Stochastic control on the half-line and applications to the optimal dividend/consumption problem

TL;DR

This paper studies a stochastic control problem with barrier constraints, proves smooth solutions to the associated Hamilton-Jacobi-Bellman equations, and applies these results to optimize dividend and consumption strategies.

Contribution

It introduces a method to solve barrier-constrained stochastic control problems and applies it to optimal dividend and consumption issues.

Findings

Proved existence of smooth solutions to Hamilton-Jacobi-Bellman equations under general conditions.

Developed a fixed point approach using stochastic representations for linear equations.

Applied the theoretical results to derive optimal strategies in dividend and consumption problems.

Abstract

We consider a stochastic control problem with the assumption that the system is controlled until the state process breaks the fixed barrier. Assuming some general conditions, it is proved that the resulting Hamilton Jacobi Bellman equations has smooth solution. The aforementioned result is used to solve the optimal dividend and consumption problem. In the proof we use a fixed point type argument, with an operator which is based on the stochastic representation for a linear equation.

Equations

$$D_t+\frac{1}{2}{\sigma }^2(x,t)D_x^{2}u+\underset{\delta \in D}{\max }\left(i(x,t,\delta )D_{x}u+h(x,t,\delta )u+f(x,t,\delta )\right)=0,(x,t)\in (0,+\infty )\times [0,T)$$

$$\{\begin{array}{ll}{D}_t+\frac{1}{2}{\sigma }^2(x,t)D_x^{2}u+H(D_{x}u,u,x,t)=0,& (x,t)\in (0,+\infty )\times [0,T),\\ u(x,t)=\beta (x,t),& (x,t)\in \partial \left((0,+\infty )\times [0,T)\right).\end{array}$$

$$d{X}_t=\sigma (X_t,t)d{W}_t$$

$$\tau (x,t):=\inf \{k\ge 0|X_k(x,t)=0\}.$$

$$\displaystyle|H(p,u,x,t)|\leq K\bigl{(}1+|u|+|p|\bigr{)},$$

$$\displaystyle|H(p,u,x,t)-H(\bar{p},\bar{u},x,t)|\leq K\bigl{(}|u-\bar{u}|+|p-\bar{p}|\bigr{)}.$$

$$\left|\mathbb{E}\tau (x,t)\wedge T-\mathbb{E}\tau (\bar{x},t)\wedge T\right|\le L|x-\bar{x}|.$$

$$\Vert u{\Vert }_{\kappa }:=\underset{(x,t)\in [0,+\infty )\times [0,T]}{\sup }{e}^{-\kappa (T-t)}|u(x,t)|+\underset{(x,t)\in (0,+\infty )\times [0,T)}{\sup }{e}^{-\kappa (T-t)}|D_{x}u(x,t)|.$$

$$\Vert u{\Vert }_{\kappa }^0=\underset{(x,t)\in [0,+\infty )\times [0,T]}{\sup }{e}^{-\kappa (T-t)}|u(x,t)|.$$

$$\{\begin{array}{ll}{D}_t+\frac{1}{2}{\sigma }^2(x,t)D_x^{2}u+f(x,t)=0,& (x,t)\in (0,+\infty )\times [0,T),\\ u(x,t)=0,& (x,t)\in \partial \left((0,+\infty )\times [0,T)\right).\end{array}$$

$$\Vert u_f{\Vert }_{\kappa }\le \frac{M}{\kappa }\Vert f{\Vert }_{\kappa }^0.$$

$$u_f(x,t)={\mathbb{E}}_{x,t}{\int }_t^{\tau (x,t)\wedge T}f(X_s,s)ds.$$

$$\begin{gathered} e^{-\kappa (T-t)}{u}_f(x,t)={\mathbb{E}}_{x,t}{e}^{-\kappa (T-t)}{\int }_t^{\tau (x,t)\wedge T}{e}^{\kappa (T-s)}{e}^{-\kappa (T-s)}f(X_s,s)ds \\ \le \Vert f{\Vert }_{\kappa }^0{e}^{-\kappa (T-t)}{\int }_t^T{e}^{\kappa (T-s)}ds=\frac{1}{\kappa }\Vert f{\Vert }_{\kappa }^0{e}^{-\kappa (T-t)}(e^{\kappa (T-t)}-1)\le \frac{1}{\kappa }\Vert f{\Vert }_{\kappa }^0. \end{gathered}$$

$$\begin{gathered} |u(x,t)-u(\bar{x},t)|\le \left|\mathbb{E}{\int }_t^{\tau (x,t)\wedge T}f(X_s(x,t))ds-\mathbb{E}{\int }_t^{\tau (x,t)\wedge T}f(X_s(\bar{x},t))ds\right| \\ +\left|\mathbb{E}{\int }_t^{\tau (x,t)\wedge T}f(X_s(\bar{x},t))-{\int }_t^{\tau (\bar{x},t)\wedge T}f(X_s(\bar{x},t))\right|=:I_1+I_2 \end{gathered}$$

$$|\Gamma (x,t,y,s)|$$

$$|D_x\Gamma (x,t,y,s)|$$

$$\begin{gathered} I_1=\left|\mathbb{E}{\int }_t^{\tau (x,t)\wedge T}f(X_s(x,t),s)ds\right| \\ \le {\int }_t^T\left|\mathbb{E}{\chi }_{\{\tau (x,t)>s\}}[f(X_s(x,t),s)-f(X_s(\bar{x},t),s)]\right|ds. \end{gathered}$$

$$I_1$$

$$\le {\int }_t^T{\int }_{\mathbb{R}}|f(z,s)|\left|\Gamma (x,t,z,s)-\Gamma (\bar{x},t,z,s)\right|dzds$$

$$=|x-\bar{x}|{\int }_t^T{\int }_{\mathbb{R}}|f(z,s)|\left|\Gamma (x^{\ast },t,z,s)\right|dzds$$

$$=|x-\bar{x}|{\int }_t^T{\int }_{\mathbb{R}}|f(z,s)|\frac{C}{(s-t)}\exp \left(-c\frac{|z-x^{\ast }{|}^2}{s-t}\right)dzds$$

$$=|x-\bar{x}|{\int }_t^T\frac{1}{\sqrt{t-s}}{\int }_{\mathbb{R}}|f(z,s)|\frac{C}{\sqrt{s-t}}\exp \left(-c\frac{|z-x^{\ast }{|}^2}{s-t}\right)dzds$$

$$\le \frac{C\sqrt{2\pi }}{\sqrt{2c}}\Vert f{\Vert }_{\kappa }^0|x-\bar{x}|{\int }_t^T\frac{1}{\sqrt{t-s}}{e}^{\kappa (T-s)}ds.$$

$$e^{-\kappa (T-t)}{I}_1\le \frac{C\sqrt{2\pi }}{\sqrt{2c}}\Vert f{\Vert }_{\kappa }^0|x-\bar{x}|{\int }_t^T\frac{1}{\sqrt{t-s}}{e}^{\kappa (t-s)}ds\le \frac{C\sqrt{2\pi }}{\sqrt{2c}}\Vert f{\Vert }_{\kappa }^0|x-\bar{x}|{\int }_t^T\frac{1}{\sqrt{t-s}}ds.$$

$$\begin{gathered} I_2:=\left|\mathbb{E}{\int }_{\tau (\bar{x},t)\wedge T}^{\tau (x,t)\wedge T}f(X_s(\bar{x},t)\right| \\ \le \Vert f{\Vert }_{\kappa }^0\mathbb{E}{\int }_{\tau (\bar{x},t)\wedge T}^{\tau (x,t)\wedge T}{e}^{\kappa (T-s)}ds=\frac{1}{\kappa }\Vert f{\Vert }_{\kappa }^0\mathbb{E}\left[e^{\kappa (T-T\wedge \tau (\bar{x},t))}-e^{\kappa (T-T\wedge \tau (x,t))}\right] \end{gathered}$$

$$\begin{gathered} e^{-\kappa (T-t)}{I}_2\le \frac{1}{\kappa }\Vert f{\Vert }_{\kappa }^0\mathbb{E}\left[e^{\kappa (t-T\wedge \tau (\bar{x},t))}-e^{\kappa (t-T\wedge \tau (x,t))}\right] \\ \le \frac{L}{\kappa }\Vert f{\Vert }_{\kappa }^0\mathbb{E}\left|T\wedge \tau (\bar{x},t)-T\wedge \tau (x,t)\right|\le \frac{L}{\kappa }\Vert f{\Vert }_{\kappa }^0. \end{gathered}$$

$$e^{-\kappa (T-t)}|D_x{u}_f(x,t)|\le \frac{L}{\kappa }\Vert f{\Vert }_{\kappa }^0,$$

$$\Vert u{\Vert }_{\kappa }\le \frac{M}{\kappa }\Vert f{\Vert }_{\kappa }^0.$$

$$\mathcal{T}u(x,t):={\mathbb{E}}_{x,t}\left[\beta (X_{T\wedge \tau (x,t)},T\wedge \tau (x,t))+{\int }_t^{T\wedge \tau (x,t)}H(D_{x}u(X_s,s),u(X_s,s),X_s,s)ds\right].$$

$$w(x,t)=\mathbb{E}\left[{\int }_0^{T\wedge \tau (x,t)}H(D_{x}u(X_s,s),u(X_s,s),X_s,s)ds+\beta (X_{T\wedge \tau (x,t)}(x,t),T\wedge \tau (x,t))\right].$$

$$\begin{gathered} \mathbb{E}\left|\beta (X_{T\wedge \tau (x,t)}(x,t),T\wedge \tau (x,t))-\beta (X_{T\wedge \tau (\bar{x},t)}(\bar{x},t),T\wedge \tau (\bar{x},t))\right| \\ \le K\mathbb{E}\left|X_{T\wedge \tau (x,t)}(x,t)-X_{T\wedge \tau (\bar{x},t)}(\bar{x},t)\right|+K\mathbb{E}\left|T\wedge \tau (x,t)-T\wedge \tau (\bar{x},t)\right|. \end{gathered}$$

$$\mathbb{E}\left|X_{T\wedge \tau (x,t)}(x,t)-X_{T\wedge \tau (\bar{x},t)}(\bar{x},t)\right|=\mathbb{E}{X}_{T\wedge \tau (x,t)}(x,t)-\mathbb{E}{X}_{T\wedge \tau (\bar{x},t)}(\bar{x},t)=x-y.$$

$$\{\begin{array}{ll}{D}_t+\frac{1}{2}a(x,t)D_x^{2}u+f(x,t)=0,& (x,t)\in (0,+\infty )\times [0,T),\\ u(x,t)=\beta (x,t),& (x,t)\in \partial \left((0,+\infty )\times [0,T)\right),\end{array}$$

$$\bar{w}(x,t)=\mathcal{T}u(x,t)-\mathcal{T}v(x,t).$$

Full text

1. Introduction

Our main motivation is to prove general existence theorem for the classical solution

($\mathcal{C}^{2,1}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$ to the parabolic HJB equation of the form

[TABLE]

with the boundary condition $u(x,t)=\beta(x,t)$, for $(x,t)\in\partial\left((0,+\infty)\times[0,T)\right)$. The set $D\subset\mathbb{R}^{l}$ is assumed to be compact. Such equation appears naturally in finite time stochastic control problems where the system is stopped when the controlled process hits the barrier. In this paper we would like to put the emphasis on problems connected to dividend optimization and consumption-investment problems. We treat the aforementioned HJB equations as semilinear equation and prove our main result under more general setting. We believe that this will open the gate to consider many singular and ergodic problems. Due to our knowledge such problem has never been solved under such general setting.

Our work is structured as follows. In Section 2 we consider general semilinear equation and prove our main theorem using a fixed point approach. In Section 3 we present the application of the result to stochastic control problems including stochastic control for dividend problems. Section 4 is dedicated to applications our main result to some unrestricted consumption-investment problems, which were introduced, in some specific examples, by Korn and Kraft [7], Kraft and Steffensen [8].

2. General results

Our main objective here is to prove the existence result for a smooth solution to the equation

[TABLE]

We find it helpful to associate equation (2.1) with the one dimensional diffusion given by

[TABLE]

where $(W_{t},\;t\geq 0)$ is a one dimensional Brownian motion. The symbol $\mathbb{E}_{x,t}f(X_{s})$ is used to denote the expected value when the system starts at time $t$ from the state $x>0$. For a notational convenience we sometimes use $\mathbb{E}f(X_{s})$ as well. Let $\tau(x,t)$ denote now the first time the process hits 0 i.e.

[TABLE]

We make the following assumptions.

Assumption 1**.**

---

• A1)

The coefficient $\sigma>\varepsilon>0$ is uniformly bounded, Lipschitz continuous on compact subsets in $[0,+\infty)\times[0,T]$, and Lipschitz continuous in $x$ uniformly with respect to $t$.

• A2)

The function $\beta$ is bounded and Lipschitz continuous.

• A3)

The Hamiltonian $H$ is Hölder continuous on compact subsets of $\mathbb{R}^{2}\times[0,+\infty)\times[0,T]$. Moreover, let there exists $K>0$ such that for all $(p,u,x,t),\;(\bar{p},\bar{u},x,t)\in\mathbb{R}^{2}\times[0,+\infty)\times[0,T]$

[TABLE]

• A4)

There exists a constant $L>0$ that for all $(x,t),(\bar{x},t)\in[0,+\infty)\times[0,T]$

[TABLE]

Let $C_{b}^{1,0}$ stands for the space of all functions that are continuous, bounded and have the first derivative with respect to $x$, which is also continuous and bounded. The space is equipped with the family of norms:

[TABLE]

Note that the space $C_{b}^{1,0}$ together with $\|\cdot\|_{\kappa}$ forms a Banach space. The norm is inspired by previous works on the parabolic Cauchy problems and is usualy applied to the consumption -investment problem: Becherer and Schweizer [4], Delong and Klüppelberg [6], Berdjane and Pergamenshchikov [5] and Zawisza [14]. But only the last paper have used such type of norm to consider equations with nonlinearities in the gradient part.

We introduce as well the subspace $C_{b,h}^{1,0}$ consisting of all functions belonging to $C_{b}^{1,0}$ for which the function $D_{x}u(x,t)$ is locally Hölder continuous in $x$ uniformly with respect to $t\in U$, for every compact $U\subset[0,T)$.

Additionally, there is also a need to use the space $C_{b}^{0,0}$ consisting of all functions which are continuous and bounded. This space is considered together with the norm

[TABLE]

We consider first the linear equation

[TABLE]

Proposition 2.1**.**

Suppose that condition A1 is satisfied and let the function $f$ be Hölder continuous on compact subsets of $[0,+\infty)\times[0,T]$ and bounded. Then there exists ($u\in\mathcal{C}^{2,1}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$), a classical solution to (2.4).

Proof.

If the function $f$ is Lipschitz continuous in $x$ and Hölder continuous in $t$ on compact subsets of $[0,+\infty)\times[0,T]$, then the claim was proved by Rubio [9, Theorem 3.1]. To extend it to weaker condition we take the sequence of mollifiers $(\varphi_{n},n\in\mathbb{N})$ and approximate the function $f$ by the sequence $f_{n}=f\star\varphi_{n}$ and use (E10) from Fleming and Rishel [10] to prove uniform bounds for Hölder norms of the corresponding sequence of smooth solutions. The standard application of Arzela-Ascoli’s Lemma ends the proof. ∎

The first step in our reasoning is to prove estimates for $\|u\|_{\kappa}$ where $u$ is a solution to (2.4).

Proposition 2.2**.**

Let $u_{f}\in\mathcal{C}^{2,1}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$ denote the classical solution to (2.4) and assume that conditions A1 and A4 are satisfied. Then, there exists a constant $M>0$ such that for all functions $f$, being bounded and Hölder continuous on compact subsets of $(0,+\infty)\times[0,T)$, we have

[TABLE]

Proof.

Due to the Feynman - Kac representation we have

[TABLE]

Hence,

[TABLE]

The derivative $D_{x}u_{f}$ is estimated using the Lipschitz constant. Fix $x,\bar{x}\in[0,+\infty]$ and assume that $x>\bar{x}$. In particular, this assumption implies that $\tau(x,t)>\tau(\bar{x},t)$.

We have

[TABLE]

The first integral can be estimated using the theory of fundamental solutions for parabolic equations. The fundamental solution is denoted by $\Gamma(x,t,z,s)$. Recall that there exist $c,C>0$ such that

[TABLE]

(see Friedman [11, Chapter 1, Theorem 11]).

We have

[TABLE]

If $\tau(x,t)\leq s$, then $\tau(\bar{x},t)\leq s$ and consequently $f(X_{s}(x,t),s)=f(X_{s}(\bar{x},t),s)$.

Therefore,

[TABLE]

Thus, by multiplying both sides by $e^{-\kappa(T-t)}$, we obtain

[TABLE]

For the second integral we have

[TABLE]

and consequently, there exists $L>0$ such that

[TABLE]

All inequalities can be now summarized into

[TABLE]

which confirms that there exists a constant $M>0$ such that

[TABLE]

∎

For $u\in\mathcal{C}_{b,h}^{1,0}$, we can define the mapping

[TABLE]

Proposition 2.3**.**

If all conditions of Assumption 1 are satisfied, then the operator $\mathcal{T}$ maps $C_{b,h}^{1,0}$ into $C_{b,h}^{1,0}$ and there exists a constant $\kappa>0$ that the mapping (2.5) is a contraction in the norm $\|u\|_{\kappa}$.

Proof.

We have to prove first, that the operator $\mathcal{T}$ maps $C_{b,h}^{1,0}$ into $C_{b,h}^{1,0}$. We fix the function $u\in C_{b,h}^{1,0}$ and define

[TABLE]

The function $w$ is bounded since $u$, $D_{x}u$, $\beta$ are bounded and the Hamiltonian $H$ satisfies the linear growth condition. The function $\beta$ is uniformly Lipschitz continuous, which implies that there exists a constant $K>0$ such that

[TABLE]

Moreover, since $X_{T\wedge\tau(x,t)}(x,t)>X_{T\wedge\tau(\bar{x},t)}(\bar{x},t)$, we have

[TABLE]

It is now easy to notice that the function $w$ is Lipschitz continuous in $x$ uniformly with respect to $t$, and consequently the function $D_{x}w$ is bounded. Moreover, $w$ is a classical solution to parabolic differential equation

[TABLE]

which guarantees that $D_{x}w$ is Hölder continuous on compact subsets and consequently $\mathcal{T}$ maps $C_{b,h}^{1,0}$ into $C_{b,h}^{1,0}$.

Now our aim is to prove that $\mathcal{T}$ is a contraction for sufficiently large $\kappa$. Let’s fix $u,v\in C_{b}^{1,0}$ and define

[TABLE]

Note that $\bar{w}$ is a classical solution to

[TABLE]

After applying Proposition 2.2 we get that there exists a constant $M>0$, that

[TABLE]

This completes the proof. ∎

Theorem 2.4**.**

Assume that all conditions from Assumption 1 are satisfied. Then there exist a classical solution to (2.1), which belongs to the class $u\in\mathcal{C}^{2,1}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$ and in addition is bounded together with $D_{x}u$.

Proof.

The proof is analogous to the proof of [14, Theorem 2.2], but we repeat it for the reader’s convenience. The reasoning is based on a fixed point type argument for the mapping $\mathcal{T}$. We take any $u_{1}\in C^{1,0}_{b,h}$ and define recursively the sequence

[TABLE]

There exists $\kappa>0$ such that the mapping $\mathcal{T}$ is a contraction in $\|\cdot\|$ and this implies that the sequence $u_{n}$ is convergent to some fixed point $u$. But $u$ belongs to $C^{1,0}_{b}$ and we have to prove that $u$ belongs also to the class $C^{1,0}_{b,h}$. Let us note first that functions $u_{n}$ and $D_{x}u_{n}$ are convergent in $\|\cdot\|_{\kappa}$ (for $\kappa$ large enough), thus they are bounded uniformly with respect to $n$. We can now exploits (E8), (E9), (E10) from Fleming and Rishel [10] and prove uniform bound on compact subsets for Hölder norm of $D_{x}u_{n}$. Therefore $D_{x}u$ is Hölder continuous in $x$ on compact subsets uniformly with respect to $t$. This confirms that the fixed point $u$ belongs to the class $\mathcal{C}^{2,1}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$ and satisfies equation (2.1). ∎

Proposition 2.5**.**

Let the function $\sigma:(0,+\infty)\to\mathbb{R}$ be bounded, bounded away from zero and uniformly Lipschitz continuous. Then condition (2.3) is satisfied for the hitting time $\tau(x,t)$ of the following SDE

[TABLE]

Proof.

The proof consists of four parts.

Step 1 First, we consider trivial dynamics of the form

[TABLE]

Note that

[TABLE]

Therefore,

[TABLE]

Step 2 In the next step we consider SDE of the form

[TABLE]

where the function $b$ is bounded and Lipschitz continuous on compact subsets of $[0,+\infty)$. In Proposition 2.4 we proved that the equation

[TABLE]

admits a classical solution $u$ with bounded derivative $D_{x}u$. The standard verification theorem ensures that $u(x,t)=\mathbb{E}T\wedge\tau(x,t)-t$. So the condition (2.3) is satisfied for (2.10) as well.

Step 3 Suppose now, that $\sigma\in C^{1+1}_{b,loc}$ and consider the dynamics

[TABLE]

We need as well the function

[TABLE]

which belongs to the class $\mathcal{C}^{2}$. By the Itô, formula we get that $X_{t}=\zeta(Y_{t})$ is the unique strong solution to

[TABLE]

We have

[TABLE]

Condition ($\eqref{property:L}$) is satisfied for the process $Y_{t}$ and using the fact that $\zeta^{-1}$ is a Lipschitz continuous function we get the same for the process $X$.

Step 4 In the fourth step we consider $\sigma$ Lipschitz continuous, bounded and bounded away from zero together with the sequence of mollifiers $(\zeta_{n}|\;n\in\mathbb{N})$ and define the sequence

[TABLE]

and the sequence of diffusions

[TABLE]

and finally the sequence of stopping times

[TABLE]

We deduce from the proof of Theorem 2.4 that it is constant $K^{\prime}>0$ such that for all $n\in\mathbb{N}$

[TABLE]

where the constant $K^{\prime}$ is independent of $n$. Passing to the limit, we get

[TABLE]

∎

3. Stochastic control applications

Here we adapt our result to be applicable for stochastic control problems. We consider the HJB equation of the form

[TABLE]

with the boundary condition $u(x,t)=\beta(x,t)$, for $(x,t)\in\partial\left((0,+\infty)\times[0,T)\right)$.

Assumption 2**.**

---

• B1)

The coefficient $\sigma>\varepsilon>0$ is bounded, Lipschitz continuous on compact subsets in $[0,+\infty)\times[0,T]$, and Lipschitz continuous in $x$ uniformly with respect to $t$.

• B2)

Functions $f$, $h$, $i$ are continuous and bounded and there exists a constant $L>0$ such that for all $\zeta=f,h,i$ and for all $\delta\in D$, $(x,t)\in[0,+\infty)\times[0,T]$

[TABLE]

• B3)

The function $\beta$ is Lipschitz continuous.

• B4)

There exists a constant $L>0$ that for all $(x,t),(\bar{x},t)\in\mathbb{R}\times[0,T]$

[TABLE]

Now we can give the immediate consequence of the Theorem 2.4.

Proposition 3.1**.**

Assume that all conditions of Assumption 2 are satisfied. Then there exists smooth solution to the problem (3.1)

Optimal restricted dividend problem We consider an insurance company and its surplus of the form:

[TABLE]

where the process $(c_{t},0\leq t\geq T)$ denotes the stream of dividends. In the literature we can find variety of problems of the form:

[TABLE]

The function $U$ we can interpret as the utility function and $r>0$ is the discount rate.

The insurance company wants to maximize $J^{c}(x,t)$ over the set of progressively measurable processes $(c_{t},0\leq t\geq T)$ taking values in a fixed compact set $[m_{1},m_{2}]$. Here, we can use the HJB of the form:

[TABLE]

.

For the discussion about recent advances of theory of dividend problems see Avanzi [3] or Zhu [15] .

4. The optimal consumption problem with uncertain horizon

Our investor has an access to two securities: a bank account $(B_{t},0\leq t<+\infty)$ and a share $(S_{t},0\leq t<+\infty)$. We assume also that the price of the share depends on one non-tradable (but observable) factor $(Y_{t},0\leq t<+\infty)$. This factor can represent an additional source of an uncertainty, here we can assume that this process will determine the investment horizon. Namely, let us define

[TABLE]

Processes mentioned above are solutions to the system of stochastic differential equations

[TABLE]

The dynamics of the investors wealth process $(X^{\pi,c}_{t},0\leq t\leq T)$ is given by the stochastic differential equation

[TABLE]

where $x$ denotes the current wealth of the investor, $(\pi_{k},t\leq k\leq T)$ is part of the wealth invested in $S_{t}$, $(c_{k},t\leq k\leq T)$ is the consumption intensity process. The objective for the investor looks as follows

[TABLE]

where $U(x)=\frac{x^{\gamma}}{\gamma}$. The investor’s aim is to maximize $J^{\pi,c}(x,y,t)$ with respect to $(\pi,c)\in\mathcal{A}$, which is not described here in detail.

To solve it we use a HJB equation of the form

[TABLE]

with boundary conditions $V(x,y,T)=\frac{1}{\gamma}x^{\gamma}$, $V(x,y_{0},t)=\frac{1}{\gamma}x^{\gamma}$.

Calculating both maxima and plugging into the equation we get

[TABLE]

with boundary conditions $F(y,T)=1$, $F(y_{0},t)=1$. Moreover, the optimal portfolio/consumption candidate is given by

[TABLE]

To simplify the equation we follow Zariphopoulou [13] and use the transformation

[TABLE]

This will reduce the equation to the form

[TABLE]

Note, that $0<1-\frac{\delta}{1-\gamma}<1$ and there exists $\alpha\in(0,1)$ that

[TABLE]

and consequently

[TABLE]

Therefore, it is reasonable to consider first HJB equations of the form

[TABLE]

with the boundary condition $G(y,t)=1$, for $(y,t)\in\partial\left((y_{0},+\infty)\times[0,T)\right)$.

Proposition 4.1**.**

Suppose that functions $\sigma$, $h$, $i$ are Lipschitz continuous and bounded and in addition let $\sigma$ be bounded away from zero. Then there exists $G\in\mathcal{C}^{2}((0,+\infty)\times[0,T))\cap\mathcal{C}([0,+\infty)\times[0,T])$ which satisfies (4.5) and is bounded together with its first derivative with respect to $y$.

Proof.

Thanks to our theorem we know that equation (4.6) has a bounded classical solution which is bounded together with the first derivative $G_{y}$.

First we need to obtain uniform bounds for the function $G$. The standard verification theorem guarantees that

[TABLE]

Since the function $h$ is bounded, there exists a constant $D>0$ such that for all $(y,t)\in[0,+\infty)\times[0,T]$

[TABLE]

Furthermore, note that

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Inserting $c\equiv 1$ and $q\equiv 0$, we get

[TABLE]

where $K<0$ is any constant such that $K\leq\inf_{y}(h(y)-\theta\alpha)$.

Thus, there exist $p,P>0$ such that

[TABLE]

This ensures that there exist a pair $m_{1}^{*}$, $m_{2}^{*}$ such that

[TABLE]

[TABLE]

and finally $G:=G_{m_{1}^{*},m_{2}^{*}}$ is a solution to (4.5). The boundedness condition for $D_{y}G$ is proved by finding the uniform bound for the Lipschitz constant. We have

[TABLE]

The function $h$ is Lipschitz continuous in $y$ and this implies that

[TABLE]

Thus,

[TABLE]

and

[TABLE]

Since $c_{s}\leq m_{1}^{*}$, we can notice that there exists a constant $M^{\prime}>0$ such that

[TABLE]

The conclusion of Theorem 2.4 and standard estimates for stochastic differential equations ensure that $D_{y}G$ is uniformly bounded and this completes the proof. ∎