2. The Knightian intertemporal utility maximization problem

Let $\Omega=C_0^d([0,\infty))$ be the space of all continuous, $\mathbb{R}^d$ -valued functions defined on $[0,\infty)$ , equipped with the distance

\begin{equation*} d\big(\omega^1,\omega^2\big)=\sum_{n=1}^{\infty}\frac{1}{2^n}\max_{0\leq t\leq n}\!\Big(|\omega^1_t-\omega^2_t|\wedge 1\Big).\end{equation*}

Let $\mathbb{P}_0$ be the Wiener measure, under which the canonical process B is a d-dimensional Brownian motion, and denote by $\mathbb{F}=(\mathcal{F}_t)_{t\geq0}$ the filtration generated by the Brownian motion B. Throughout this paper, $\mathbb{E}[{\cdot}]$ will denote the expectation taken under the probability $\mathbb{P}_0$ , and measurability properties (like progressive measurability or adaptedness) will be always with respect to $\mathbb{F}$ , unless otherwise stated.

We aim to study the optimal consumption choice of an agent facing Knightian ambiguity and whose preferences are of the Hindy–Huang–Kreps (HHK) type. We assume that the agent is ambiguity-averse and also that her consumption is financed via investment in a financial market which possibly exhibits frictions. Those characteristics of the agent and the market can be well modeled through the use of nonlinear expectations—namely, via the g-expectation $\mathcal{E}^g[{\cdot}]$ initiated by S. Peng in [Reference Peng29], which will be explained in detail in Section 6. The g-expectation is formally defined as the first component of the solution to a backward stochastic differential equation (BSDE) with driver $g\,:\,\Omega\times[0,T]\times \mathbb{R}^d\rightarrow\mathbb{R}$ satisfying the following requirements:

1. (A1) for any $z\in\mathbb{R}^d$ , $g(\cdot,\cdot,z)$ is progressively measurable and

   \begin{equation*} \mathbb{E}\bigg[\int_0^T |g(t,z)|^2dt\bigg]<\infty; \end{equation*}

2. (A2) there exists some constant $\kappa>0$ such that, for any $(\omega,t)\in \Omega\times[0,T]$ and $z,z^{\prime}\in\mathbb{R}^d$ ,

   \begin{equation*} |g(\omega,t,z)-g(\omega,t,z^{\prime})|\leq \kappa|z-z^{\prime}|; \end{equation*}

3. (A3) $g(\omega, t, \cdot)$ is concave for any $(\omega,t)\in\Omega\times [0,T]$ ;

4. (A4) $g(\omega,t,0)=0$ for any $(\omega,t)\in\Omega\times [0,T]$ .

The Knightian intertemporal optimal consumption problem under study is defined as follows. Introduce the set

\begin{align*}\mathcal{X}\,:\!=\,\big\{C\,\big|\,& C \textrm{ is the distribution function of a nonnegative}\\ &\textrm{optional random measure on [0,T]}\big\},\end{align*}

let $r=\{r_t\}_{t\in[0,T]}$ be a bounded, progressively measurable process representing the instantaneous interest rate, and let g and h be two functions satisfying (A1)–(A4). An agent with initial wealth $w>0$ will pick a consumption plan from the budget-feasible set

(2.1) \begin{equation}\mathcal{A}_h(w)\,:\!=\,\bigg\{C\in\mathcal{X}\,|\,\Psi(C)\,:\!=\,\widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d} C_t\bigg]\leq w\bigg\},\end{equation}

where $\gamma_t\,:\!=\,\exp\!{\big({-}\int_0^t r_sds\big)}$ and $\widetilde{\mathcal{E}}^h[{\cdot}]=-\mathcal{E}^h[{-\cdot}]=\mathcal{E}^{\widetilde{h}}[{\cdot}]$ $\big($ we define $\widetilde{h}(t,z)\,:\!=\,-h(t,-z)\big)$ . It is easy to check that $\mathcal{A}_h(w)$ is nonempty when h satisfies (A1)–(A4), since $\widetilde{\mathcal{E}}^h[0]=0$ .

For a fixed consumption plan $C \in \mathcal{X}$ , the agent’s level of satisfaction is defined by

(2.2) \begin{equation}Y_t^C=\eta_t+\int_0^t \theta_{t,s}\text{d} C_s,\end{equation}

where $\eta\,:\,[0,T]\rightarrow\mathbb{R}_+$ and $\theta\,:\,[0,T]^2\rightarrow\mathbb{R}_+$ are exogenously given continuous functions related to the agent’s preferences. More precisely, the quantity $\theta_{t,s}$ can be seen as the weight assigned at time t to consumption made at time $s\leq t$ , and $\eta_t$ describes the level of satisfaction that the agent has at time t, in absence of consumption over the time period [0, t]. In (2.2), and in the following, we interpret the integrals with respect to the optional random measure $\text{d} C$ on [0, T] in the Lebesgue–Stieltjes sense as $\int_0^{\cdot} ({\cdot})\text{d} C_t = \int_{[0,\cdot]} ({\cdot})\text{d} C_t$ . In this way, a possible initial jump of the process C (i.e. an initial consumption gulp) is taken into account in the integral.

We assume that the agent’s utility is of the HHK type and depends on the current level of satisfaction (hence on past consumption as well) via a certain instantaneous felicity function $u\,:\,\Omega \times [0,T]\times\mathbb{R}_+\rightarrow\mathbb{R}$ . In particular, we have

\begin{equation*}U(C)=\int_0^T u\big(t,Y_t^C\big)\text{d} t,\end{equation*}

where $u(\omega, \cdot,\cdot)$ is continuous, with $u(\omega, t,\cdot)$ being increasing and concave and such that, for any $y \in \mathbb{R}_+$ , $(\omega,t) \mapsto u(\omega,t,y)$ is progressively measurable. The goal of the agent is then that of maximizing her expected utility over all budget-feasible consumption plans—that is, to find

(2.3) \begin{equation}v_{g, h}(w)\,:\!=\,\sup_{C\in\mathcal{A}_h(w)}V(C)=\sup_{C\in\mathcal{A}_h(w)}\mathcal{E}^{g}[U(C)].\end{equation}

In order to simplify notation, in the following we shall omit the subscripts g, h in $v_{g, h}$ , as well as its dependence on w. In (2.3) the use of the nonlinear expectation $\mathcal{E}^{g}$ is justified by thinking that the agent is ambiguity-averse and therefore employs variational preferences in order to evaluate the utility from a consumption plan. The g-expectation is indeed one of the most prominent examples of variational preferences. We also refer to Section 2.4 in [Reference Beissner and Riedel6] for a discussion on utility functionals under (Knightian) uncertainty.

3. Existence and uniqueness of the optimal consumption plan

In this section, we prove existence and uniqueness of a consumption plan solving the problem (2.3). For this purpose, we need the following assumption on the budget-feasible set:

1. (H1) The family of budget-feasible utilities $\{U(C), C\in\mathcal{A}_h(w)\}$ is bounded in $L^p$ with a $p>2$ .

Notice that thanks to Assumption (H1) and a priori estimates for BSDEs (cf. Proposition 2.1 in [Reference El Karoui, Peng and Quenez14]), one has

\begin{equation*} v =\sup_{C\in\mathcal{A}_h(w)}V(C)<\infty. \end{equation*}

Before moving on with the existence and uniqueness result, we recall the following technical lemma from [Reference Bank and Riedel5].

A sufficient condition for Assumption (H1) to hold is given by the next result, whose proof can be found in Appendix B.

Lemma 3.2. Suppose that the function h satisfies (A1)–(A4). The family of budget-feasible utilities $\{U(C), \, C\in\mathcal{A}_h(w)\}$ is bounded in $L^p$ with a $p>2$ if there exist $\alpha\in\big(0,\frac{1}{2}\big)$ and $K>0$ such that the felicity function u satisfies the power growth condition

\begin{equation*} \sup_{t\in[0,T]}|u(t,y)|\leq K\big(1+y^\alpha\big) \quad \textrm{ for all } y\geq 0. \end{equation*}

Proof. Let $\{C^n\}_{n=1}^\infty\subset \mathcal{A}_h(w)$ be such that $\sup_{C\in\mathcal{A}(w)}V(C)=\lim_{n\rightarrow\infty}V\big(C^n\big)$ . Denote by $\ell$ the convex dual of h. Since the interest rate is bounded, applying the representation for g-expectation (see Proposition A.1 in Appendix A), there exists a constant $K>0$ such that for any $\xi\in D_h$ (hence, $|\xi_t|\leq \kappa$ for all $t\geq0$ a.s.) we have

\begin{align*} \sup_{C\in\mathcal{A}(w)}\mathbb{E}^{\mathbb{P}^\xi}[C_T]&\leq K\sup_{C\in\mathcal{A}(w)}\mathbb{E}^{\mathbb{P}^\xi}\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]\\ & \leq K\bigg\{\sup_{C\in\mathcal{A}(w)}\widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]+\mathbb{E}^{\mathbb{P}^\xi}\bigg[\int_0^T \ell(s,\xi_s)\text{d} s\bigg]\bigg\}<\infty, \end{align*}

where we have employed (B.3) in the last inequality. Then, by a version of Komlós’ theorem due to Y. Kabanov (Lemma 3.5 in [Reference Kabanov23]), there exists a subsequence, for simplicity still denoted by $\{C^n\}$ , such that $\mathbb{P}^\xi$ -a.s.

\begin{equation*} \widetilde{C}^n_t\,:\!=\,\frac{1}{n}\sum_{k=1}^{n}C^k_t\rightarrow C^*_t \quad \textrm{as }n\rightarrow\infty \end{equation*}

for $t=T$ and for every point of continuity t of $C^*$ . Since $\mathbb{P}^\xi$ is equivalent to $\mathbb{P}_0$ , the above convergence holds $\mathbb{P}_0$ -a.s. We claim that $\big\{\widetilde{C}^n\big\}$ is also a maximizing sequence for the problem (2.3). Indeed, the convexity of $\widetilde{\mathcal{E}}^h[{\cdot}]$ implies that $\widetilde{C}^n\in \mathcal{A}_h(w)$ for any $n\in\mathbb{N}$ . Therefore, we have $V\big(\widetilde{C}^n\big)\leq \sup_{C\in\mathcal{A}_h(w)}V(C)$ . On the other hand, it is easy to check that

\begin{equation*} Y^{\widetilde{C}^n}_t=\frac{1}{n}\sum_{k=1}^{n}Y_t^{C^k}, \end{equation*}

and since $u(\omega, t,\cdot)$ and $\mathcal{E}^{g}[{\cdot}]$ are both concave, we then obtain that

\begin{equation*} V\big(\widetilde{C}^n\big)=\mathcal{E}^{g}[U\big(\widetilde{C}^n\big)]\geq \frac{1}{n}\sum_{k=1}^n \mathcal{E}^{g}\big[U\big(C^k\big)\big]=\frac{1}{n}\sum_{k=1}^n V\big(C^k\big). \end{equation*}

Noting that $\{C^n\}$ is a maximizing sequence, it follows that $\liminf_{n\rightarrow\infty} V\big(\widetilde{C}^n\big)\geq \sup_{C\in\mathcal{A}_h(w)}V(C)$ , and the claim thus holds.

We then show that $C^*$ is optimal for the problem (2.3). Since $\gamma$ is continuous, we have by the portmanteau theorem that

\begin{equation*} \lim_{n\rightarrow\infty}\int_0^T \gamma_t\text{d}\widetilde{C}^n_t=\int_0^T \gamma_t\text{d} C^*_t \qquad \mathbb{P}_0\textrm{-a.s.}, \end{equation*}

and by Fatou’s lemma and the convexity of $\widetilde{\mathcal{E}}^h[{\cdot}]$ it follows that

\begin{equation*} \widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d} C^*_t\bigg]\leq \liminf_{n\rightarrow\infty}\widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d}\widetilde{C}^n_t\bigg]\leq \liminf_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n\widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d} C^n_t\bigg]\leq w, \end{equation*}

which implies that $C^*\in\mathcal{A}_h(w)$ . Furthermore, by Lemma 3.1, we have $U\big(\widetilde{C}^n\big)\rightarrow U\big(C^*\big)$ , $\mathbb{P}_0$ -a.s. Now, invoking Assumption (H1) yields that $\mathbb{E}\Big[\big|U\big(\widetilde{C}^n\big)-U\big(C^*\big)\big|^2\Big]\rightarrow 0$ , and by estimates for BSDEs (cf. Proposition 2.1 in [Reference El Karoui, Peng and Quenez14]), we obtain that

\begin{equation*} V\big(\widetilde{C}^n\big)=\mathcal{E}^{g}\big[U\big(\widetilde{C}^n\big)\big]\rightarrow \mathcal{E}^{g}\big[U\big(C^*\big)\big]=V\big(C^*\big). \end{equation*}

Recalling that $\big\{\widetilde{C}^n\big\}$ is a maximizing sequence of the problem (2.3), it follows that $C^*$ is optimal.

It remains to prove uniqueness. If $C_1$ and $C_2$ are both optimal and they are not indistinguishable, then their associated levels of satisfaction $Y^1=Y^{C^1}$ and $Y^2=Y^{C^2}$ are not indistinguishable either. By arguments similar to those employed in the proof of Theorem 2.3 in [Reference Bank and Riedel5], on a set with positive probability, $Y^1$ and $Y^2$ differ on an open time interval. By the strict concavity of $u(\omega,t,\cdot)$ and the strict comparison theorem for solutions to BSDEs (cf. Therorem 2.2 in [Reference El Karoui, Peng and Quenez14]), this gives

\begin{align*}V\bigg(\frac{1}{2}\big(C^1+C^2\big)\bigg) &=\mathcal{E}^{g}\bigg[\int_0^T u\bigg(t,\frac{1}{2}\Big(Y^1_t+Y^2_t\Big)\bigg)\text{d} t\bigg]\\ & >\mathcal{E}^{g}\bigg[\int_0^T \frac{1}{2}\Big(u\big(t,Y^1_t\big)+u\big(t,Y^2_t\big)\Big)\text{d} t\bigg]\\ &\geq \frac{1}{2}\bigg(\mathcal{E}^{g}\bigg[\int_0^T u\big(t,Y^1_t\big)\text{d} t\bigg]+\mathcal{E}^{g}\bigg[\int_0^T u\big(t,Y^2_t\big)\text{d} t\bigg]\bigg)\\ &=\frac{1}{2}\big(V\big(C^1\big)+V\big(C^2\big)\big)=\sup_{C\in\mathcal{A}_h(w)}V(C), \end{align*}

which gives the desired contradiction.

Remark 3.2. We could also consider the utility maximization problem under the general nonlinear expectations

\begin{equation*}\begin{split}&\mathcal{E}[X]\,:\!=\,\inf_{\mathbb{P}\in\mathcal{P}_1}\!\Big(\mathbb{E}^\mathbb{P}[X]+c_1(\mathbb{P})\Big) \quad \text{and} \quad \widetilde{\mathcal{E}}[Y]\,:\!=\,\sup_{\mathbb{P}\in\mathcal{P}_2}\!\Big(\mathbb{E}^\mathbb{P}[Y]-c_2(\mathbb{P})\Big).\end{split}\end{equation*}

In fact, $c_1, c_2$ are two penalty functions, which can be interpreted as an index of ambiguity aversion (see [Reference Maccheroni, Marinacci and Rustichini25]). For this purpose, for any fixed constant $p>1$ , we assume that the multiple priors and penalty functions satisfy the assumptions

1. (i) $\sup_{\mathbb{P}\in\mathcal{P}_1}\mathbb{E}\Big[\Big|\frac{\text{d}\mathbb{P}}{\text{d}\mathbb{P}_0}\Big|^p\Big]<\infty$ , where $p>1$ , and

2. (ii) $0\leq \inf_{\mathbb{P}\in\mathcal{P}_2}c_2(\mathbb{P})\leq \sup_{\mathbb{P}\in\mathcal{P}_2}c_2(\mathbb{P})<\infty$ ,

and we define the budget-feasible set as

\begin{equation*} \widehat{\mathcal{A}}(w)\,:\!=\,\bigg\{C\in\mathcal{X}\,\big|\, \widetilde{\mathcal{E}}\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]\leq w\bigg\}.\end{equation*}

Here, $\gamma$ is the discount factor associated with a bounded, progressively measurable interest rate r and $\mathcal{X}$ is the collection of all distribution functions of nonnegative optional random measures as in Section 2. By Assumption (ii), $\mathcal{A}(w)$ is nonempty for any given initial wealth $w>0$ . The level of satisfaction $Y^C$ and the agent’s utility U(C) are as in Section 2. The agent aims to maximize her expected utility over all budget-feasible consumption plans, and the value function is now defined as

(3.1) \begin{equation} \widehat{v}\,:\!=\,\sup_{C\in \widehat{\mathcal{A}}(w)}\mathcal{E}[U(C)].\end{equation}

Supposing that the family of budget-feasible utilities $\big\{U(C), C\in\widehat{\mathcal{A}}(w)\big\}$ is bounded in $L^{p^*}$ , where $p^*>p/(p-1)$ , we can still show that the utility maximization problem (3.1) has a solution. However, owing to the lack of the strict comparison property for $\mathcal{E}[{\cdot}]$ , we lose uniqueness.

4. Sufficient first-order conditions for optimality

This section is devoted to the proof of FOCs for optimality. For any two functions g, h satisfying (A1)–(A3), let $f,\ell$ denote their respective convex duals. Now, for any budget-feasible consumption plan C for the problem (2.3), set

\begin{align*}\mathcal{P}_1(C)&\,:\!=\,\bigg\{\mathbb{P}^\xi\,\big|\,\xi\in D_g, \mathcal{E}^{g}[U(C)]=\mathbb{E}^{\mathbb{P}^\xi}\bigg[U(C)+\int_0^T f(s,\xi_s)\text{d} s\bigg]\bigg\},\\\mathcal{P}_2(C)&\,:\!=\,\bigg\{\mathbb{P}^\xi\,\big|\,\xi\in D_h, \widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]=\mathbb{E}^{\mathbb{P}^\xi}\bigg[\int_0^T \gamma_t\text{d} C_t-\int_0^T \ell(s,\xi_s)\text{d} s\bigg]\bigg\}.\end{align*}

In fact, in light of Proposition A.1 in Appendix A, $\mathcal{P}_1(C)$ can be regarded as the collection of lowest-utility probabilities and $\mathcal{P}_2(C)$ the collection of largest-cost probabilities for the consumption plan C. In order to obtain the FOC, we need the following additional assumption on the felicity function:

1. (H2) The felicity function u is such that $u(\omega,t,\cdot)$ is strictly concave and differentiable for any $(\omega,t) \in \Omega \times [0,T]$ .

Proof. Assume that $C^*$ satisfies the conditions (1)–(3), and consider another budget-feasible consumption plan $C\in\mathcal{A}_h(w)$ . For simplicity, set $Y\,:\!=\,Y^C$ , $Y^*\,:\!=\,Y^{C^*}$ , and let

\begin{align*} I&\,:\!=\,{\mathbb{E}}^{\mathbb{P}_1}\bigg[\int_0^T \bigg(\int_0^s \partial_yu\big(s,Y_s^*\big)\theta_{s,t}\text{d} C_t^*\bigg) \text{d} s\bigg]={\mathbb{E}}^{\mathbb{P}_1}\bigg[\int_0^T \bigg(\int_t^T \partial_yu\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg) \text{d} C_t^* \bigg],\\ II&\,:\!=\,{\mathbb{E}}^{\mathbb{P}_1}\bigg[\int_0^T \bigg(\int_0^s \partial_yu\big(s,Y_s^*\big)\theta_{s,t}\text{d} C_t\bigg) \text{d} s\bigg]={\mathbb{E}}^{\mathbb{P}_1}\bigg[\int_0^T \bigg(\int_t^T \partial_yu\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg) \text{d} C_t \bigg], \end{align*}

where we have used Fubini’s theorem. Noting that $\mathbb{P}_2\in\mathcal{P}_2\big(C^*\big)$ , it is easy to check that

\begin{align*} I&=M \mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \gamma_t\text{d} C^*_t\bigg]=M \mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \gamma_t\text{d} C^*_t+\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s-\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg]\\ &=M\widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d} C^*_t\bigg]+M\mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg]=Mw+M\mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg], \end{align*}

and

\begin{align*} II & = {\mathbb{E}}^{\mathbb{P}_2}\bigg[\int_0^T \frac{\text{d}\mathbb{P}_1}{\text{d}{\mathbb{P}}_2} \bigg(\int_t^T \partial_yu\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg)\text{d} C_t\bigg]\\ & = {\mathbb{E}}^{\mathbb{P}_2}\bigg[\int_0^T {\mathbb{E}}^{\mathbb{P}_2}_t\bigg[\frac{\text{d}\mathbb{P}_1}{\text{d}{\mathbb{P}}_2}\int_t^T \partial_y u\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg]\text{d} C_t\bigg]\\ & \leq M\mathbb{E}^{{\mathbb{P}}_2}\bigg[\int_0^T \gamma_t\text{d} C_t+\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s-\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg]\\ & = M \widetilde{\mathcal{E}}^h\bigg[\int_0^T \gamma_t\text{d} C_t\bigg]+M\mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg]\\ & \leq Mw+M\mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \ell\big(s,\xi^2_s\big)\text{d} s\bigg], \end{align*}

where we have used Theorem 1.33 in [Reference Jacod21] to obtain the second equality above. Noting that $u(\omega,t,\cdot)$ is strictly concave, we finally have

\begin{align*} V\big(C^*\big)-V(C) & \geq \mathbb{E}^{\mathbb{P}_1}\bigg[U\big(C^*\big)+\int_0^T f\big(s,\xi^1_s\big)\text{d} s\bigg]-\mathbb{E}^{\mathbb{P}_1}\bigg[U(C)+\int_0^T f\big(s,\xi^1_s\big)\text{d} s\bigg]\\ & =\mathbb{E}^{\mathbb{P}_1}\bigg[\int_0^T \big(u\big(s,Y^*_s\big)-u\big(s,Y_s\big)\big)\text{d} s\bigg] \geq \mathbb{E}^{\mathbb{P}_1}\bigg[\int_0^T \partial_y u\big(s,Y_s^*\big)\big(Y_s^*-Y_s\big)\text{d} s\bigg]\\ & =\mathbb{E}^{P_1}\bigg[\int_0^T \bigg(\int_0^s \partial_y u\big(s,Y_s^*\big)\theta_{s,t}\big(\text{d} C_t^*-\text{d} C_t\big)\bigg)\text{d} s\bigg] = I-II\geq 0, \end{align*}

which clearly completes the proof.

4.1. Some remarks

We conclude this section with some comments.

Remark 4.1.

1. (i) It is easily seen that only the representation property of the g-expectation (see Proposition A.1) is used in the previous proof. Therefore, if g does not satisfy Assumption (A4), Theorem 4.1 still holds.

2. (ii) Let $\phi_t\,:\!=\,{\mathbb{E}}^{\mathbb{P}_1}_t\Big[\int_t^T \partial_y u\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\Big]$ . Applying the continuous-time Bayes’ rule, the condition (2) of Theorem 4.1 is equivalent to

   \begin{equation*} \phi_t\leq M\gamma_t \mathbb{E}^{\mathbb{P}_1}_t\bigg[\frac{\text{d}\mathbb{P}_2}{\text{d}\mathbb{P}_1}\bigg], \ \ t\in[0,T], \end{equation*}
   which is similar to the condition (ii) in Theorem 3.2 in [Reference Bank and Riedel5]. Here $\gamma_t \mathbb{E}^{\mathbb{P}_1}_t\Big[\frac{\text{d}\mathbb{P}_2}{\text{d}\mathbb{P}_1}\Big]$ can be regarded as the state-price density.

Remark 4.2. Consider the general utility maximization problem introduced in Remark 3.2. For this problem, too, we can establish a set of sufficient conditions for optimality. For any budget-feasible consumption plan $C\in\widehat{\mathcal{A}}(w)$ , we define

\begin{align*} &\mathcal{P}^1(C)\,:\!=\,\Big\{\mathbb{P}\in\mathcal{P}_1\,\big|\,\mathcal{E}[U(C)]=\mathbb{E}^\mathbb{P}[U(C)]+c_1(\mathbb{P})\Big\},\\ &\mathcal{P}^2(C)\,:\!=\,\Big\{\mathbb{P}\in\mathcal{P}_2\,\big|\,\widetilde{\mathcal{E}}\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]=\mathbb{E}^\mathbb{P}\bigg[\int_0^T \gamma_t \text{d} C_t\bigg]-c_2(\mathbb{P})\Big\},\end{align*}

and let Assumption (H2) hold. Suppose also that the family of budget-feasible utilities $\big\{U(C), C\in\widehat{\mathcal{A}}(w)\big\}$ is bounded in $L^{p^*}$ (where $p^*>p/(p-1)$ ). Then a consumption plan $C^*$ solves the utility maximization problem (3.1) if there exist some $\mathbb{P}_i\in\mathcal{P}^i\big(C^*\big)$ , $i=1,2$ , such that

1. (1) $ \displaystyle \widetilde{\mathcal{E}}\bigg[\int_0^T \gamma_t\text{d} C^*_t\bigg]=w$ ,

2. (2) $\displaystyle {\mathbb{E}}^{\mathbb{P}_2}_t\bigg[\frac{\text{d}\mathbb{P}_1}{\text{d}\mathbb{P}_2}\bigg(\int_t^T \partial_y u\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg)\bigg]\leq M\gamma_t$ for any $t\in[0,T]$ a.s., and

3. (3) $\displaystyle {\mathbb{E}}^{\mathbb{P}_1}\bigg[\int_0^T\bigg(\int_t^T \partial_y u\big(s,Y_s^*\big)\theta_{s,t}\text{d} s\bigg) \text{d} C^*_t\bigg]=M \mathbb{E}^{\mathbb{P}_2}\bigg[\int_0^T \gamma_t\text{d} C^*_t\bigg]$ ,

where $M>0$ is a finite Lagrange multiplier and $Y^*=Y^{C^*}$ .

Remark 4.3. Recall that in the linear case treated in [Reference Bank and Riedel5], the FOCs are also shown to be necessary for optimality (cf. Theorem 3.2 in [Reference Bank and Riedel5]). In [Reference Bank and Riedel5] the proof of this is organized as follows. First it is shown that the optimal consumption plan $C^*$ solves an auxiliary problem linearized around $C^*$ . Then a characterization of any solution to such a linearized problem is provided.

In our setting, consider the following assumption:

1. (A5) For each $\omega\in\Omega$ , $t\in[0,T]$ , and $z\in\mathbb{R}$ , the equation $g(\omega,t,z)-xz=f(\omega,t,x)$ admits a unique solution $x\in[{-}\kappa,\kappa]$ , denoted by $x(\omega,t,z)$ . Furthermore, $z \mapsto x(\omega,t,z)$ is continuous, for any $(\omega,t) \in\Omega\times [0,T]$ .

Assuming (A5) and arguing as in the proof of Lemma B3 in [Reference Ferrari, Li and Riedel17], we can show that there exists some $\mathbb{P}_1\in\mathcal{P}_1(C^\star)$ such that the optimal $C^*$ for (2.3) also solves

(4.1) \begin{equation}\sup_{C\in\mathcal{A}_h(w)}\mathbb{E}^{\mathbb{P}_1}\left[\int_0^T \mathbb{E}^{\mathbb{P}_1}_t\left[\int_t^T \partial_y u\big(s,Y^\star_s\big)\theta_{s,t}\text{d} s\right]\text{d} C_t\right],\end{equation}

where $Y^*\,:\!=\,Y^{C^*}$ . However, the fact that the expectation arising in the set $\mathcal{A}_h(w)$ is nonlinear gives rise to technical difficulties when trying to characterize conditions for the above problem (4.1). Fortunately, we shall see in the next sections that the sufficiency of the FOCs does actually suffice as a verification tool for checking the optimality of a given candidate consumption plan.

5. Time-consistency and structure of the optimal consumption plan

In this section, we first study the optimal consumption problem dynamically and prove a version of the dynamic programming principle, which indicates that if a consumption plan is optimal at time zero, then it is also optimal at any later time. We then show how to construct the optimal consumption plan through an auxiliary backward equation.

Proposition 5.1. Suppose that the functions g, h satisfy (A1)–(A4). Let $S\leq T$ be a stopping time, let $C^*$ be the optimal consumption plan for the utility maximization problem (2.3), and set

\begin{equation*} \mathcal{A}_S\big(C^*\big)\,:\!=\,\Big\{C\in\mathcal{X}\,\big|\,C|_{[0,S)}\equiv C^*|_{[0,S)}, \Psi_S(C)\leq \Psi_S\big(C^*\big)\Big\}, \end{equation*}

where

\begin{equation*} \Psi_S(C)\,:\!=\,\frac{1}{\gamma_S}\widetilde{\mathcal{E}}^h_S\bigg[\int_S^T\gamma_t\text{d} C_t\bigg]. \end{equation*}

Consider then the optimal consumption problem starting at time S,

(5.1) \begin{equation} v_S\,:\!=\,\mathop{{\textrm{ess}}\,{\textrm{sup}}}_{C\in\mathcal{A}_S\big(C^*\big)}\mathcal{E}^{g}_S[U(C)], \end{equation}

and assume that the felicity function u satisfies $u(\omega,t,0)=0$ for any $(\omega,t)\in\Omega \times [0,T]$ . Then the process v is an $\mathcal{E}^{g}$ -supermartingale in the strong sense. In addition, $C^*$ is optimal for (5.1).

Proof. Step 1. Let $S\leq T$ be a stopping time. We first show that the family $\big\{\mathcal{E}^{g}_S[U(C)], C\in\mathcal{A}_S\big(C^*\big)\big\}$ is upward-directed. For any $C^i\in \mathcal{A}_S\big(C^*\big)$ , $i=1,2$ , set $C=C^1 \unicode{x1D7D9}_A+C^2 \unicode{x1D7D9}_{A^c}$ , where $A\,:\!=\,\big\{\mathcal{E}^{g}_S\big[U\big(C^1\big)\big]\geq \mathcal{E}^{g}_S\big[U\big(C^2\big)\big]\big\}$ is $\mathcal{F}_S$ -measurable. Note that

\begin{equation*} \Psi_S(C) =\frac{1}{\gamma_S}\widetilde{\mathcal{E}}^h_S\bigg[\int_S^T \gamma_r \text{d} C^1_r\bigg]\unicode{x1D7D9}_A+\frac{1}{\gamma_S}\widetilde{\mathcal{E}}^h_S\bigg[\int_S^T \gamma_r \text{d} C^2_r\bigg]\unicode{x1D7D9}_{A^c}\leq \Psi_S\big(C^*\big), \end{equation*}

which implies that $C\in \mathcal{A}_S\big(C^*\big)$ . It is also easy to check that for any $s\in[0,T]$ , $Y^C_s=Y^{C^1}_s \unicode{x1D7D9}_A+Y^{C^2}_s \unicode{x1D7D9}_{A^c}$ . Since $u(s,0)=0$ , it follows that $u\big(s,Y_s^c\big)=u\Big(s,Y^{C^1}_s\Big) \unicode{x1D7D9}_A+u\Big(s,Y^{C^2}_s\Big)\unicode{x1D7D9}_{A^c}$ . Therefore, we have

\begin{align*} \mathcal{E}^{g}_S[U(C)] & =\mathcal{E}^{g}_S\big[U\big(C^1\big)\unicode{x1D7D9}_A+U\big(C^2\big)\unicode{x1D7D9}_{A^c}\big]=\mathcal{E}^{g}_S\big[U\big(C^1\big)\big]\unicode{x1D7D9}_A+\mathcal{E}^{g}_S\big[U\big(C^2\big)\big]\unicode{x1D7D9}_{A^c}\\& =\mathcal{E}^{g}_S\big[U\big(C^1\big)\big]\vee\mathcal{E}^{g}_S\big[U\big(C^2\big)\big]; \end{align*}

that is, the family $\Big\{\mathcal{E}^{g}_S[U(C)], C\in\mathcal{A}_S\big(C^*\big)\Big\}$ is upward-directed. As a consequence, there exists an increasing sequence $\Big\{\mathcal{E}^{g}_S\big[U\big(C^n\big)\big]\Big\}_{n=1}^\infty$ such that

(5.2) \begin{equation} v_S=\lim_{n\rightarrow\infty} \mathcal{E}^{g}_S\big[U\big(C^n\big)\big], \end{equation}

where $\{C^n\}_{n=1}^\infty\subset \mathcal{A}_S\big(C^*\big)$ .

Step 2. For any stopping times $\tau,\sigma$ with $\tau\leq \sigma$ , we have $\mathcal{A}_\sigma\big(C^*\big)\subset \mathcal{A}_\tau\big(C^*\big)$ . Indeed, for any $C\in \mathcal{A}_\sigma\big(C^*\big)$ , a simple calculation yields that

\begin{align*} \Psi_\tau(C)=&\frac{1}{\gamma_\tau}\widetilde{\mathcal{E}}^h_\tau\bigg[\int_\tau^T \gamma_r\text{d} C_r\bigg]=\frac{1}{\gamma_\tau}\widetilde{\mathcal{E}}^h_\tau\bigg[\int_\tau^\sigma \gamma_r\text{d} C^*_r+\gamma_\sigma\Psi_\sigma(C)\bigg]\\ \leq &\frac{1}{\gamma_\tau}\widetilde{\mathcal{E}}^h_\tau\bigg[\int_\tau^\sigma \gamma_r\text{d} C^*_r+\gamma_\sigma\Psi_\sigma\big(C^*\big)\bigg]=\Psi_\tau\big(C^*\big). \end{align*}

Now, recalling (5.2) from Step 1 above, for any $\tau\leq S$ , it is easy to check that

\begin{align*} \mathcal{E}^{g}_\tau[v_S]&=\mathcal{E}^{g}_\tau\Big[\lim_{n\rightarrow\infty}\mathcal{E}_S^{g}\big[U\big(C^n\big)\big]\Big]=\lim_{n\rightarrow\infty}\mathcal{E}^{g}_\tau\big[\mathcal{E}^{g}_S\big[U\big(C^n\big)\big]\big]\\& =\lim_{n\rightarrow\infty}\mathcal{E}_\tau^{g}\big[U\big(C^n\big)\big]\leq \mathop{{\textrm{ess}}\,{\textrm{sup}}}_{C\in\mathcal{A}_\tau\big(C^*\big)}\mathcal{E}^{g}_\tau[U(C)]=v_\tau. \end{align*}

Step 3. It remains to show that $C^*$ is optimal for the problem (5.1). Proceeding as in the proof of Theorem 3.1, there exists a unique consumption plan $\widehat{C}^{S}$ which is optimal for the problem (5.1). Suppose that $\widehat{C}^{S}$ and $C^*$ are distinguishable on [S, T], so that

\begin{equation*}\mathcal{E}^g_S\big[U\big(\widehat{C}^{S}\big)\big]>\mathcal{E}^g_S\big[U\big(C^*\big)\big].\end{equation*}

By invoking the strict comparison theorem for g-expectation (cf. Theorem 2.2 in [Reference El Karoui, Peng and Quenez14]), we find

\begin{equation*}\mathcal{E}^g\big[U\big(\widehat{C}^{S}\big)\big]=\mathcal{E}^g\big[\mathcal{E}^g_S\big[U\big(\widehat{C}^{S}\big)\big]\big]\,>\mathcal{E}^g\big[\mathcal{E}^g_S\big[U\big(C^*\big)\big]\big]=\mathcal{E}^g\big[U\big(C^*\big)\big],\end{equation*}

which leads to a contradiction.

We now move on by studying the structure of the optimal consumption plan. As a matter of fact, Theorem 3.1 indicates that the optimal consumption plan $C^*$ exists, but it does not give an explicit form for $C^*$ . Inspired by [Reference Bank and Riedel5] and the sufficiency of the FOCs for optimality, we now construct $C^*$ through a progressively measurable process L, called the minimal level of satisfaction, which is the solution to a backward equation (see (5.3) below).

In the rest of this section we assume the following dynamics for the level of satisfaction:

1. (H3) The functions $\eta\,:\,[0,T]\rightarrow\mathbb{R}$ and $\theta\,:\,[0,T]^2\rightarrow \mathbb{R}$ are of following forms:

   \begin{equation*} \eta_t=\eta \exp\!\bigg({-}\int_0^t \beta_s \text{d} s\bigg),\ \ \theta_{t,s}=\beta_s \exp\!\bigg({-}\int_s^t\beta_r\text{d} r\bigg), \ \ 0\leq s\leq t\leq T, \end{equation*}
   where $\beta=\{\beta_s\}_{s\in[0,T]}$ is a strictly positive, continuous function and $\eta\geq 0$ .

For each fixed $\xi^1\in D_g$ , $\xi^2\in D_h$ , $M>0$ and stopping time $\tau<T$ , consider the backward equation

(5.3) \begin{equation}\mathbb{E}_\tau\bigg[\int_\tau^T \frac{\text{d} \mathbb{P}^1}{\text{d} \mathbb{P}_0}\bigg|_{\mathcal{F}_t}\partial_yu\bigg(t,\sup_{\tau\leq v\leq t}\bigg\{L_v\exp\!\bigg({-}\int_v^t \beta_s\text{d} s\bigg)\bigg\}\bigg)\theta_{t,\tau}\text{d} t\bigg]=M\gamma_\tau \frac{\text{d}{\mathbb{P}^2}}{\text{d}\mathbb{P}^0}\bigg|_{\mathcal{F}_\tau},\end{equation}

where $\mathbb{P}^i$ is the probability measure whose Girsanov kernel (with respect to $\mathbb{P}_0$ ) is given by $\xi^i$ , $i=1,2$ . By employing Theorem 3 in [Reference Bank and El Karoui2], it can be shown that for any $\tau < T$ the above equation admits a unique progressively measurable process $L=L^{M,\mathbb{P}^1,{\mathbb{P}^2}}$ with upper right-continuous paths and such that $L_T=0$ .

Starting from this we can then construct two processes by setting

\begin{align*}&Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t\,:\!=\,\exp\!\bigg({-}\int_0^t \beta_s\text{d} s\bigg)\bigg(\eta\vee\sup_{0\leq v\leq t}\bigg\{L_v^{M,\mathbb{P}^1,{\mathbb{P}^2}}\exp\!\bigg(\int_0^v \beta_s\text{d} s\bigg)\bigg\}\bigg), \quad t\in[0,T],\\&C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t\,:\!=\int_0^t Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_s\text{d} s+\int_0^t \beta^{-1}_s\text{d} Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_s, \quad t\in[0,T], \quad C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_{0-}=0.\end{align*}

In fact, the optimal consumption plan of an agent is to consume ‘just enough’ to ensure that the induced level of satisfaction never falls below the lower bound $L^{M,\mathbb{P}^1,{\mathbb{P}^2}}$ . Mathematically, according to Lemma 3.9 in [Reference Bank and Riedel5], one has the following:

1. (i) $Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}$ is an adapted process that is right-continuous with lower limits and of bounded variation, with $Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}\geq L^{M,\mathbb{P}^1,{\mathbb{P}^2}}$ ;

2. (ii) $C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}$ is right-continuous, nondecreasing, and adapted; in other words, $C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}\in\mathcal{X}$ ;

3. (iii) the level of satisfaction induced by $C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}$ , denoted by $Y^{C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}}$ , coincides with $Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}$ and is minimal in the following sense:

   \begin{equation*} Y^{C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}}_t=Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t=\inf_{C\in\mathcal{X},Y^C\geq L}Y^C_t, \ \ t\in[0,T]. \end{equation*}

Following the terminology of [Reference Bank and Riedel5], in the sequel we shall say that the process $C^L$ constructed above is the consumption plan that tracks the level process L.

Theorem 5.1. Recall that $f,\ell$ denote the convex duals of the drivers g and h, respectively. Suppose that the functions g, h satisfy (A1)–(A4), and let Assumptions (H1)–(H3) hold. Let $L^{M,\mathbb{P}^1,\mathbb{P}^2}$ be the solution to (5.3). If we can find some $\mathbb{P}^i$ with Girsanov kernels (with respect to $\mathbb{P}_0$ ) $\xi^i$ , $i=1,2$ , such that

\begin{align*} \mathcal{E}^{g}\bigg[\int_0^T u\Big(t,Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t\Big)\text{d} t\bigg]&=\mathbb{E}^{\mathbb{P}^1}\bigg[\int_0^T u\Big(t,Y^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t\Big)\text{d} t+\int_0^T f\big(r,\xi^1_r\big)\text{d} r\bigg],\\[2pt] \widetilde{\mathcal{E}}^h\bigg[\int_0^T\gamma_t\text{d} C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t\bigg]&=\mathbb{E}^{{\mathbb{P}^2}}\bigg[\int_0^T \gamma_t\text{d} C^{L^{M,\mathbb{P}^1,{\mathbb{P}^2}}}_t-\int_0^T \ell\big(r,\xi^2_r\big)\text{d} r\bigg], \end{align*}

then the consumption plan $C^{L^{M,\mathbb{P}^1,\mathbb{P}^2}}$ is optimal for the utility maximization problem (2.3) with given initial capital $w=\Psi\Big(C^{L^{M,\mathbb{P}^1,\mathbb{P}^2}}\Big)$ .

Remark 5.4. Existence of the desired minimal level of satisfaction leads to a challenging fixed point problem that we discuss in the following and whose study is left for future research.

Fix the Lagrange multiplier M in Equation (5.3). Choose $\xi^{1,1}\in D_g$ , $\xi^{2,1}\in D_h$ (recall that $D_g$ and $D_h$ are defined in Proposition A.1), and let $\mathbb{P}^{i,1}$ be the probability measure whose Girsanov kernel is given by $\xi^{i,1}$ , $i=1,2$ . Then there exists a unique solution $L^1$ to Equation (5.3) with $\mathbb{P}^i=\mathbb{P}^{i,1}$ , $i=1,2$ . Let $C^1$ be the consumption plan which tracks $L^1$ , and let $\Gamma\big(C^1\big)\,:\!=\,\int_0^T \gamma_t \text{d} C_t^1$ . Consider the BSDEs

\begin{align*} &Y^{1,1}_t=U\big(C^1\big)+\int_t^T g\big(s,Z_s^{1,1}\big) \text{d} s-\int_t^T Z_s^{1,1} \text{d} B_s,\\[3pt] &Y^{2,1}_t=\Gamma\big(C^1\big)-\int_t^T h\big(s,-Z_s^{2,1}\big) \text{d} s-\int_t^T Z_s^{2,1} \text{d} B_s, \end{align*}

and let $\xi^{1,2}$ and $\xi^{2,2}$ be the solutions to

\begin{align*} &g\big(s,Z_s^{1,1}\big)-Z_s^{1,1}\xi^{1,2}_s=f\big(s,\xi^{1,2}_s\big) \quad \text{and} \quad -h\big(s,-Z_s^{2,1}\big)-Z_s^{2,1}\xi^{2,2}_s=-\ell\big(s,\xi^{2,2}_s\big), \end{align*}

respectively. Then by Girsanov’s theorem it is easy to check that

\begin{align*} &\mathcal{E}^g\big[U\big(C^1\big)\big]=\mathbb{E}^{\mathbb{P}^{1,2}}\bigg[U\big(C^1\big)+\int_0^T f\big(s,\xi^{1,2}_s\big)\text{d} s\bigg], \\ &\widetilde{\mathcal{E}}^h\big[\Gamma\big(C^1\big)\big]=\mathbb{E}^{\mathbb{P}^{2,2}}\bigg[\Gamma\big(C^1\big)-\int_0^T \ell\big(s,\xi^{2,2}_s\big)\text{d} s\bigg], \end{align*}

where $\mathbb{P}^{i,2}$ is the probability measure with Girsanov kernel $\xi^{i,2}$ , $i=1,2$ , with respect to $\mathbb{P}_0$ .

Defining the mapping $\mathcal{T}\,:\,D_g\times D_h\rightarrow D_g\times D_h$ as

\begin{equation*} \mathcal{T}\big(\xi^{1,1},\xi^{2,1}\big)=\big(\xi^{1,2},\xi^{2,2}\big), \end{equation*}

we see that if $\mathcal{T}$ has a fixed point $\big(\xi^1,\xi^2\big)$ , then $C^{\xi^1,\xi^2}$ tracking $L^{\xi^1,\xi^2}$ is optimal for (2.3), where $L^{\xi^1,\xi^2}$ is the solution to (5.3) with $\mathbb{P}^i=\mathbb{P}^{\xi^i}$ , $i=1,2$ .

6. Explicit solution in a stationary homogeneous setting

6.1. Setting and main result

One can easily be convinced that the sufficient FOCs for optimality previously determined still hold when $T=+\infty$ . Indeed, their proof employs the linear structure of $C \mapsto Y^C$ and the concavity of the instantaneous felicity function, which are clearly not affected by the length of the time interval considered. In this section, we shall use the sufficient optimality conditions to provide the explicit solution in a homogeneous setting.

Consider a financial market following the Samuelson model. One of the assets is a risk-free bond, whose price $S^0$ evolves according to the equation

(6.1) \begin{equation}\text{d} S^0_t=r S^0_t \text{d} t,\end{equation}

where $r>0$ is the interest rate. The price of the stock is denoted by S and satisfies the stochastic differential equation

(6.2) \begin{equation}\text{d} S_t=S_t\big[\mu \text{d} t+ \sigma \text{d} B_t\big],\end{equation}

where $\mu$ represents the stock appreciation rates, $\sigma>0$ is the volatility, and $B_t$ is a one-dimensional $(\mathcal{F}_t)_t$ -Brownian motion. Clearly, there exists a constant $\vartheta\in\mathbb{R}$ such that

\begin{equation*} \mu-r=\sigma \vartheta. \end{equation*}

The constant $\vartheta$ is usually referred to as the market price of risk.

Imagine now that our agent invests in the financial market and thus selects a portfolio $\pi_t$ at time t, where $\pi_t$ is the proportion of her wealth $V_t$ invested in the stock and $\pi^0_t=1- \pi_t$ is the proportion of her wealth invested in the bond. We assume that $\pi$ is $\mathbb{F}$ -predictable, since the agent can only make decisions on the basis of the currently available information, and that it guarantees ${V}_t \geq 0$ a.s. for all $t\geq0$ . In the following, we shall refer to portfolio strategies satisfying the previous requirements as admissible.

The agent can also choose a consumption plan $C \in \mathcal{X}_\infty$ , where $C_t$ represents the total amount of consumption up to time t and

\begin{align*}\mathcal{X}_\infty\,:\!=\,\big\{C\,\big|\,& C \textrm{ is the distribution function of a nonnegative}\\&\textrm{optional random measure on $[0,\,\infty)$}\big\}.\end{align*}

Assume that the risk premia for long and short positions are different, and let $a^{\prime}-a$ be the difference in excess return between long and short positions (see the second part of Example 1.1 in [Reference El Karoui, Peng and Quenez14] or Section 3 in [Reference Jouini and Kallal22]). Then the wealth V associated to the portfolio $\pi$ and consumption plan C evolves for $t\geq0$ as

(6.3) \begin{equation}\text{d} V_t= r V_t \text{d} t -\sigma a \pi_t V_t \text{d} t+\sigma \big(a^{\prime}-a\big)\pi^-_t V_t \text{d} t +\sigma \pi_t V_t \text{d} B_t - \text{d} C_t, \quad V_0=w,\end{equation}

which, setting $\bar{V}_t\,:\!=\,e^{-rt}V_t$ , gives

(6.4) \begin{equation}\text{d} \bar{V}_t= -\sigma a \pi_t \bar{V}_t \text{d} t+\sigma \big(a^{\prime}-a\big)\pi^-_t \bar{V}_t \text{d} t +\sigma \pi_t \bar{V}_t \text{d} B_t - e^{-rs}\text{d} C_s, \quad \bar{V}_0=w.\end{equation}

Now, for any bounded, adapted process $\xi$ , let $\varepsilon^\xi_t\,:\!=\,\varepsilon^{1,\xi}_t$ , where $\varepsilon^{x_0,\xi}_t\,:\!=\,x_0\exp\!\Big(\int_0^t \xi_s \text{d} B_s-\frac{1}{2}\int_0^t \xi^2_s\text{d} s\Big)$ for $x_0>0$ , and define the set of priors

(6.5) \begin{equation}\mathcal{P}^2\,:\!=\,\bigg\{\mathbb{P}^\xi\,\bigg|\, \xi \textrm{ adapted with values in } [a^{\prime},a],\, \frac{\text{d}\mathbb{P}^\xi}{\text{d}\mathbb{P}_0}\bigg|_{\mathcal{F}_t}=\varepsilon^\xi_t, 0<t<\infty\bigg\}.\end{equation}

By Girsanov’s theorem, under any $\mathbb{P}^\xi \in \mathcal{P}^2$ we have that $B^{\xi}_t\,:\!=\, B_t - \int_0^t \xi_s \text{d} s$ is an $\mathbb{F}$ -Brownian motion. Hence, by standard manipulations using that $\pi_t = \pi^+_t - \pi^-_t$ , under $\mathbb{P}^\xi$ the dynamics of $\bar{V}$ can be rewritten as

(6.6) \begin{equation}\bar{V}_t - w = \int_0^t \sigma \bar{V}_s \big(\pi^-_s \big(a^{\prime} - \xi_s\big) - \pi^+_s(a - \xi_s)\big) \text{d} s + \int_0^t \sigma \bar{V}_s \pi_s \text{d} B^{\xi}_s - \int_0^t e^{-rs}\text{d} C_s, \quad t\geq0.\end{equation}

Notice now that the integrand in the first integral above is a.s. negative for any $s \geq 0$ , while $\bar{V}_t$ is a.s. nonnegative for any $t\geq0$ by admissibility of the portfolio strategy. Hence, by making use of a standard localizing argument, taking expectations in (6.6) and letting $t\uparrow \infty$ by monotone convergence, we obtain

\begin{equation*}\mathbb{E}^{\mathbb{P}^{\xi}}\bigg[\int_0^{\infty} e^{-rs}\text{d} C_s\bigg] \leq w.\end{equation*}

Given the arbitrariness of $\mathbb{P}^\xi \in \mathcal{P}^2$ , we are therefore left to focus on the set of admissible consumption plans

(6.7) \begin{equation}\mathcal{A}(w)\,:\!=\,\bigg\{C\in \mathcal{X}_\infty\,\big|\,\mathcal{E}^2\bigg[\int_0^\infty e^{-rt} \text{d} C_t\bigg]\leq w\bigg\},\end{equation}

where we have defined

\begin{equation*}\mathcal{E}^2[X]\,:\!=\,\sup_{\mathbb{P}\in\mathcal{P}^2}\mathbb{E}^\mathbb{P}[X].\end{equation*}

We now move on by describing the agent’s preferences. We assume that the felicity function is deterministic and given by

\begin{equation*}u(t,y)=e^{-\delta t}\frac{1}{\alpha}y^\alpha,\end{equation*}

where $\delta>0$ and $\alpha\in(0,1)$ , and $\gamma_t=e^{-rt}$ . Moreover, we suppose that the level of satisfaction has dynamics

(6.8) \begin{equation}Y_t^C=\eta e^{-\beta t}+\int_0^t \beta e^{-\beta(t-s)} \text{d} C_s,\end{equation}

where $\eta,\beta>0$ .

For given constants b and b ^′, we then define the set of priors for the utility as

(6.9) \begin{equation}\mathcal{P}^1\,:\!=\,\bigg\{\mathbb{P}^\xi\,\big|\, \xi \textrm{ adapted with values in } [b,b^{\prime}],\, \frac{\text{d}\mathbb{P}^\xi}{\text{d}\mathbb{P}_0}\bigg|_{\mathcal{F}_t}=\varepsilon^\xi_t, 0<t<\infty\bigg\},\end{equation}

and, accordingly, introduce the nonlinear expectation

\begin{equation*} \mathcal{E}^1[X]\,:\!=\,\inf_{\mathbb{P}\in\mathcal{P}^1}\mathbb{E}^\mathbb{P}[X]. \end{equation*}

Recalling that $\kappa$ is the Lipschitz constant in Assumption (A2), we assume that a, b, a ^′, b ^′ are such that $ -\kappa\leq a^{\prime}< a<b<b^{\prime}\leq \kappa$ . The aim is to solve

(6.10) \begin{equation} \sup_{C\in\mathcal{A}(w)}\mathcal{E}^1\bigg[\int_0^\infty u\big(t,Y_t^C\big)\text{d} t\bigg],\end{equation}

with $\mathcal{A}(w)$ as in (6.7).

As a matter of fact, we can see from Appendix A that, by taking $g(z)=b z^+ - b^{\prime} z^-$ and $h(z)=a^{\prime} z^+ - az^-$ , the representation of g-expectation yields

\begin{equation*}\mathcal{E}^1[X]=\mathcal{E}^g[X] \quad \text{and} \quad \mathcal{E}^2[X]=\widetilde{\mathcal{E}}^h[X].\end{equation*}

In our analysis a crucial role will be played by the backward equation

(6.11) \begin{equation}\mathbb{E}_\tau\bigg[\int_\tau^\infty \beta\exp({-}\beta(t-\tau))\varepsilon^{\xi^1}_t \partial_y u\bigg(t,\sup_{\tau\leq v\leq t}\bigg\{L_v \exp({-}\beta(t-v))\bigg\}\bigg)\text{d} t\bigg]= M e^{-r\tau} \varepsilon^{\xi^2}_\tau,\end{equation}

where $\tau$ is any finite stopping time, $M>0$ is a given constant, and $\xi^i$ are the Girsanov kernels (with respect to $\mathbb{P}_0$ ) such that $\mathbb{P}_i\,:\!=\,\mathbb{P}^{\xi^i}\in\mathcal{P}^i$ , $i=1,2$ . The solution to (6.11) (if it does exist) is denoted by $L^{M,\mathbb{P}_1,\mathbb{P}_2}$ . Moreover, we denote by $C^{{M,\mathbb{P}_1,\mathbb{P}_2}}$ the consumption plan which tracks the level process $L^{M,\mathbb{P}_1,\mathbb{P}_2}$ and by $Y^{{M,\mathbb{P}_1,\mathbb{P}_2}}$ the corresponding level of satisfaction. Similarly to Theorem 5.1, we have the following result.

Theorem 6.1. Suppose that $\mathbb{P}_i\in \mathcal{P}^i\big(C^{M,\mathbb{P}_1,\mathbb{P}_2}\big)$ , $i=1,2$ , where for any admissible consumption plan

\begin{align*} &\mathcal{P}^1(C)\,:\!=\,\bigg\{\mathbb{P}\in\mathcal{P}^1\,\big|\,\mathcal{E}^1\bigg[\int_0^\infty u\big(t,Y^C_t\big)\text{d} t\bigg]=\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty u\big(t,Y^C_t\big)\text{d} t\bigg]\bigg\},\\[3pt] &\mathcal{P}^2(C)\,:\!=\,\bigg\{\mathbb{P}\in\mathcal{P}^2\,\big|\,\mathcal{E}^2\bigg[\int_0^\infty e^{-rt} \text{d} C_t\bigg]=\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty e^{-rt} \text{d} C_t\bigg]\bigg\}. \end{align*}

Then $C^{M,\mathbb{P}_1,\mathbb{P}_2}$ is an optimal consumption plan for (6.10) with $w=\mathcal{E}^2\Big[\int_0^\infty e^{-rt} \text{d} C^{M,\mathbb{P}_1,\mathbb{P}_2}_t\Big]$ . Consequently, $L^{M,\mathbb{P}_1,\mathbb{P}_2}$ is the optimal minimal level of satisfaction.

We can now provide the main result of this section.

Theorem 6.2. Suppose that $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ . The measures $\mathbb{P}^b$ and $\mathbb{P}^a$ realize the worst-case scenarios for utility and cost of consumption, respectively. Moreover, the optimal consumption plan is such that

\begin{equation*}C^*_t \,:\!=\, C^K_t = \int_0^t e^{-\beta s} \text{d} \overline{C}^*_s,\end{equation*}

where

\begin{equation*}\overline{C}^*_t \,:\!=\, \sup_{0 \leq s \leq t}\Big(\frac{L_s^K - \eta e^{-\beta s}}{e^{-\beta s}}\Big) \vee 0,\qquad \overline{C}^*_{0-}=0,\end{equation*}

with $L^K$ —the optimal minimal level of satisfaction—being given by

\begin{equation*}L_t^K=\bigg(K e^{(\delta-r)t}\frac{\varepsilon^{a}_t}{\varepsilon^b_t}\bigg)^{\frac{1}{\alpha-1}}=K^{\frac{1}{\alpha-1}}\exp\!\bigg\{\frac{a-b}{\alpha-1}B_t-\frac{1}{2(\alpha-1)}\big[\big(a^2-b^2\big)-2(\delta-r)\big]t\bigg\}.\end{equation*}

Here, K is a suitable constant depending on the model’s parameters and uniquely determined through the initial wealth w via the equation

\begin{equation*}w = \mathbb{E}^a\left[\int_0^\infty e^{-rt}\text{d} C^K_t\right].\end{equation*}

Consequently, the optimal level of satisfaction is

\begin{equation*}Y^*_t \,:\!=\,Y^K_t = e^{-\beta t} \bigg( \eta \vee \sup_{0 \leq s \leq t}\!\Big(L^K_s e^{\beta s}\Big)\bigg).\end{equation*}

Remark 6.1. We observe that our condition $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ is consistent with that posed in Theorem 4.7 of [Reference Bank and Riedel5], where

(6.12) \begin{equation}\delta> \alpha r+(1-\alpha)\pi\left(\frac{\alpha\theta^{\prime}}{1-\alpha}\right)+\alpha\pi\big({-}\theta^{\prime}\big),\end{equation}

with $\pi({\cdot})$ being the Laplace exponent of a Lévy process X and $\theta^{\prime}$ the market price of risk. In order to see this, notice that, under a measure $\mathbb{P}^{\xi_1} \in \mathcal{P}^1$ , (6.11) can be rewritten as

(6.13) \begin{equation}\mathbb{E}^{\xi_1}_\tau\bigg[\int_\tau^\infty \beta\exp({-}\beta(t-\tau))\partial_y u\bigg(t,\sup_{\tau\leq v\leq t}\bigg\{L_v \exp({-}\beta(t-v))\bigg\}\bigg)\text{d} t\bigg]= M e^{-r\tau} \frac{\varepsilon^{\xi^2}_\tau}{\varepsilon^{\xi^1}_\tau}.\end{equation}

Then, taking $\xi_1\equiv b$ and $\xi_2\equiv a$ , the latter is exactly of the form of Equation (17) in [Reference Bank and Riedel5], upon setting (now under $\mathbb{P}^b$ )

\begin{equation*}\psi_t = e^{-rt + (a-b)B^b_t - \frac{1}{2}(a-b)^2 t}, \quad t\geq0.\end{equation*}

Here, $B^b$ is a standard Brownian motion under $\mathbb{P}^b$ . In particular, it follows that in our case $\theta^{\prime}$ of (6.12) is such that $\theta^{\prime}=a-b$ , and simple algebra shows that the right-hand side of (6.12) indeed becomes our expression $\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ .

As a further consequence, we can argue as in the proof of Theorem 4.7 of [Reference Bank and Riedel5] and prove that the condition $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ is in fact also necessary to have a well-posed optimal consumption problem (in the sense that, without that condition, one can construct a consumption plan that has finite cost but induces infinite utility).

6.2. On the proof of Theorem 6.2

The proof of Theorem 6.2 will be organized as follows. First, for any fixed $\mathbb{P}^{\xi^i}\in\mathcal{P}^i$ with $\xi^i$ being a constant, $i=1,2$ , under the constellation of parameters considered, we solve the backward equation (6.11) explicitly. Then we verify that $\mathbb{P}^a$ is the largest-cost probability and $P^b$ is the lowest-utility probability for the consumption $C^K$ that tracks $L^K$ as defined in Theorem 6.2.

The proof of the next result exploits the time-homogeneity of our setting. It appears in Appendix B for the sake of completeness.

Lemma 6.1. For any fixed constants M, $\xi^i$ and probability measures $\mathbb{P}^i\,:\!=\,\mathbb{P}^{\xi^i}$ , $i=1,2$ , the solution to Equation (6.11) is

\begin{equation*} L_t\,:\!=\,L^{M,\xi^1,\xi^2}_t=\Bigg(K e^{(\delta-r)t}\frac{\varepsilon^{\xi^2}_t}{\varepsilon^{\xi^1}_t}\Bigg)^{\frac{1}{\alpha-1}}, \end{equation*}

where $K=K(M)$ is the constant satisfying

\begin{equation*} K\beta \mathbb{E}\Bigg[\int_0^\infty e^{-(\delta+\alpha\beta)t}\inf_{0\leq v\leq t}\Bigg\{e^{(\delta+\beta(\alpha-1)-r)v}\varepsilon^{\xi^2}_v \frac{\varepsilon^{\xi^1}_t}{\varepsilon^{\xi^1}_v}\Bigg\}\text{d} t\Bigg]=M. \end{equation*}

In order to simplify the notation, set

\begin{equation*}\theta\,:\!=\,\frac{b-a}{1-\alpha}, \qquad \lambda\,:\!=\,\frac{1}{2(\alpha-1)}\big[\big(a^2-b^2\big)-2(\delta-r)\big],\end{equation*}

and notice that $\theta>0$ .

For any constant $\xi$ , let $x^{\alpha}_+(\xi)$ and $x_+(\xi)$ be the largest solutions to the equations $h^{\alpha,\xi}(x)=0$ and $h^\xi(x)=0$ , respectively, where

(6.14) \begin{equation}h^{\alpha,\xi}(x)\,:\!=\, \frac{1}{2}\alpha^2\theta^2x^2-\alpha(\lambda-\beta-\theta\xi)x-(\delta+\alpha\beta), \quad x \in \mathbb{R},\end{equation}

and

(6.15) \begin{equation}h^\xi(x)=\frac{1}{2}\theta^2x^2-(\lambda-\beta-\theta\xi)x-(r+\beta), \quad x \in \mathbb{R}.\end{equation}

Let $C^K$ be the consumption plan which tracks $L^K$ as defined in Theorem 6.2, and let $Y^K$ be the corresponding level of satisfaction. By the necessary characterization of the optimal consumption plan under a single prior (cf. [Reference Bank and Riedel5]), the plan $C^K$ is optimal for the linear problem

(6.16) \begin{equation}v^{a,b}\,:\!=\,\sup_{C\in \mathcal{A}^a(w)}\mathbb{E}^b\left[\int_0^\infty u\big(t,Y^C_t\big)\text{d} t\right],\end{equation}

where

(6.17) \begin{equation}\mathcal{A}^a(w)\,:\!=\,\bigg\{C\in\mathcal{X}\,\big|\,\mathbb{E}^a\bigg[\int_0^\infty e^{-rt} \text{d} C_t\bigg]\leq w\bigg\},\end{equation}

and $\mathbb{E}^a$ and $\mathbb{E}^b$ are the expectations taken under $\mathbb{P}^a$ , $\mathbb{P}^b$ , respectively. Furthermore, proceeding similarly to [Reference Bank and Riedel5] $\big($ see Equation (38) therein and the following equation for $V\big(C^K\big)\big)$ , the expected utility and expected cost associated with the consumption plan $C^K$ can be explicitly given by

(6.18) \begin{equation} \begin{split} \phi^b(\eta)&\,:\!=\,\mathbb{E}^b\left[\int_0^\infty u\big(t,Y^K_t\big)\text{d} t\right]\\ &=\frac{1}{\alpha(\delta+\alpha\beta)}\begin{cases} \eta^\alpha+\frac{1}{x^\alpha_+(b)-1}K^{\frac{\alpha x_+^\alpha(b)}{\alpha-1}}\eta^{\alpha\big(1-x_+^\alpha(b)\big)}, &\eta> K^{\frac{1}{\alpha-1}};\\[5pt] \frac{x^\alpha_+(b)}{x^\alpha_+(b)-1}K^{\frac{\alpha}{\alpha-1}}, &\eta\leq K^{\frac{1}{\alpha-1}}, \end{cases} \end{split} \end{equation}

and

(6.19) \begin{equation} \psi^a(\eta)\,:\!=\,\mathbb{E}^a\left[\int_0^\infty e^{-rt}\text{d} C^K_t\right]=\frac{1}{\beta}\begin{cases} \frac{1}{x_+(a)-1}K^{\frac{x_+(a)}{\alpha-1}}\eta^{1-x_+(a)}, &\eta> K^{\frac{1}{\alpha-1}};\\[7pt] \frac{x_+(a)}{x_+(a)-1}K^{\frac{1}{\alpha-1}}-\eta, &\eta\leq K^{\frac{1}{\alpha-1}}. \end{cases} \end{equation}

For any $\eta>0$ , set

\begin{equation*}\phi^{\xi^1}(\eta)\,:\!=\,\mathbb{E}\bigg[\int_0^\infty \varepsilon^{\xi^1}_t u\big(t,Y_t^K\big)\text{d} t\bigg] \quad \text{and} \quad \psi^{\xi^2}(\eta)\,:\!=\,\mathbb{E}\bigg[\int_0^\infty e^{-rt}\varepsilon^{\xi^2}_t \text{d} C_t^K\bigg].\end{equation*}

Lemma 6.2. Under the assumptions of Theorem 6.2, we have

\begin{align*} \phi^b(\eta)=\inf_{\xi^1\in [b,b^{\prime}], \textrm{adapted}}\phi^{\xi^1}(\eta), \qquad \psi^a(\eta)=\sup_{\xi^2\in [a^{\prime},a], \textrm{adapted}}\psi^{\xi^2}(\eta). \end{align*}

Furthermore, the values are finite.

Proof. Notice that

\begin{align*}Y^K_t&=e^{-\beta t}\bigg\{\eta\vee\sup_{0\leq v\leq t}\Big(L^K_ve^{\beta v}\Big)\bigg\}\\&=e^{-\beta t}\bigg\{\eta\vee\sup_{0\leq v\leq t}K^{\frac{1}{\alpha-1}}\exp\!\Big(\theta B_v-(\lambda-\beta)v\Big)\bigg\}.\end{align*}

Applying Girsanov’s theorem and Tonelli’s theorem, it is easy to check that

\begin{align*} \phi^{\xi^1}(\eta)=&\int_0^\infty \frac{1}{\alpha}e^{-(\delta+\alpha\beta)t}\mathbb{E}\bigg[\varepsilon^{\xi^1}_t \bigg\{\eta^\alpha\vee\sup_{0\leq v\leq t}K^{\frac{\alpha}{\alpha-1}}\exp(\alpha\theta B_v-\alpha(\lambda-\beta)v)\bigg\}\bigg]\text{d} t\\ =&\int_0^\infty \frac{1}{\alpha}e^{-(\delta+\alpha\beta)t}\mathbb{E}\bigg[ \bigg\{\eta^\alpha\vee\sup_{0\leq v\leq t}K^{\frac{\alpha}{\alpha-1}}\exp(\alpha\theta B_v-\alpha(\lambda-\beta)v+\alpha\theta\int_0^v \xi^1_s\text{d} s)\bigg\}\bigg]\text{d} t.\end{align*}

Clearly, for any adapted $\xi^1$ taking values between b ^′ and b, we have $\phi^b(\eta)\leq \phi^{\xi^1}(\eta)\leq \phi^{b^{\prime}}(\eta)$ . The assumption $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ implies that $h^{\alpha,b}(1)<0$ . Hence, $x^\alpha_+(b)>1$ . Recalling Equation (6.18), $\phi^{b}(\eta)$ is finite.

Next we calculate $\psi^{\xi^2}(\eta)$ . Noting that $\text{d} C_t^K=Y_t^K\text{d} t+\frac{1}{\beta}\text{d} Y_t^K=\frac{1}{\beta}e^{-\beta t}\text{d} \big(e^{\beta t}Y_t^K\big)$ , a simple calculation yields that

(6.20) \begin{align} \int_0^T e^{-rt}\varepsilon^{\xi^2}_t \text{d} C^K_t & = \int_0^T \frac{1}{\beta}e^{-(r+\beta)t}\varepsilon^{\xi^2}_t \text{d} \big(e^{\beta t}Y^K_t\big)\nonumber\\ & =\frac{1}{\beta}e^{-rT}\varepsilon^{\xi^2}_T Y^K_T-\frac{\eta}{\beta}-\frac{1}{\beta}\int_0^T e^{\beta t}Y_t^K\text{d} \Big(e^{-(r+\beta)t}\varepsilon^{\xi^2}_t\Big)\\ & =\frac{1}{\beta}e^{-rT}\varepsilon^{\xi^2}_T Y^K_T-\frac{\eta}{\beta}+\Big(1+\frac{r}{\beta}\Big)\int_0^Te^{-rt}\varepsilon^{\xi^2}_t Y^K_t\text{d} t-\frac{1}{\beta}\int_0^Te^{-rt}\xi^2_t\varepsilon^{\xi^2}_t Y^K_t \text{d} B_t.\nonumber\end{align}

Now, set

\begin{equation*}\widetilde{\psi}^{\xi^2}(\eta)\,:\!=\,\mathbb{E}\bigg[\int_0^\infty e^{-rt}\varepsilon^{\xi^2}_t Y^K_t\text{d} t\bigg].\end{equation*}

Proceeding similarly to the evaluation of $\phi^{\xi^1}(\eta)$ , for any adapted process $\xi^2$ taking values in [a, a ^′] we have

\begin{equation*}\widetilde{\psi}^{\xi^2}(\eta)=\int_0^\infty e^{-(r+\beta)t}\mathbb{E}\bigg[\eta\vee\sup_{0\leq v\leq t}K^{\frac{1}{\alpha-1}}\exp\!\bigg(\theta B_v-(\lambda-\beta)v+\theta\int_0^v \xi^2_s\text{d} s\bigg)\bigg]\text{d} t,\end{equation*}

and therefore $\widetilde{\psi}^{a^{\prime}}(\eta)\leq \widetilde{\psi}^{\xi_2}(\eta)\leq \widetilde{\psi}^{a}(\eta)$ . Calculations analogous to those performed in the proof of Proposition 5.8 in [Reference Ferrari, Li and Riedel17] imply that, for $\xi^2\in \{a,a^{\prime}\}$ ,

\begin{equation*} \widetilde{\psi}^{\xi^2}(\eta)=\frac{1}{\beta+r}\begin{cases} \frac{1}{x_+\big(\xi^2\big)-1}K^{\frac{x_+\big(\xi^2\big)}{\alpha-1}}\eta^{1-x_+\big(\xi^2\big)}+\eta, &\eta> K^{\frac{1}{\alpha-1}};\\[12pt] \frac{x_+\big(\xi^2\big)}{x_+\big(\xi^2\big)-1}K^{\frac{1}{\alpha-1}}, &\eta\leq K^{\frac{1}{\alpha-1}}. \end{cases} \end{equation*}

The assumption $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ yields that $h^a(1)<0$ and $h^{a^{\prime}}(1)<0$ . Therefore, we have $x_+(a)>1$ and $x_+(a^{\prime})>1$ . Hence, the quantity $\widetilde{\psi}^{\xi^2}(\eta)$ is positive and finite for any adapted $\xi^2\in[a,a^{\prime}]$ . Since by following the steps of the proof of Lemma 4.9 in [Reference Bank and Riedel5] we can show that

\begin{equation*} \lim_{T\rightarrow \infty}e^{-rT}\varepsilon^{\xi^2}_T Y^K_T=0,\end{equation*}

taking the expectation on both sides of (6.20) and letting T to infinity, we obtain

\begin{align*} \psi^{\xi^2}(\eta)=\Big(1+\frac{r}{\beta}\Big)\widetilde{\psi}^{\xi^2}(\eta)-\frac{\eta}{\beta}.\end{align*}

It is now straightforward to see that $\psi^{a^{\prime}}(\eta)\leq \psi^{\xi^2}(\eta)\leq \psi^a(\eta)$ , which completes the proof.

The results collected so far allow us finally to prove Theorem 6.2.

Proof of Theorem 6.2. By Lemmas 6.1 and 6.2, we know that $L^K$ is the solution to (6.11) with $\mathbb{P}_1=\mathbb{P}^b$ , $\mathbb{P}_2=\mathbb{P}^a$ and $\mathbb{P}^b\in \mathcal{P}^1\big(C^K\big)$ , $\mathbb{P}^a\in \mathcal{P}^2\big(C^K\big)$ . By Theorem 6.1, it remains to find an appropriate constant K such that $w=\psi^{a}(\eta)$ , where $\psi^a(\eta)$ is given by (6.19). By simple calculations it can be shown that the needed K is given by

\begin{equation*}K=\begin{cases}\left(\frac{x_+(a)-1}{{x_+(a)}}(\beta w+\eta)\right)^{\alpha-1}, & w\geq \frac{\eta}{\beta(x_+(a)-1)};\\[3pt]\big(\beta(x_+(a)-1)\eta^{x_+(a)-1}w\big)^{\frac{\alpha-1}{x_+(a)}} &\textrm{otherwise}.\end{cases}\end{equation*}

6.3. On the portfolio financing the optimal consumption plan

In this section, we will find the portfolio process $\pi$ needed to finance the optimal consumption plan derived in the previous section. For this purpose, let $V^a_t$ , $\widehat{V}_t$ be the present values of the future consumption $C^K$ taken under the probability $\mathbb{P}^a$ and the set of multiple priors $\mathcal{P}^2$ , respectively; i.e.,

\begin{align*}&V^a_t\,:\!=\,\mathbb{E}^a_t\left[\int_t^\infty e^{-r(s-t)}\text{d} C^K_s\right]=\mathbb{E}_t\left[\int_t^\infty \frac{\varepsilon^a_s}{\varepsilon_t^a}e^{-r(s-t)}\text{d} C^K_s\right],\\&\widehat{V}_t\,:\!=\,\mathop{{\textrm{ess}}\,{\textrm{sup}}}_{\mathbb{P}^\xi\in\mathcal{P}^2}\mathbb{E}^\xi_t\left[\int_t^\infty e^{-r(s-t)}\text{d} C^K_s\right]=\mathop{{\textrm{ess}}\,{\textrm{sup}}}_{\mathbb{P}^\xi\in\mathcal{P}^2}\mathbb{E}_t\left[\int_t^\infty \frac{\varepsilon^\xi_s}{\varepsilon_t^\xi}e^{-r(s-t)}\text{d} C^K_s\right].\end{align*}

Lemma 6.3. Recall $\psi^a$ as in (6.19). One has

\begin{equation*}\widehat{V}_t=V^a_t=e^{\theta B_t-\lambda t}\psi^a\big(e^{-\theta B_t+\lambda t}Y_t^K\big).\end{equation*}

Proof. By Equation (6.20), we have

\begin{equation*}V_t^a=\frac{(\beta+r)e^{rt}}{\beta\varepsilon^a_t}\mathbb{E}_t\left[\int_t^\infty e^{-rs}\varepsilon^a_s Y^K_s\text{d} s\right]-\frac{Y^K_t}{\beta}\,=\!:\,\widetilde{V}^a_t-\frac{Y^K_t}{\beta}.\end{equation*}

On the other hand, the Markov property implies that

\begin{align*}\widetilde{V}_t^a & = \frac{\tilde{\beta}e^{rt}}{\beta\varepsilon^a_t}\mathbb{E}_t\left[\int_t^\infty e^{-\tilde{\beta}s}\varepsilon^a_s\left\{\eta\vee \sup_{0\leq v\leq s}K^{\frac{1}{\alpha-1}}\exp\!\big(\theta B_v-(\lambda-\beta)v\big)\right\}\text{d} s\right]\\[3pt]& =\frac{\tilde{\beta}}{\beta}\mathbb{E}_t\left[\int_0^\infty e^{-\tilde{\beta}s}\frac{\varepsilon^a_{s+t}}{\varepsilon_t^a}\left\{Y^K_t\vee e^{-\beta t}\sup_{t\leq v\leq s+t}K^{\frac{1}{\alpha-1}}\exp\!\big(\theta B_v-(\lambda-\beta)v\big)\right\}\text{d} s\right]\\[3pt]& =\frac{\tilde{\beta}}{\beta}\mathbb{E}_t\left[\int_0^\infty e^{-\tilde{\beta}s}\frac{\varepsilon^a_{s+t}}{\varepsilon_t^a}\left\{Y^K_t\vee e^{-\lambda t}\sup_{0\leq v\leq s}K^{\frac{1}{\alpha-1}}\exp\!\big(\theta B_{v+t}-(\lambda-\beta)v\big)\right\}\text{d} s\right]\\[3pt]& =\frac{\tilde{\beta}}{\beta}e^{\theta B_t-\lambda t}\mathbb{E}_t\left[\int_0^\infty e^{-\tilde{\beta}s}\frac{\varepsilon^a_{s+t}}{\varepsilon_t^a}\left\{\eta\vee \sup_{0\leq v\leq s}K^{\frac{1}{\alpha-1}}\exp\!\big(\theta B_{v+t}^t-(\lambda-\beta)v\big)\right\}\text{d} s\right]\bigg|_{\eta=\widetilde{Y}^K_t}\\[3pt]& =\frac{\tilde{\beta}}{\beta}e^{\theta B_t-\lambda t}\mathbb{E}\left[\int_0^\infty e^{-\tilde{\beta}s}\varepsilon_s^a\left\{\eta\vee \sup_{0\leq v\leq s}K^{\frac{1}{\alpha-1}}\exp\!\big(\theta B_v-(\lambda-\beta)v\big)\right\}\text{d} s\right]\bigg|_{\eta=\widetilde{Y}^K_t}\\[3pt]& =e^{\theta B_t-\lambda t}\left(\psi^a\big(e^{-\theta B_t+\lambda t}Y^K_t\big)+\frac{1}{\beta}e^{-\theta B_t+\lambda t}Y^K_t\right),\end{align*}

where $\widetilde{Y}^K_t=e^{-\theta B_t+\lambda t}Y^K_t$ , $\tilde{\beta}=\beta+r$ , and $B^{t}_{v+t}=B_{v+t}-B_t$ . The above analysis yields that $V^a_t=e^{\theta B_t-\lambda t}\psi^a\big(e^{-\theta B_t+\lambda t}Y_t^K\big)$ . Then, following the arguments developed in the proof of Theorem 5.11 in [Reference Ferrari, Li and Riedel17], we conclude that $\widehat{V}_t=V_t^a$ .

Proposition 6.1. Under the assumptions of Theorem 6.1, we have

(6.21) \begin{equation}\pi_t=\frac{\theta x_+(a)}{\sigma} \,=\!:\,\pi \quad {for\ all}\,\, t \geq 0.\end{equation}

Proof. It is easy to check that for any $t\geq 0$ , $e^{-\theta B_t+\lambda t}Y^K_t\geq K^{\frac{1}{\alpha-1}}$ . By Lemma 6.3 and (6.19), applying Itô’s formula, we find

\begin{equation*}\text{d} V^a_t=\big(r-a\theta x_+(a)\big) V^a_t\text{d} t+\theta x_+(a) V^a_t \text{d} B_t - \text{d} C^K_t.\end{equation*}

Compared with (6.3), the latter gives that the portfolio is actually constant and such that $\pi_t=\theta x_+(a)/\sigma$ for all $t\geq0$ . In particular, the determined portfolio $\pi$ is admissible given the nonnegativity of $V^a$ .

Remark 6.2. By Theorem 6.2 and the analysis in Section 4 of [Reference Bank and Riedel5], the optimal policy $C^K$ for the utility maximization problem (6.10) is also optimal for the linear problem (6.16). Hence, the portfolio process (6.21) is also such as to finance the consumption plan appearing in (6.17) and thus is consistent with that found in Theorem 4.14 of [Reference Bank and Riedel5].

Recall that $x_+(a)$ is the largest solution to the equation $h^a(x)=0$ , where $h^a$ is defined in (6.15). We now study the dependence of the portfolio (6.21) with respect to $b-a$ (measuring in some sense the discrepancy between the beliefs of the agents related to the utility and cost from consumption), the volatility $\sigma$ , and the parameter of relative risk aversion $1-\alpha$ .

Proposition 6.2. Recall $\pi$ from (6.21). One has the following:

1. 1. The process $\pi$ is decreasing with respect to $\sigma$ .

2. 2. The process $\pi$ is decreasing with respect to $1-\alpha$ .

3. 3. Let $\hat{\delta}\,:\!=\,\beta + \frac{\delta - r}{\alpha-1}$ . Then the following hold:

   1. (i) If $\hat{\delta}\geq 0$ , then $\pi$ is increasing with respect to $b-a$ .

   2. (ii) If $\hat{\delta}<0$ and $\delta-2\alpha\delta+\alpha^2 r+\beta\alpha(1-\alpha)\leq 0$ , then $\pi$ is decreasing with respect to $b-a$ .

   3. (iii) If $\hat{\delta}<0$ and $\delta-2\alpha\delta+\alpha^2 r+\beta\alpha(1-\alpha)> 0$ , then $\pi$ is decreasing with respect to $b-a$ when $(b-a)^2\in\big(0, - 2(1-\alpha)\hat{\delta}\big)$ and increasing when $(b-a)^2\in \left({-}2(1-\alpha)\hat{\delta}, (1-\alpha)\frac{2(\delta-\alpha r)}{\alpha} \right)$ .

Proof. Since it readily follows that $\pi$ is decreasing with respect to $\sigma$ , we proceed directly to proving the second claim of the proposition.

Setting

\begin{equation*} \widetilde{\delta}\,:\!=\,(a-b)^2+2(\delta-r), \qquad I(\alpha)\,:\!=\,\frac{\beta}{(a-b)}(1-\alpha)+\frac{\widetilde{\delta}}{2(b-a)}, \end{equation*}

it is easy to check that

\begin{equation*} \theta x_+(a)=I(\alpha)+\sqrt{I^2(\alpha)+2(r+\beta)}\,=\!:\,H(\alpha) \end{equation*}

and

\begin{equation*} H^{\prime}(\alpha)=\left(1+\frac{I(\alpha)}{\sqrt{I^2(\alpha)+2(r+\beta)}}\right)I^{\prime}(\alpha)\geq 0. \end{equation*}

From the latter, it follows that $\pi$ is increasing in $\alpha$ , and this proves the second claim.

As for the third claim, recall that $\theta=(b-a)/(1-\alpha)$ and notice that

\begin{equation*}\theta x_+(a)=\frac{1-\alpha}{2}\theta-\frac{\hat{\delta}}{\theta}+\sqrt{\bigg(\frac{1-\alpha}{2}\theta-\frac{\hat{\delta}}{\theta}\bigg)^2+2(r+\beta)}\,:\!=\,F(\theta).\end{equation*}

Defining $G(\theta)\,:\!=\,\frac{1-\alpha}{2}\theta-\frac{\hat{\delta}}{\theta}$ , we find from simple calculations that

\begin{equation*}F^{\prime}(\theta)=\frac{\sqrt{G^2(\theta)+2(r+\beta)}+G(\theta)}{\sqrt{G^2(\theta)+2(r+\beta)}}G^{\prime}(\theta),\end{equation*}

and $G^{\prime}(\theta)=(1-\alpha)/2+\hat{\delta}\theta^{-2}$ . Since $F^{\prime}\geq0$ if and only if $G^{\prime}\geq0$ , it follows that the monotonicity of $\pi$ with respect to $b-a$ is the same as the monotonicity of G with respect to $\theta$ . Now, the requirement $\delta>\alpha r+\frac{\alpha(a-b)^2}{2(1-\alpha)}$ implies that $0<\theta^2<\frac{2(\delta-\alpha r)}{\alpha(1-\alpha)}$ . We then have the following three cases, completing the proof.

Case (i). Suppose that $\hat{\delta}\geq 0$ . Then we have $G^{\prime}(\theta)\geq 0$ for any $\theta^2\in\left(0,\frac{2(\delta-\alpha r)}{\alpha(1-\alpha)}\right)$ .

Case (ii). Suppose that $\hat{\delta}<0$ and $\delta-2\alpha\delta+\alpha^2 r+\beta\alpha(1-\alpha)\leq 0$ . It is easy to check that for any $\theta^2\in \left(0,\frac{2(\delta-\alpha r)}{\alpha(1-\alpha)} \right)$ , $G^{\prime}(\theta)\leq 0$ .

Case (iii). Suppose that $\hat{\delta}<0$ and $\delta-2\alpha\delta+\alpha^2 r+\beta\alpha(1-\alpha)> 0$ . By simple calculations, we have $G^{\prime}(\theta)< 0$ when $\theta^2\in \left(0,\frac{2\hat{\delta}}{\alpha-1} \right)$ and $G^{\prime}(\theta)>0$ when $\theta^2\in \left(\frac{2\hat{\delta}}{\alpha-1},\frac{2(\delta-\alpha r)}{\alpha(1-\alpha)} \right)$ .

Let us comment on the comparative statics of the optimal portfolio. First, we obtain the intuitive result that an increase in volatility $\sigma$ reduces investment in the risky asset. More risk-averse agents also invest less in the risky assets than their less risk-averse counterparts. These results are well in line with the usual comparative statics in asset pricing models.

Knightian uncertainty and market friction are well described by the difference of the parameters $b-a$ in our model because these parameters describe the respective worst-case models for the agent’s minimal expected utility and maximal expected cost. Interestingly, it is not always the case that an increase of this term, i.e. a better expected return for investing in the risky asset, leads to more exposure in the risky asset. We have to distinguish two cases here. When $\hat{\delta}=\beta + \frac{\delta - r}{\alpha-1} \ge 0$ , the optimal level of satisfaction diverges to infinity (compare also the discussion in [Reference Bank and Riedel4] in the frictionless case). In this case, the optimal portfolio is indeed decreasing in market friction. In the opposite case, we have $(1-\alpha) \beta < \delta - r $ . When interest rates r are quite low, impatience $\delta$ is high, and the depreciation rate $\beta$ is low, i.e. the agent enjoys past consumption for a long time; the agent optimally consumes a lot in early years and thus does not invest for higher future consumption, as he enjoys his past consumption for a long time and as he is relatively impatient. He is thus not interested in the prospect of higher returns in the long run; he may even decrease his investment in the risky asset if the returns increase.

6.4. Abstention from the asset market

We have already seen in Remark 6.1 that the requirement on $\delta$ imposed in the statement of Theorem 6.2 is necessary in order to have a well-posed optimization problem. We now want to better understand the role of the other assumption made within this section—namely, the requirement $-\kappa\leq a^{\prime}< a < b <b^{\prime}\leq \kappa$ imposed on the parameters defining the sets of priors $\mathcal{P}_1$ and $\mathcal{P}_2$ (cf. (6.9) and (6.5)).

Proposition 6.3. Suppose that there exists some probability $\mathbb{P}$ such that $\mathbb{P}\in \mathcal{P}^1\cap \mathcal{P}^2$ , and let $\delta > \alpha r$ . Then the optimal consumption plan for the utility maximization problem (6.10) is deterministic.

Remark 6.3. The condition $\delta>\alpha r$ arises from the condition $\delta > \alpha r + \frac{\alpha(a-b)^2}{2(1-\alpha)}$ needed to avoid an ill-posed problem in Theorem 6.2. Indeed, in the case $\mathcal{P}^1\cap \mathcal{P}^2 \neq \emptyset$ , we can simply take $a=b$ , thus leading to $\delta > \alpha r$ .

Proof. It is easy to check that $L^K_v=\big(Ke^{(\delta-r)v}\big)^{\frac{1}{\alpha-1}}$ is the solution to (6.11) with $\xi^1=\xi^2$ such that $\mathbb{P}^{\xi^1}=\mathbb{P}^{\xi^2}=\mathbb{P}$ and the Lagrange multiplier $M=\frac{K\beta}{r+\beta}$ . Clearly, the consumption plan that tracks such $L^K$ is a deterministic function, and hence the utility and the cost induced by this consumption plan are deterministic as well. Therefore, we have

\begin{align*} &\inf_{\mathbb{P}\in\mathcal{P}^1}\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty u\big(t,Y_t^K\big)\text{d} t\bigg]=\int_0^\infty u\big(t,Y_t^K\big)\text{d} t=\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty u\big(t,Y_t^K\big)\text{d} t\bigg] ,\\ &\sup_{P\in\mathcal{P}^2}\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty e^{-rt} \text{d} C^K_t\bigg]=\int_0^\infty e^{-rt} \text{d} C^K_t=\mathbb{E}^\mathbb{P}\bigg[\int_0^\infty e^{-rt} \text{d} C^K_t\bigg], \end{align*}

which implies that $\mathbb{P}\in \mathcal{P}^1\big(C^K\big)\cap \mathcal{P}^2\big(C^K\big)$ . Hence, $\mathbb{P}$ is indeed the worst-case scenario for the utility and cost associated to the deterministic plan $C^K$ .

In the following, we complete the proof by determining the appropriate constant K such that the induced cost equals the initial wealth w.

Case 1. Suppose that $\delta<r+(1-\alpha)\beta$ . In this case,

\begin{equation*}Y^K_t=\big(e^{-\beta t}\eta\big)\vee \Big(K^{\frac{1}{\alpha-1}}e^{\frac{\delta-r}{\alpha-1}t}\Big)=\Big(e^{-\beta t}\eta\Big)\vee L^K_t.\end{equation*}

It remains to find K such that the cost $\int_0^\infty e^{-rt} \text{d} C_t^K$ equals w, where

\begin{equation*} \int_0^\infty e^{-rt}\text{d} C_t^K=\begin{cases} K^{\frac{1}{\alpha-1}}\frac{(1-\alpha)(\beta+r)}{\beta(\delta-\alpha r)}-\frac{\eta}{\beta}, & \eta \leq K^{\frac{1}{\alpha-1}};\\[7pt]\frac{(1-\alpha)\beta+r-\delta}{\beta (\delta-\alpha r)}\eta^{\frac{\alpha r-\delta}{(1-\alpha)\beta+r-\delta}}K^{-\frac{r+\beta}{(1-\alpha)\beta+r-\delta}}, & \eta > K^{\frac{1}{\alpha-1}}.\end{cases} \end{equation*}

Simple calculations imply that

\begin{equation*}K=\begin{cases}\left[\frac{(\eta+\beta w)}{(1-\alpha)(\beta+r)}\right]^{\alpha-1}, &w\geq\frac{\beta\eta(1-\alpha)+\eta(r-\delta)}{\beta(\delta-\alpha r)}; \\[8pt]\left[\frac{\beta w(\delta-\alpha r)}{(1-\alpha)\beta+r-\delta}\right]^{-\frac{(1-\alpha)\beta+r-\delta}{r+\beta}}\eta^{\frac{\delta-\alpha r}{\delta+r}} &\textrm{otherwise}.\end{cases}\end{equation*}

Case 2. Suppose that $\delta\geq r+(1-\alpha)\beta$ . In this case,

\begin{equation*}Y^K_t=e^{-\beta t}\big(\eta\vee L_0^K\big)=e^{-\beta t}\big(\eta\vee K^{\frac{1}{\alpha-1}}\big).\end{equation*}

Recalling the dynamics of the level of satisfaction (6.8), it is easy to check that $K=(\eta+\beta w)^{\alpha-1}$ and the optimal consumption $C^*$ should be such that $C^*_t=C^*_0=w$ . That is, the agent will consume all his initial wealth at the original time.