Study of a degenerate elliptic equation in an optimal consumption problem under partial information

Abstract

We consider an optimal consumption/investment problem under partial information. By using a known reduction scheme, the problem can be transformed to a standard stochastic control problem. The related Hamilton–Jacobi–Bellman (HJB) equation is, after suitable transformation, a second-order linear degenerate elliptic equation. Although the problem is somewhat simple and classical, and the method of solution is well known, it is not obvious that the related HJB equation has a unique classical solution. Here, we prove the existence, uniqueness, and boundary regularity of the solution to the HJB equation. The main components of the proof are the construction of an appropriate weak solution and the verification of the Hörmander condition. Using the classical solution, we also prove the verification theorem for the original optimal consumption/investment problem.

A Proof of Lemma 1

Proof

Step 1. Let $\xi _i(t):=\mathbf {1}_{\{\xi (t)=\mathbf {e}_i\}}$ and

$$\begin{aligned} z_i(t) := \bar{E}[H(t)\xi _i(t) |\mathcal {Y}_{t}], \qquad i=1,\ldots ,d. \end{aligned}$$

(57)

Then, the process $z(t) = (z_1(t),\ldots ,z_d(t))$ satisfies the following equation.

$$\begin{aligned} dz_i(t)&= \gamma \eta (\mathbf {e}_{i},h(t),c(t))z_i(t)dt + z_i(t) ( g(\mathbf {e}_i) + \gamma \sigma ^*h(t) ) \cdot dY(t)\nonumber \\&+ (Q^{*}z(t))_idt, \\ z_i(0)&= \mathbf {p}_i(0), \qquad i=1,\ldots ,d \nonumber \end{aligned}$$

(58)

We only outline the proof of (58). For technical details, see [3, Sects. 8.2–8.3] and [1, Chap. 3]. The dynamics of $\xi (t)$ is given by

$$\begin{aligned} d\xi (t) = Q^{*}\xi (t)dt + dM(t), \end{aligned}$$

where $M(t)=(M_1(t),\ldots ,M_d(t))$ is an $\mathcal {F}_t$-martingale of pure jump type taking values in $\{\mathbf {e}_1,\ldots ,\mathbf {e}_d\}$. Recalling (4) and using Itô’ s formula, we obtain

$$\begin{aligned} H(t)\xi _i(t)&= \xi _i(0) + \int _0^t H(s)dM_i(s) + \int _0^t \gamma \eta (\xi (s),h(s),c(s))H(s)\xi _i(s)ds \\&+ \int _0^t H(s)\xi _i(s) ( g(\xi (s)) + \gamma \sigma ^* h(s) ) \cdot dY(s)\\&+ \int _0^t (Q^* H(s)\xi (s))_ids. \end{aligned}$$

We apply $\bar{E}[\cdot |\mathcal {Y}_t]$ to both sides:

$$\begin{aligned} \bar{E}[\xi _i(0)|\mathcal {Y}_t]&= \bar{E}[\xi _i(0)] = E[\xi _i(0)] = \mathbf {p}_i(0),\\ \bar{E}[\int _0^t H(s)dM_i(s) |\mathcal {Y}_t]&= 0, \\ \bar{E}[\int _0^t \gamma \eta (\xi (s),h(s),c(s))H(s)\xi _i(s)ds|\mathcal {Y}_t]&= \int _0^t \bar{E}[\gamma \eta (\xi (s),h(s),c(s))H(s)\xi _i(s)|\mathcal {Y}_s]ds\\&= \int _0^t \gamma \eta (\mathbf {e}_{i},h(s),c(s))z_i(s)ds,\\ \bar{E}[ \int _0^t H(s)\xi _i(s) ( g(\xi (t)) + \gamma \sigma ^*h(s) ) \cdot dY(s)|\mathcal {Y}_t]&= \int _0^t \bar{E}[H(s)\xi _i(s) ( g(\xi (t)) + \gamma \sigma ^* h(s) |\mathcal {Y}_s]\cdot dY(s)\\&= \int _0^t z_i(s) ( g(\mathbf {e}_i) + \gamma \sigma ^*h(s))\cdot dY(s),\\ \bar{E}[\int _0^t (Q^* H(s)\xi (s))_ids|\mathcal {Y}_t]&= \int _0^t \bar{E}[(Q^* H(s)\xi (s))_i|\mathcal {Y}_s]ds = \int _0^t (Q^*z(s))_ids. \end{aligned}$$

Combining these, we obtain (58).

Step 2. Next, we claim that

$$\begin{aligned} z_i(t) = \bar{H}(t)x_i(t), \qquad i=1,\ldots ,d. \end{aligned}$$

(59)

Indeed, if we compute $d(\bar{H}(t)x_i(t))$ by using Itô’s formula, we see that $\tilde{q}_i(t):= \bar{H}(t)x_i(t)$ also satisfies (58). A similar computation is seen in the proof of Proposition 2.1 in [18].

Step 3. From (5), (57), (59), and that $\sum _{i=1}^d x_i(t) = 1$, we deduce

$$\begin{aligned} E[(c(t)V^{h,c}(t))^\gamma ]&= v_0^\gamma \bar{E}[c(t)^\gamma H(t)]\\&= v_0^\gamma \bar{E}[c(t)^\gamma \bar{E}[H(t)|\mathcal {Y}_{t}]] = v_0^\gamma \bar{E}[ c(t)^\gamma \sum _{i=1}^{d}\bar{E}[H(t)\xi _i(t)|\mathcal {Y}_{t}]]\\&= v_0^\gamma \bar{E}[c(t)^\gamma \sum _{i=1}^{d}z_i(t)]= v_0^\gamma \bar{E}[ c(t)^\gamma \sum _{i=1}^{d}\bar{H}(t)x_i(t)] \\&= v_0^\gamma \bar{E}[ c(t)^\gamma \bar{H}(t)]. \end{aligned}$$

In a similar way, we obtain $E[(V^{h,c}(t))^\gamma ] = v_0^\gamma \bar{E}[\bar{H}(t)]$. $\square$