H–J–B Equations of Optimal Consumption-Investment and Verification Theorems

Abstract

We consider a consumption-investment problem on infinite time horizon maximizing discounted expected HARA utility for a general incomplete market model. Based on dynamic programming approach we derive the relevant H–J–B equation and study the existence and uniqueness of the solution to the nonlinear partial differential equation. By using the smooth solution we construct the optimal consumption rate and portfolio strategy and then prove the verification theorems under certain general settings.

Appendices

Appendix 1

Here, we give the proof of the existence of a solution to (2.11) along the line of Hata and Sheu [7], which complete the proofs of Theorems 4.1 and 4.2. We rewrite the equation (2.11) as follows.

$$\begin{aligned} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+b(x,z,Dz;\gamma )=0, \quad \text{ in }\;\; R^n, \end{aligned}$$

(6.1)

where

$$\begin{aligned} b(x,z,p;\gamma )=\beta _{\gamma }(x)^*p+\frac{1}{2}p^*\lambda N_{\gamma }^{-1}\lambda ^*p+U_{\gamma }(x)+(1-\gamma )e^{-\frac{z}{1-\gamma }}-\rho . \end{aligned}$$

To prove the existence of a solution to (6.1), we first consider the Dirichlet problem on $B_R$:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+b(x,z,Dz;\gamma )=0, &{} x\in B_R\\ z={\underline{z}},&{} x\in \partial B_R, \end{array}\right. \end{aligned}$$

(6.2)

where ${\underline{z}}$ is a smooth sub-solution to (2.11). For $\gamma <0$, ${\underline{z}}$ is the sub-solution appeared in Lemma 3.1 and for $0<\gamma <1$, ${\underline{z}}=-C$ with sufficiently large $C$. In proving the existence of a solution to (6.2) we appeal to the following theorem, a modification of the Leray-Schauder fixed point theorem, according to the scheme due to Hata and Sheu ([7]).

Theorem 6.1

(Hata and Sheu [7]) Let ${\mathcal B}$ be a Banach space with the norm $\parallel \cdot \parallel _{{\mathcal B}}$ and $T$ a continuous, compact operator from ${\mathcal B}\times [0,1]$ to ${\mathcal B}$. Assume that there exists a constant $M>0$ such that $\parallel \xi \parallel _{{\mathcal B}}<M$ for all $(\xi ,\tau )$ satisfying $\xi =T(\xi ,\tau )$, or $\xi =\tau T(\xi ,0)$. Then, there exists a fixed point $\xi \in {\mathcal B}$: $\xi = T(\xi ,1)$.

In applying this theorem, we consider the following problem for each $\tau \in [0,1]$:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+b(x,z,Dz;\tau \gamma )=0, &{} x\in B_R\\ z=\tau {\underline{z}},&{} x\in \partial B_R. \end{array}\right. \end{aligned}$$

(6.3)

We note that, under assumptions (2.13)–(2.15), for $x\in B_R,\; |z|\le M$, we have

$$\begin{aligned}&c_1|\xi |^2\le \frac{1}{2}\xi ^*\lambda \lambda ^*(x)\xi \le c_2|\xi |^2,\quad \xi \in R^n\\&|b(x,z,p;\tau \gamma )|+|\frac{\frac{1}{2}\partial (\lambda \lambda ^*)^{ij}(x)}{\partial x_j}|\le c_3(1+|p|^2), \end{aligned}$$

where $c_1,c_2$ and $c_3$ are positive constants. To study (A.3), we consider the linear partial differential equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+b(x,w,Dw;\tau \gamma )=0,&{} x\in B_R\\ z=\tau {\underline{z}}, &{} x\in \partial B_R. \end{array}\right. \end{aligned}$$

(6.4)

for a given function $w\in C^{1,\mu }({\overline{B}_R})$, $1>\mu >0$. Under assumptions (2.13)–(2.15), we have a unique solution $z\in C^{2,\mu \mu '}({\overline{B}_R})$ since ${\underline{z}}\in C^{2,\mu }({\overline{B}_R})$ (cf. [13]). Thus, we can define a continuous, compact mapping $T(w,\tau )$ of ${\mathcal B}\times [0,1]$ into ${\mathcal B}$ as $T(w,\tau )=z$, where $z$ is the solution to (6.4) for a given function $w\in {\mathcal B}:=C^{1,\mu }({\overline{B}_R})$ and $\tau \in [0,1]$. Indeed, since $b(x,w,Dw;\tau \gamma )\in C^{0,\mu }({\overline{B}_R})$ for $w\in C^{1,\mu }({\overline{B}_R})$, and we assume that $\tau {\underline{z}} \in C^{2,\mu }({\overline{B}_R})$, $T(w,\tau )$, for every $\tau$, transforms the function $w\in C^{1,\mu }({\overline{B}_R})$ into $z(x;\tau )$ in $C^{2,\mu '\mu }({\overline{B}_R})$. Further, we have

$$\begin{aligned} \parallel z \parallel _{C^{2,\mu '\mu }({\overline{B}_R})}\le f(\parallel w\parallel _{C^{1,\mu }({\overline{B}_R})}), \end{aligned}$$

where $f$ is a continuous monotonically increasing function of $t\in [0,\infty )$. Since an arbitrary bounded set in $C^{2,\mu '\mu }({\overline{B}_R})$ is compact in the space $C^{1,\mu }({\overline{B}_R})$, $T(w,\tau )$ maps each bounded set of the pairs $(w,\tau )$ in $C^{1,\mu }({\overline{B}_R})\times [0,1]$ into a compact set in $C^{1,\mu }({\overline{B}_R})$. On the other hand, when $\parallel w_1-w_2\parallel _{C^{1,\mu }({\overline{B}_R})}$ goes to $0$, $\parallel b(x,w_1,Dw_1;\tau \gamma )- b(x,w_2,Dw_2;\tau \gamma )\parallel _{C^{0,\mu }({\overline{B}_R})}$ tends to $0$ uniformly with respect to $\tau$. Moreover, $T(w,\tau )$ is continuous in $\tau$ uniformly with respect to $(x, w, Dw) \in {\overline{B}_R}\times \{u\in R^n;|u|\le c\}\times \{p\in R^n; |p|\le c\}$. Therefore $T(w,\tau )$ is a continuous map of $(w,\tau )\in C^{1,\mu }({\overline{B}_R})\times [0,1]$ into $C^{1,\mu }({\overline{B}_R})$.

Note that a fixed point $z^{(\tau )}$ of $T$, $z^{(\tau )}=T(z^{(\tau )},\tau )$, is a solution to (6.3) and $z^{(1)}$ is a solution to (6.2). On the other hand, if we set $z_0=T(w,0)$, then $z_0$ satisfies

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+b(x,w,Dw;0)=0, &{} x\in B_R\\ z=0,&{} x\in \partial B_R, \end{array}\right. \end{aligned}$$

(6.5)

with

$$\begin{aligned} b(x,w,Dw;0)=\beta (x)^*Dw+\frac{1}{2}(Dw)^*\lambda \lambda ^*Dw+e^{-w}-\rho . \end{aligned}$$

Therefore, $z_0^{(\tau )}=\tau T(w;0)$ turns out to be a solution to the equation

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+\tau b(x,w,Dw;0)=0, &{} x\in B_R\\ z=0,&{} x\in \partial B_R. \end{array}\right. \end{aligned}$$

(6.6)

Hence, a fixed point ${\hat{z}}_0^{\tau }=\tau T({\hat{z}}_0^{\tau };0)$ is a solution to

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2z]+\tau b(x,z,Dz;0)=0, &{} x\in B_R\\ z=0,&{} x\in \partial B_R. \end{array}\right. \end{aligned}$$

(6.7)

Now, let us give the proof of the existence of a solution to (6.1).

Step 1. Proof of the existence of a solution to (6.2).

Owing to Theorem 3.8 in Hata and Sheu [7], we have the estimate for $z^{(\tau )}$:

$$\begin{aligned} e^{z^{(\tau )}(x)}\le \tau e^{{\overline{z}}(x)}+(1-\tau )f_0(x), \end{aligned}$$

(6.8)

where $f_0(x)$ is the solution to

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}\text{ tr }[\lambda \lambda ^* D^2f_0]+\beta (x)Df_0-\rho +1=0, &{} x\in B_R\\ f_0=1, &{} x\in \partial B_R. \end{array}\right. \end{aligned}$$

(6.9)

Moreover, we see that

$$\begin{aligned} z^{(\tau )}(x) \ge -C \end{aligned}$$

(6.10)

for sufficiently large $C>0$. Indeed, when $\gamma <0$, for $x\in \{x; {\underline{z}}(x)\le 0\}$ $\tau {\underline{z}}(x)\ge {\underline{z}}(x)>-C$ for each $\tau \in (0,1]$ since ${\underline{z}}(x)$ is bounded in ${\overline{B}_R}$. $\tau {\underline{z}}(x)>-C$ holds as well in $x\in \{x; {\underline{z}}(x)\le 0\}^c$. Moreover, $-C$ becomes a sub-solution of (6.3) by taking $C$ to be sufficiently large. Therefore we see (6.10) owing to Lemma 3.6 in [7]. Similarly, (6.10) holds also in the case of $0<\gamma <1$.

Further, owing to Theorem 3.9 in Hata and Sheu [7], we have

$$\begin{aligned} -\rho E[\sigma _R]\le z_0^{(\tau )}\le \log (1+E[\sigma _R]), \end{aligned}$$

(6.11)

where $\sigma _R=\inf \{ t; |X_t|=R\}$, and $X_t$ is the solution to the stochastic differential equation:

$$\begin{aligned} dX_t=\tau \beta (X_t)dt+\lambda (X_t)dW_t,\quad X_0=x. \end{aligned}$$

Therefore, from Theorem 4.1 and 6.1, Chapter 4 in [13], we obtain the estimates

$$\begin{aligned} \sup _{B_R}|z^{(\tau )} |\le M,\quad \sup _{B_R}|z_0^{(\tau )} | \le M \end{aligned}$$

for a positive constant $M$ independent of $\tau$, $z^{(\tau )}$ and $z_0^{(\tau )}$. Then, from Theorem 4.1 and 6.1, Chapter 4 in [13], we obtain

$$\begin{aligned}&\sup _{B_R}|\nabla z^{(\tau )}|\le M_1,\;\sup _{B_R}|\nabla z_0^{(\tau )}|\le M_1,\\&\sum _{i=1}^n\parallel D_iz^{(\tau )}\parallel _{C^{1,\mu '}({\overline{B}_R})}\le M_2,\; \sum _{i=1}^n\parallel D_iz_0^{(\tau )}\parallel _{C^{1,\mu '}({\overline{B}_R})}\le M_2, \end{aligned}$$

where the constants $M_1$, $M_2$ and $\mu '$ are determined by $n$, $M$, $c_1$, $c_2$, and $c_3$. Thus, we can consider the operator $T(w;\tau )$ only on the space

$$\begin{aligned} Z:=\{z\in C^{1,\mu }({\overline{B}_R}); \sup _{B_R}|z |&\le M+\epsilon ,\; \sup _{B_R}|\nabla z|\le M_1\nonumber \\&\quad +\,\epsilon ,\;\sum _{i=1}^n\parallel D_iz\parallel _{C^{1,\mu '}({\overline{B}_R})}\le M_2+\epsilon \} \end{aligned}$$

for $\epsilon >0$. Then, we see that there exists ${\tilde{M}}$ such that

$$\begin{aligned} \parallel z^{(\tau )}\parallel _{C^{1,\mu '}({\overline{B}_R})}, \;\parallel z_0^{(\tau )}\parallel _{C^{1,\mu '}({\overline{B}_R})}<{\tilde{M}} \end{aligned}$$

for any fixed points $z^{(\tau )}$ and $z_0^{(\tau )}$. Hence, Theorem 6.1 applies and we see that $z\in C^{2,\mu \mu '}({\overline{B}_R})$ such that $z=T(z,1)$.

Step 2. Proof of the existence of the solution to (6.1).

Let us take a sequence $\{R_n\}$ such that $R_n\rightarrow \infty$, as $n\rightarrow \infty$, and a sequence of solutions $z_{R_n}$ to (6.2). Since ${\underline{z}}$ is a subsolution to (6.1), we can see that $z_{R_n}$ is nondecreasing because of the maximum principle. Further, we can see that $z_{R_n}$ is dominated by the super-solution ${\overline{z}}(x)$ again by the maximum principle. Therefore, there exists $z(x)$ to which $z_{R_n}$ converges as $n\rightarrow \infty$. Note that there exists a constant $M$ independent of $n$ such that

$$\begin{aligned} \sup _{x\in B_r}|\nabla z_{R_n}|\le M(1+r) \end{aligned}$$

for $r<R_n$, which can be seen in a similar manner to the proof of Proposition 3.2 in [17] (cf. Appendix 2 and also (3.8)). Therefore, we can see that $z_{R_n}\rightarrow z$, $W^{1,p}_{loc}$ weakly $\forall p>1$ by taking a subsequence if necessary. The convergence can be strengthen as $\nabla z_{R_n}$ converges in $L^2_{loc}$ strongly to $\nabla z$. As a result we can see that $z\in W^{1,p}_{loc}$ is a weak solution to (6.1). Then, from the regularity theorem we see that $z\in C^{2,\mu ''}$ and that it is a classical solution to (6.1). $\square$

Appendix 2

Let us give the gradient estimates for the solution to H–J–B equation (2.11).

Lemma 7.1

Under assumptions(2.13)–(2.15), the solution $z$ to (2.11) satisfies the following estimate.

$$\begin{aligned} |\nabla z|\le C(1+|x|) \end{aligned}$$

(7.1)

for some positive constant $C>0$.

The proof of this estimate is almost the same as the one of Proposition 3.2 in [17]. Here we only give some remarks that the proof could proceed in almost parallel to it. One could see [17] to be more precise.

Proof

Set $Q^{ij}:=(\lambda N_{\gamma }^{-1}\lambda ^*)^{ij}$ and differentiate (2.11). Then, we have

$$\begin{aligned} 0&= \frac{1}{2}(\lambda \lambda ^*)^{ij}D_{ijk}z+\beta _{\gamma }^iD_{ik}z+(D_iz)Q^{ij}D_{jk}z+D_kU_{\gamma }-D_kze^{-\frac{z}{1-\gamma }}\nonumber \\&\quad +\frac{1}{2}D_k(\lambda \lambda ^*)^{ij}D_{ij}z+(D_k\beta _{\gamma }^i)D_iz+\frac{1}{2}(D_iz)(D_kQ^{ij})D_jz \end{aligned}$$

(7.2)

Set $F=|\nabla z|^2=\sum _{k=1}^n|D_kz|^2$ and

$$\begin{aligned} \Gamma (F):=\frac{1}{2}(\lambda \lambda ^*)^{ij}D_{ij}F+\beta _{\gamma }^iD_iF+Q^{ij}D_izD_jF. \end{aligned}$$

Then, we have

$$\begin{aligned} \Gamma (F)&= (\lambda \lambda ^*)^{ij}D_{ik}zD_{jk}z+D_kz\{(\lambda \lambda ^*)^{ij}D_{ijk}z+2\beta _{\gamma }^iD_{ik}z+2Q^{ij}D_izD_{jk}z\}\\&= (\lambda \lambda ^*)^{ij}D_{ik}zD_{jk}z+D_kz\{-2D_kU_{\gamma }+2D_kze^{-\frac{z}{1-\gamma }}-D_k(\lambda \lambda ^*)^{ij}D_{ij}z\\&\quad -2(D_k\beta _{\gamma }^i)D_iz-(D_iz)(D_kQ^{ij})D_jz\}\\&\ge \frac{1}{2nc_2}\{(\lambda \lambda ^*)^{ij}D_{ij}z\}^2+\frac{1}{2}(\lambda \lambda ^*)^{ij}D_{ik}zD_{jk}z-\frac{|\nabla z|^2}{2\delta }-\frac{\delta C}{2}|D^2z|^2\\&\quad -2|\nabla z||\nabla U_{\gamma }|+2|\nabla z|^2e^{-\frac{z}{1-\gamma }}-2|\nabla z|^2|\nabla \beta _{\gamma }|-|\nabla z|^3|\nabla Q|\\&\ge \frac{1}{2nc_2}\{(\lambda \lambda ^*)^{ij}D_{ij}z\}^2-\frac{|\nabla z|^2}{2\delta }-2|\nabla z||\nabla U_{\gamma }|+2|\nabla z|^2e^{-\frac{z}{1-\gamma }}\\&\quad -2|\nabla z|^2|\nabla \beta _{\gamma }|-|\nabla z|^3|\nabla Q|. \end{aligned}$$

Here we have used (7.2) and the matrix inequality $(\text{ tr }[AB])^2\le nC\text{ tr }[AB^2]$, for symmetric matrix $B$ and nonnegative definite symmetric matrix A having the maximum eigenvalue $C$. Thus, in a similar manner to the proof of Proposition 3.2 in [17] (cf. also [9, 10, 15]) we can obtain estimate (7.1) by using H–J–B equation (2.11) again in the last line in the above. Although we have the term $(1-\gamma )e^{-\frac{z}{1-\gamma }}$ in the equation it does not affect the proof since it is nonnegative. $\square$