
Kaise, H., Sheu, S.J.: On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34(1), 284–320 (2006)
Tunaru, R.: Dividend derivatives. Quant. Finance 18(1), 1–19 (2018)
Geske, R.: The pricing of options with stochastic dividend yield. J. Finance 33(2), 617–625 (1978)
Lioui, A.: Black–Scholes–Merton revisited under stochastic dividend yields. J. Futures Mark. 26(7), 703732 (2006)
Pang, T., Varga, K.: Optimal investment and consumption for a portfolio with stochastic dividends. J. Res. Finance Manag. 1(2), 1–22 (2015)
Chevalier, E., Vath, V.L., Scotti, S.: An optimal dividend and investment control problem under debt constraints. SIAM J. Financ. Math. 4, 297–326 (2013)
Fleming, W.H., Pang, T.: A stochastic control model of investment, production and consumption. Q. Appl. Math. 63, 71–87 (2005)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (1993)
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Appendices
Appendices
Appendix A: Proof of Lemma 2.1
Proof
First consider \(\rho \in [0,1]\). It is easy to get that \(q(y,z)> \sigma _2^2(z) \ge \tilde{\sigma }_2^2>q_0> 0\). Now consider \(\rho _0 \le \rho <0 \). The minimum of q with respect to y is given by the function \(\sigma _2^2(z)(1-\rho ^2)\). So we have
So (24) holds for all \(\rho \in [\rho _0,1]. \)\(\square \)
Appendix B: Proof of Lemma 2.2
Proof
To prove this, we consider two cases. First, let \(0 \le \rho \le 1\). Then, by virtue of (26) and (15), we can get
If \((\mu - r)^2 - \frac{b^2\tilde{\sigma }_2^2}{\sigma _1^2} \le 0\), then, we have \(\varPsi (y,z) \le \frac{b^2}{\sigma _1^2(1-\gamma )}\). Otherwise, we have
Thus, if \(0\le \rho \le 1\), we have
We can get a similar bound if \(\rho \) is negative. If \(\rho _0 \le \rho < 0,\) then \(\rho (\sigma _1 e^{-y} - \sigma _2(z))^2 \le 0\) implies
and so we can get
Let
Then, we can get that \(0 \le \varPsi (y,z) \le {\bar{\varPsi }} < \infty \). Thus, \(\varPsi (y,z)\) is bounded. \(\square \)
Appendix C: Proof of Lemma 3.2
Proof
Recall from (25) that for \(k^*> 0\), G is defined by
or in the trivial case when \(k^*=0\), \(G\equiv 0\) otherwise. Consider the expression for G given above. We can expand G to a quadratic form in p:
where
By the definition of q(y, z) [see (15)], it is not hard to show that \(|g_2|, |g_1|,\) and \(|g_0|\) are bounded for all \((y,z) \in \mathbb {R}^2\). Then, we can get (36) very easily. \(\square \)
Appendix D: Proof of Lemma 3.3
Proof
Suppose \(\displaystyle \sup _{(y,z) \in \bar{B}_R} (\tilde{Q} - \hat{Q})(y,z) > 0\). Since \(\tilde{Q} \le \hat{Q}\) on \(\partial B_R,\) this implies that \(\tilde{Q} - \hat{Q}\) reaches its maximum at \((y_0, z_0) \in B_R\). So we have that
By (37) we have that, for \((y,z) \in B_R,\)
Evaluating this inequality at \((y_0,z_0)\) and applying (92), (93) and (32), we can get
or equivalently \(\hat{Q}(y_0,z_0) > \tilde{Q}(y_0,z_0),\) which is a contradiction. Therefore, \(\tilde{Q} \le \hat{Q}\) on \({\bar{B}}_R\). \(\square \)
Appendix E: Proof of Corollary 3.1
Proof
Suppose \(Q^{(1)}, Q^{(2)} \in C^2({\bar{B}}_R)\) are both solutions of (34). Then, \(Q^{(1)} = Q^{(2)}\) on \(\partial B_R\). Further, for \((y,z)\in B_R\), we have
Now we assume that \(\displaystyle \sup _{(y,z)\in B_R} (Q^{(2)}-Q^{(1)}) > 0\). Then, \(Q^{(2)} - Q^{(1)}\) attains its maximum at some \((y_0,z_0) \in B_R\). Then,
Evaluating (94) at \((y_0,z_0)\) and applying (96) and (95), we arrive at
or \(Q^{(2)}(y_0,z_0) < Q^{(1)}(y_0,z_0),\) which is a contradiction. Therefore, we must have \(Q^{(2)} \le Q^{(1)}\) in \(B_R\). Similarly, we can show that \(Q^{(2)} \ge Q^{(1)}\) in \(B_R\) by assuming that
Therefore, \(Q^{(1)} \equiv Q^{(2)}\) on \({\bar{B}}_R\). \(\square \)
Appendix F: Proof of Lemma 3.4
Proof
By the Cauchy–Schwarz Inequality, we have
This completes the proof. \(\square \)
Appendix G: Proof of Theorem 3.3
Proof
We first prove inequality (41). Let
Then, it is easy to show that \(\tilde{Q}^\tau \) is a subsolution of (34). We also have that \(\tilde{Q}^\tau \le \tau \psi = Q^\tau \) on \(\partial B_R\). Therefore, \(\tilde{Q}^\tau \) and \(Q^\tau \) satisfy
Then, by Lemma 3.3, \(\tilde{Q}^\tau \le Q^\tau \) holds in \({\bar{B}}_R,\) which gives us (41). Next we prove (38). It is equivalent to show that
Define \( V(x,y,z) \equiv \frac{1}{\gamma }x^\gamma e^{\hat{Q}(y,z)} \quad \text {and} \quad V^\tau (x,y,z) \equiv \frac{1}{\tau \gamma }x^{\tau \gamma } e^{Q^\tau (y,z)}\). Then, we have
Now, we define
Since f does not depend on x, we have that \((V_0)_x = V_x\) and \((V_0^\tau )_x = V^\tau _x\). Thus, we can get
Define the operator \(\mathcal {L}^{k, c}\) as
Noting that f satisfies (39) and \(\hat{Q}\) is a supersolution of (34) for \(\tau =1\), we can get that, for \((y, z) \in B_R\),
Similarly, for \(V_0^\tau \), we have
Suppose that the optimal controls for the above equation are given by \({\tilde{k}}\) and \({\tilde{c}}\). Then, we can rewrite (101) as
At the same time, from (100) we obtain
Define \( g(\tau ;\theta ) = \frac{1}{\tau \gamma }(\theta ^\tau - 1), \quad \forall 0<\tau \le 1. \) Then, the difference between Eqs. (102) and (103) is the function \(g (\tau ; \theta )\), with \(\theta = (cx)^\gamma > 0\) for \(\tau =1\) in (102) and \(\tau <1\) in (103). It is not hard to verify that \(g'(\tau )>0\), so \(g(\tau )\) is a nonincreasing function with respect to \(\tau \). Therefore,
or equivalently,
Therefore, from (103) we can get
Subtracting the above inequality from (102), we have
Note that (106) holds for \(x > 0, (y,z) \in \bar{B}_R\). This equation is used later in this proof to show a contradiction.
To prove the estimate in (99), we wish to show that for \(x >0, (y,z) \in {\bar{B}}_R,\)
We then take \(x = 1\) to get the desired result.
From the definition of f [see (40)], we can get
Hence \(f > 0\). Therefore,
On the boundary \((y,z) \in \partial B_R\), we have that \(\hat{Q} \ge \psi \). Using (104) with \(\theta = x^\gamma e^{\psi }\), we can get that
Next we prove that \(V_0^\tau \le V_0\) for \((y,z) \in {\bar{B}}_R\). Suppose on the contrary that
Then, the maximum is attained at some \((x_0,y_0,z_0)\), where \(x_0 > 0\) and \(|(y_0,z_0)| < R\). So at \((x_0,y_0,z_0)\) we have the following:
and \(D^2(V_0^\tau - V_0)\) is negative semi-definite at \((x_0,y_0,z_0)\), where \(D^2\) is the \(3 \times 3\) matrix operator of second derivatives (the Hessian). That is, for any \(\eta \in \mathbb {R}^3,\)
or in expanded form, at \((x_0, y_0, z_0)\) for any \(\eta _1, \eta _2, \eta _3 \in \mathbb {R}\),
Note that this implies
at the point \((x_0,y_0,z_0)\). We also note that
Now we evaluate (106) at \((x_0,y_0,z_0)\) and apply conditions (111) and (112). Then, at the point \((x_0,y_0,z_0)\), we have
at \((x_0,y_0,z_0)\). Taking \(\eta _1 = \frac{{\tilde{k}} x}{\sqrt{2}}(\sigma _2 + \rho \sigma _1 e^{-y}), \eta _2 = \frac{\sigma _2(z)}{\sqrt{2}}\), and \(\eta _3 = 0\), we get a contradiction to (114). Therefore, \(V_0^\tau \le V_0\) for \((y,z) \in {\bar{B}}_R\). Thus, (107) holds. We take \(x=1\) in (107) to arrive at (38). \(\square \)
Appendix H: Proof of Theorem 3.4
Proof
Since \(Q_\tau ^0\) is a solution of (42), by virtue of the definition of H, we have that
Then, we can get
Applying Ito’s rule to \(Q_\tau ^0(\hat{Y}_t,\hat{Z}_t)\) and using (118), we have that, for \(0 \le t \le {\bar{t}}_R,\)
Integrate it from 0 to \(\bar{t}_R\), and we can get
Since \(Q^0_\tau = 0\) on \(\partial B_R\), we can take expectations to the above inequality and get
which proves the first inequality in (43). Define \(\phi (y,z) = e^{Q_\tau ^0(y,z)}\). Then,
Then, it is easy to check that \(\phi \) satisfies
Since the terms \(\displaystyle \frac{1-\tau }{2\phi } (\sigma _2^2(z) \phi ^2_y + \sigma _3^2 \phi ^2_z)\) and \(\tau \beta \phi \) are nonnegative, we can get an inequality:
Next, we can apply Ito’s rule to \(\phi (\hat{Y}_t,\hat{Z}_t)\) and use the above inequality to obtain
The integral form is
Using the boundary condition that \(\phi =1\) on \(\partial B_R\), we can get that \(1 - \phi (y,z) \ge -\tau \mathbf {E}[{\bar{t}}_R]\), which implies
This proves the second inequality in (43). \(\square \)
Appendix I: Proof of Lemma 4.1
Proof
By the definitions of \((k^*, c^*)\), Theorem 3.6, and the fact that \({\tilde{Q}}\) is bounded, it is easy to show that \(k^*\) and \(c^*\) are bounded. Further, by the definition of \(k^*\) and q(y, z) [see (15)], we can get that \(e^{-y} k^*\) is bounded. Therefore, we can assume that there is a constant \(\varLambda \) such that
From Eq. (9), we can get that
Using the above equation and (121), we can get (71). \(\square \)
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Pang, T., Varga, K. Portfolio Optimization for Assets with Stochastic Yields and Stochastic Volatility. J Optim Theory Appl 182, 691–729 (2019). https://doi.org/10.1007/s10957-019-01513-y
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DOI: https://doi.org/10.1007/s10957-019-01513-y