# Constrained portfolio-consumption strategies with uncertain parameters and borrowing costs

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## Abstract

This paper studies the properties of the optimal portfolio-consumption strategies in a finite horizon robust utility maximization framework with different borrowing and lending rates. In particular, we allow for constraints on both investment and consumption strategies, and model uncertainty on both drift and volatility. With the help of explicit solutions, we quantify the impacts of uncertain market parameters, portfolio-consumption constraints and borrowing costs on the optimal strategies and their time monotone properties.

## Keywords

Robust utility maximization Explicit solutions Portfolio-consumption constraints Different borrowing and lending rates Model uncertainty## AMS subject classifications

35R60 47J20 93E20## JEL Classifications

C61 G11## 1 Introduction

*c*and terminal wealth \(X_T^{\pi ,c;\mu ,\sigma }\), respectively. The market is described by a set of parameters \((\mu ,\sigma )\)–the drift and volatility of the risky assets, and the investor’s utilities are often assumed to admit some homothetic properties (for example, power, logarithm and exponential types). Due to the market incompleteness arising from the randomness of the market parameters and the portfolio constraints, the resulting optimal portfolio is described as the sum of a myopic strategy of Merton’s type and a hedging strategy. The latter is used to partially hedge the market risk stemming from the market incompleteness. Both the hedging strategy and the optimal consumption can be described via the solution of a backward stochastic differential equation (see [7, 15]). However, the solution is in general not explicit, and consequently, there is limited information about the properties of the optimal strategies.

The purpose of this article is to study the properties of the optimal investment and consumption strategies when the investor optimally allocates her wealth among risky and riskless assets and her consumption. Our model takes consideration of several features including *model uncertainty, constraints on both investment and consumption strategies, and borrowing costs*. Under both power and logarithm utility functions, we characterize the optimal portfolio-consumption strategies and the worst-case market parameters using the solutions of nonlinear ODEs, and furthermore, derive their explicit solutions in one-dimensional setting. The explicit forms further allow us to study the impacts of uncertain market parameters, portfolio-consumption constraints and different borrowing and lending rates on the optimal strategies and their time monotone properties.

*robust utility maximization*for which the investor worries about the worst-case scenario,

^{1}and as opposed to (1), we solve the following maxmin problem

As a first contribution, we show that the functions used to construct the value processes for power and logarithm utilities [see \(F_P\) and \(F_L\) in (8)] admit saddle points (Lemma 3.1). The saddle points in turn characterize locally the optimal portfolio-consumption strategies and the worst-case parameters. Since the constraint set for the portfolios-consumption strategies may not be compact, it is not even clear *ex ante* whether a saddle point exists or not. We tackle the problem by an approximation procedure using a sequence of saddle points in compact sets to construct a saddle point in the non-compact constraint set. We further characterize the optimal strategies using the solutions of nonlinear ODEs in Theorem 3.2 (for power utility) and Theorem 3.3 (for logarithm utility). We show that even with random market parameters and portfolio-consumption constraints, the optimal strategies are however deterministic in a robust utility framework. It is due to the fact that when the investor worries about the worst-case scenario, the optimal strategies are given via a deterministic saddle point and the solution of an associated nonlinear ODE. Eventually, this leads the investor to implement myopic strategies of Merton’s type to optimize her portfolios as in a complete market. Thus, there is no need for her to enforce the hedging strategy as opposed to the incomplete market situation. A similar phenomenon also occurs in [23], where the authors considered a market driven by Lévy processes with uncertain parameters but without consumption and borrowing costs.

Furthermore, in one-dimensional setting we obtain the optimal portfolio-consumption strategies and the worst-case parameters both in closed forms. Closed-form solutions seldom exist except for the standard Merton’s model with constant market parameters without portfolio-consumption constraints. We find that the explicit solutions still exist for both power and logarithm utility functions in the general framework incorporating model uncertainty, constraints on both investment and consumption strategies, and borrowing costs.

As the first example, when the uncertain market parameters stay in an interval set, we obtain a classification of the optimal portfolio strategies in terms of borrowing and lending rates as well as the uncertain market parameters. We show that (1) when the investor is optimistic about the market, meaning that her worst estimation of the stock’s return is still better than the borrowing rate, she will implement a *borrow-to-buy strategy* to borrow as much as possible to approach the optimal strategy without constraint. (2) When her worst estimation of the stock’s return is between the borrowing and lending rates, neither borrowing nor lending are attractive, and the investor will simply put all her money in the stock, i.e. performing *a full-position strategy*. (3) When the lending rate is between the best and worst estimations of the stock’s return, the investor will simply put all her money in the bank account, i.e. performing *a no-trading strategy*. (4) When the investor is pessimistic about the market, meaning that her best estimation of the stock’s return is still lower than the lending rate, she will implement a *shortsale strategy* to short sell the stock as much as possible. See Theorem 4.2 for further details.

As the second example, when the uncertain drift and volatility are correlated, we further show that the saddle point may become an interior point of the uncertain parameter set. The worst-case parameters are then given through the explicit interior saddle point, as opposed to the bang-bang type of saddle points in the existing literature. As a result, the optimal portfolio strategy is also given through the interior saddle point, albeit still in Merton’s type. See Theorem 4.4 for further details.

The explicit solutions further allows us, for the first time, to give a systematic study of the consumption plans in various situations. We argue that the consumption should stay above a minimum level for subsistence purpose, and be dominated by a reasonable upper bound for the sake of future consumption and investment. We show that the investor’s optimal consumption will degenerate to a deterministic process when she worries about the worst-case market scenario (see Theorem 4.5 for the power case and Theorem 4.7 for the logarithm case). By virtue of the closed form solutions, we are able to obtain the time monotone properties of the optimal consumption plan (see Proposition 4.6 and Theorem 4.7), and quantify the impacts of different parameters (e.g. borrowing rate, uncertain market parameters and portfolio-consumption constraints) on the optimal consumption plan (see Propositions 5.1 and 5.2).

One of the striking results is that, for the power utility case, the optimal consumption is not necessarily increasing or decreasing when the investor lifts her upper bound for consumption. This is because the investor needs to balance her current consumption and future consumption and investment when she optimizes her consumption plans. Increasing the upper bound of consumption means the investor would consume in a larger constraint set in the future, and increase the weight of her future utility, thus the investor might decrease her current consumption level. On the other hand, lifting the upper bound for consumption also means a larger constraint set from which the investor makes her current consumption decisions, and in turn her current consumption level might increase. This two contradicting factors will offset their impacts by each other, and result in a non-monotone relationship of optimal consumption with respect to the upper bound of consumption plans.

Turning to the literature, optimal portfolio-consumption problems in continuous time were first studied by Merton in 1970s (see [22] for a summary). In a sequence of papers [16, 17] and [19], the authors developed and generalized Merton’s model. In particular, [19] is one of the first arguing that the consumption must always be above a certain subsistence level, and sometimes neither borrowing nor shortsale are allowed for trading stocks, so they imposed constraints on both consumption and investment. Following this work, the optimal consumption with constraints was further studied in [6, 27], and more recently in [18, 31] in a complete market setting with constant market parameters. On the other hand, [8, 30, 32] and [33] among others studied constrained investment problems for models of varying generality.

Equal borrowing and lending rates is often assumed in the literature, and as a consequence, the wealth equation is always linear. However, it is argued in [1] that such an assumption stands in contrast with reality. Subsequently, [11] introduced the borrowing cost for the utility maximization problem, and more recently in [3], the authors took borrowing costs into account in an optimal credit investment problem.

The early development of model uncertainty went back to [28] where the authors considered a worst-case risk management problem. Robust utility maximization in mathematical finance started with [4, 13] and [26], which mainly dealt with drift uncertainty. The problem of volatility uncertainty is much harder, and has been treated via various mathematical tools. To name a few, duality method was used in [9] where the uncertainty is specified by a family of semimartingales laws. *G*-expectation was employed in [12] in a stochastic volatility model to treat uncertain correlations. In contrast, [21] studied the robust utility maximization problem under volatility uncertainty via second-order backward stochastic differential equations, and [29] considered uncertain drift and volatility using mixed strategies and derived an explicit solution in a non-traded asset setting. More recently, the results have been further generalized in [23] to include drift, volatility and jump uncertainty, which are parameterized by a set of Lévy triples. However, consumption is not considered in the above works. Two exceptions are [20] and more recently [2], where the authors worked in a similar framework to our model, but portfolio-consumption constraints are not treated in those papers.

In summary, it seems the existing literature mainly focuses on the investment-consumption models with only parts of the above features: either with portfolio constraints and market uncertainty or with consumption constraints and borrowing costs. Although many elegant mathematical results are achieved in these papers, explicit solutions and the properties of the optimal strategies rarely exist except for some special cases. In particular, consumption constraints make it difficult to obtain explicit solutions, and almost all of the explicit solutions with consumption constraints are in the framework of infinite horizon (see [6, 19] and [27]).

In contrast, our paper systematically studies constrained portfolio-consumption strategies under model uncertainty and borrowing costs in a finite horizon, and quantifies their impacts on the optimal strategies. We obtain explicit solutions and properties of the optimal strategies. Although explicit solutions are derived under one risky asset setting, our method can be applied to study the multiple risky assets setting as in [23], and similar results will still hold, albeit with more complicated situations.

The paper is organized as follows. Section 2 presents a robust utility maximization model subject to borrowing costs and portfolio-consumption constraints in a multiple risky assets setting. Section 3 solves the associated maxmin problem via a martingale argument, and characterizes the optimal portfolio-consumption strategies and the worst-case market parameters via the solutions of nonlinear ODEs. Section 4 further obtains their closed form solutions in a single risky asset setting with different uncertain parameter sets. Section 5 studies the impacts of the various model parameters on the optimal strategies and the worst-case parameters. The proof of explicit solutions is given in the “Appendix”.

## 2 The utility maximization model

### 2.1 Uncertain parameters and borrowing costs

*d*and \(d^\prime \) be two positive integers. Let

*W*be a standard \({d^\prime }\)-dimensional Brownian motion defined on a complete probability space \((\Omega ,\mathcal{F},\mathbb P)\), and \(\mathbb {F}:=\{\mathcal{F}_t\}_{t\ge 0}\) be the augmented filtration generated by

*W*. The market consists of

*d*risky assets and a riskless bank account. The price processes of the risky assets \(S_{i}\), \(1\le i\le d\), solve

*B*, the standard assumption of equal borrowing and lending rates is in contrast with empirical evidence (see [1]). In reality, there always exists a spread between borrowing and lending rates. Let

*R*and

*r*be the constant borrowing and lending rates, respectively. When

*B*is positive, the investor lends with rate

*r*. When

*B*is negative, the investor borrows with rate

*R*. It is nature to assume that \(R\ge r\). Consequently, the bank account

*B*follows

### 2.2 Portfolio and consumption constraints

*proportion of her wealth*invested in the risky assets,

*c*be her

*consumption rate proportional to her wealth*, and \(X^{x;\pi ,c,\mu ,\Sigma }\) be the wealth process with initial value

*x*, portfolio-consumption strategies \((\pi ,c)\) and parameters \((\mu ,\Sigma )\). Using (2) and (3), it follows from the self-financing condition that

One typical example of the constraint set is \(\mathbb {A}=\bigotimes _{i=1}^d[\underline{\pi }_i,\overline{\pi }_i\,]\times [\underline{c},\overline{c}\,]\), where \(\underline{\pi }_i,\overline{\pi }_i,\underline{c},\overline{c}\) are constants satisfying \(-\infty \le \underline{\pi }_i\le 0,\;1\le \overline{\pi }_i\le +\infty ,\;0\le \underline{c}\le \overline{c}\le +\infty \) for \(i=1,\ldots ,d\). Then, the portfolio constraint cube \(\bigotimes _{i=1}^d\left[ \,\underline{\pi }_i, \overline{\pi }_i\,\right] \) has the following financial interpretations: \(\left( \sum _{i=1}^d\overline{\pi }_i-1\right) \) represents the maximum proportion of wealth that the investor is allowed to borrow to invest in the risky assets; \(\left( -\sum _{i=1}^d\underline{\pi }_i\right) \) represents the largest shortsale position that the investor is allowed to take; \(\underline{\pi }_i=0\) means prohibition of shortsale the *i*th risky asset; \(\overline{\pi }_i=1\) means prohibition of borrowing to invest in the *i*th risky asset; and \(-\underline{\pi }_i=\overline{\pi }_i=+\infty \) means no portfolio constrains on the *i*th risky asset. Moreover, the consumption constraint \([\underline{c},\overline{c}]\) means that the investor should keep a minimal consumption level \(\underline{c}\) for subsistence purpose, and at the same time, her consumption is also controlled by an upper bound \(\overline{c}\) for the sake of future consumption and investment.

### 2.3 The robust utility maximization problem

*P*,

*L*represents, respectively, the power and logarithm utility functions, i.e. \(U_P^c(x)=U_P(x)=\frac{1}{p}x^{p}\) with \(p\in (-\infty ,0)\cup (0,1)\), and \(U_L^c(x)=U_L(x)=\ln x\). Herein, \(\lambda \ge 0\) represents the weight of the intertemporal consumption relative to the final bequest at maturity

*T*, and \(\rho \ge 0\) represents the discount factor.

To robustify the optimal portfolio-consumption strategy, the inner part of the above optimization problem is played by a so called mother nature who acts maliciously to minimize the expected utility by choosing the worst-case scenario, whereas the investor aims to select the best strategy that is least affected by the mother nature’s choice. For this reason, the maxmin problem (5) is also dubbed as the *robust utility maximization problem* in the literature (see [23] for example).

## 3 Nonlinear ODE characterization of the value functions

In this section, we apply a martingale argument, firstly introduced in [7] and [15], to construct a saddle point strategy \(\left\{ (\mu ^*,\Sigma ^*),(\pi ^*,c^*)\right\} \) for the expected utility \(\mathcal{J}_i(x;\pi ,c,\mu ,\Sigma )\). This will in turn solve the original maxmin problem (5).

To this end, we aim to construct an \(\mathbb {F}\)-adapted process \(J_{i,t}^{x;\pi ,c,\mu ,\Sigma }\), \(t\in [0,T]\), satisfying the following three conditions: For any \((\pi ,c)\in \mathcal {A}\) and \((\mu ,\Sigma )\in \mathcal {B}\),

*T*,

(C3) there exist \((\pi ^*,c^*)\in \mathcal {A}\) and \((\mu ^*,\Sigma ^*)\in \mathcal {B}\) such that the process \(J_i^{x;\pi ^*,c^*,\mu ^*,\Sigma ^*}\) is a martingale, \(J_i^{x;\pi ,c,\mu ^*,\Sigma ^*}\) is a supermartingale, and \(J_i^{x;\pi ^*,c^*,\mu ,\Sigma }\) is a submartingale.

### Lemma 3.1

For \(i=P,L\), the function \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) admits the following properties.

(iii) If \(p<0\) or \(i=L\), then \((\widetilde{x}_c^*(x_q))^{-1}\) is also locally bounded in \(x_q\in \mathbb {R}\).

### Proof

*Step 1* We first prove the assertion (i) when the set \(\mathbb {A}\) is compact. Indeed, for fixed \(x_q\in \mathbb {R}\), it is clear that the function \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) is concave with respect to \((x_\pi , x_c)\), and convex (accurately linear) with respect to \((x_\mu ,x_\Sigma )\). Since \(\mathbb {A}\) and \(\mathbb {B}\) are convex and compact, we may apply the minmax theorem (see Theorem B on pp. 131 in [25] or Sect. 3 in [23]), and deduce that there exists a saddle point \((\widetilde{x}^*_\pi ,\widetilde{x}^*_c;\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma )\) such that (9) holds. Moreover, the compactness of \(\mathbb {A}\) and \(\mathbb {B}\) implies that \(\widetilde{x}^*_\pi ,\widetilde{x}^*_c,\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma \) are bounded.

*Step 2* If the set \(\mathbb {A}\) is not compact, for any positive integer *n*, let \(\mathbb {A}_n:=\mathbb {A}\cap \{(x_\pi ,x_c):|(x_\pi ,x_c)|\le n\}\). It is clear that we can choose a large enough positive integer *N* such that \(\mathbb {A}_n\) is non-empty for any \(n\ge N\) and, without loss of generality, we may suppose that \(n\ge N\) below. Thanks to Step 1, we know that the function \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) has at least one saddle point \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c;\widetilde{x}^n_\mu ,\widetilde{x}^n_\Sigma )\) in \(\mathbb {A}_n\times \mathbb {B}\), and we denote \(F_i(x_q;\widetilde{x}^n_\pi ,\widetilde{x}^n_c;x^n_\mu ,x^n_\Sigma )\) by \(F^n_i\).

*n*and has a uniformly lower bound for any \(n\ge N\). To this end, note that

*n*. Furthermore, the second equality in (11) implies that, for any \(n\ge N\) and \((x^0_\pi ,x^0_c)\in \mathbb {A}_N\),

*n*and has a uniformly lower bound for any \(n\ge N\).

*Step 3*We prove that there exists a large enough positive integer

*M*such that \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c)\in \mathbb {A}_M\) for any \(n\ge M\). Indeed, we may choose a positive constant \(\epsilon \) and a positive-definite matrix \(x^0_\Sigma \) such that \((x^0_\mu ,x^0_\Sigma )\in \mathbb {B}\) and \(x_\pi ^\mathrm{T}x_\Sigma ^0 x_\pi \ge \epsilon |x_\pi |^2\) for any \(x_\pi \in \mathbb {R}^d\). Hence, as \(x_c\rightarrow 0^+\) when \(p<0\) or \(i=L\), or \(|(x_\pi ,x_c)|\rightarrow +\infty \), the compactness of \(\mathbb {B}\) implies that

*C*is a constant independent of \(x_q,x_\pi ,x_c\), \(x_\mu \) and \(x_\Sigma \), In turn, there exists a large enough positive integer \(M\ge N\) such that for any \((x_\pi ,x_c)\in \mathbb {A}\backslash \mathbb {A}_M\), or for any \((x_\pi ,x_c)\in \mathbb {A}\) with \(x_c<1/M\) when \(p<0\) or \(i=L\),

*n*(see Step 2). Thus, the last equality in (11) implies that \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c)\in \mathbb {A}_M\) for any \(n\ge M\) and, moreover, \(\widetilde{x}^n_c\ge 1/M\) when \(p<0\) or \(i=L\).

*Step 4* We prove that the function \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) has at least one saddle point \((\widetilde{x}^*_\pi ,\widetilde{x}^*_c;\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma )\) in \(\mathbb {A}\times \mathbb {B}\). Indeed, according to Step 3, \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) has at least one saddle point \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c;\widetilde{x}^n_\mu ,\widetilde{x}^n_\Sigma )\) in \(\mathbb {A}_n\times \mathbb {B}\), and all of them belong to a compact set \(\mathbb {A}_M\times \mathbb {B}\) for any \(n\ge M\). Hence, there exists a subsequence (still denoted by itself) such that \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c;\widetilde{x}^n_\mu ,\widetilde{x}^n_\Sigma ) \rightarrow (\widetilde{x}^*_\pi ,\widetilde{x}^*_c;\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma )\in \mathbb {A}_M\times \mathbb {B} \subseteq \mathbb {A}\times \mathbb {B}\). Next, we prove that \((\widetilde{x}^*_\pi ,\widetilde{x}^*_c;\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma )\) is a saddle point of \(F_i(x_q;\cdot ,\cdot ;\cdot ,\cdot )\) in \(\mathbb {A}\times \mathbb {B}\).

*Step 5* We prove that assertions (ii) and (iii) hold. Indeed, from the proof in Step 4, we know that all saddle points \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c;\widetilde{x}^n_\mu , \widetilde{x}^n_\Sigma )\) belong to a compact set \(\mathbb {A}_M\times \mathbb {B}\) for any \(n\ge M\) and \(x_q\in \mathbb {R}\). Furthermore, it follows from (12) and (13) in Step 3 that, there exists a neighborhood of \(x_q\), say \(x_q\in (a,b)\), such that the subscript *M* in \(\mathbb {A}_M\) is independent of \(x_q\) (but may depend on *a* and *b*). Thus, for \(x_q\in (a,b)\), \((\widetilde{x}_\pi ^*(x_q),\widetilde{x}_c^*(x_q),\widetilde{x}_\mu ^*(x_q),\widetilde{x}_\Sigma ^*(x_q))\in \mathbb {A}_M\times \mathbb {B}\) which means the functions \(\widetilde{x}_\pi ^*(x_q),\widetilde{x}_c^*(x_q),\widetilde{x}_\mu ^*(x_q)\) and \(\widetilde{x}_\Sigma ^*(x_q)\) are locally bounded and, moreover, (8) and (10) imply that \(G_i(x_q)\) is also locally bounded in \(x_q\in \mathbb {R}\).

From Step 3, we know that for any \(n\ge M\), \(\widetilde{x}^n_c\ge 1/M\) when \(p<0\) or \(i=L\). Since the saddle point \((\widetilde{x}^*_\pi ,\widetilde{x}^*_c;\widetilde{x}^*_\mu ,\widetilde{x}^*_\Sigma )\) is the limit of \((\widetilde{x}^n_\pi ,\widetilde{x}^n_c;\widetilde{x}^n_\mu ,\widetilde{x}^n_\Sigma )\), we deduce that \(\widetilde{x}^*_c\ge 1/M\) in the case of \(p<0\) or \(i=L\), which means that \((\widetilde{x}^*_c(x_q))^{-1}\) is locally bounded in \(x_q\in \mathbb {R}\). \(\square \)

We are now ready to state our first main result, which is about nonlinear ODE characterization of the value functions \(J_i(\cdot )\) for \(i=P,L\). Since the conclusions for power and logarithm utility functions are different, we present their results separately.

### Theorem 3.2

### Proof

\(J_{P}^{x;\pi ,c,\mu ,\Sigma }\) in (16) obviously satisfies the conditions (C1) and (C2), so it suffices to verify the martingale property (C3).

*T*]. Together with Lemma 3.1, we deduce that \(G_P(q_P(\cdot ))\) and \(\pi ^*,c^*,\mu ,\Sigma ^*\) are all bounded, and \((c^*)^{-1}\) is also bounded when \(p<0\). It follows that the stochastic exponential \(\mathcal {E}\left( p\int _0^{\cdot }(\pi ^*_s)^\mathrm{T}\Sigma _s^*dW_s\right) \) is a uniformly integrable martingale. Moreover, from (17), we deduce that

### Theorem 3.3

### Proof

\(J_L^{x;\pi ,c,\mu ,\Sigma }\) in (20) obviously satisfies the conditions (C1) and (C2), so it suffices to verify the martingale property (C3).

*T*]. Together with Lemma 3.1, we deduce that \(G_L(q_L(\cdot ))\) and \(\pi ^*,c^*,\mu ,\Sigma ^*,(c^*)^{-1}\) are all bounded. It follows that the stochastic integral \(\int _0^\cdot (\pi _s^*)^\mathrm{T}\Sigma _s^*dW_s\) is a uniformly integrable martingale. Moreover, from (21), we deduce that

### 3.1 One-dimensional case

In the rest of this paper, we focus on one-dimensional case, and derive explicit solutions for the optimal investment-consumption strategies and the worst-case parameters. Assume that \(d=d'=1\) and \(\mathbb {A}=[\underline{\pi },\overline{\pi }]\times [\underline{c},\overline{c}]\), where \(\underline{\pi },\overline{\pi },\underline{c},\overline{c}\) are constants satisfying \(-\infty \le \underline{\pi }\le 0,\;1\le \overline{\pi }\le +\infty ,\;0\le \underline{c}\le \overline{c}\le +\infty \).

*g*for \(i=L\).

*g*in \([\underline{\pi },\overline{\pi }]\times \mathbb {B}\), i.e.

*t*,

*T*], and in the literature, \(e^{q_P(t)}\) is dubbed as a (deterministic)

*opportunity process*(see [24]).

*the consumption contributing factor*\(pf_P(q_P(\cdot ),c^*(\cdot ))\), representing the change of the opportunity process due to the consumption optimization, and including two parts: current contribution \(\lambda e^{-q_P(\cdot )}(c^*(\cdot ))^{p}/p\) and future contribution \(-c^*(\cdot )\); (ii)

*the future investment contributing factor*\(pg(x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})\), representing the change of the opportunity process due to the portfolio optimization in the remaining horizon; and (iii)

*the discount rate*\(\rho \). Increasing the consumption and future investment contributing factors or decreasing the discount rate will lead to a larger opportunity process.

The current consumption contributing factor is the only one affecting the instantanous utility, which is also reflected in the expression of the expected utility (4). The future consumption contributing factor and the future investment contributing factor determine the future consumption and terminal utility through the channel of the future wealth. The player achieves the maximum utility through balancing the risky asset and riskless asset via the investment strategy, while balancing the current utility and future utility via the consumption strategy. Moreover, the definition of \(f_P(q_P(\cdot ),c^*(\cdot ))\) implies that \(\lambda e^{-q_P(\cdot )}\) is the weight of the current consumption utility relative to the future utility, which is consistent with our intuition that increasing opportunity process will lead to a larger weight of the future utility, and decrease the current consumption.

## 4 Explicit solutions of the optimal strategies and worst-case parameters

### 4.1 The worst-case parameters and the optimal portfolios

In this section, we further compute the saddle point \((x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})\) of the function \(g(\cdot ;\cdot ,\cdot )\) given in (23). It then follows from Theorems 3.2 and 3.3 that the saddle point provides an explicit solution for the worst-case parameters and the optimal portfolio of the maxmin problem (5) by letting \((\mu ^*_s,\sigma ^*_s)=(x_{\mu }^{*},{\sqrt{x_{\sigma }^*}})\) and \(\pi ^*_s=x_{\pi }^*\) for \(s\in [0,T]\). In particular, we consider two specific examples of the uncertain parameter sets \(\mathbb {B}\).

### Assumption 4.1

Assume that \(\mathbb {B}=[\,\underline{\mu },\overline{\mu }\,]\times [\,\underline{\sigma }^2,\overline{\sigma }^2\,]\), where \(\underline{\mu }, \overline{\mu },\underline{\sigma },\overline{\sigma }\) are constants satisfying \(-\infty<\underline{\mu }\le \overline{\mu }<+\infty ,\, 0\le \underline{\sigma }\le \overline{\sigma }<+\infty \) and \(\overline{\sigma }>0\).

### Theorem 4.2

Under Assumption 4.1, the worst-case parameters \((\mu ^*,\sigma ^*)\) and the optimal portfolio \(\pi ^*\) are given as follows:

### Proof

Due to its length, the proof is postponed to “Appendix A”. \(\square \)

We note that the worst-case volatility \(\sigma ^*\) attains its upper bound \(\overline{\sigma }\). This is due to the fact that the value function is monotone in volatility \({\sigma }\) in the one-dimensional setting. A larger \(\sigma \) means the investor faces more market risks, and therefore, she will have a smaller value function.

On the other hand, the worst-case drift is a bang-bang type. By the assertion (ii) about the optimal portfolio strategies, we know that \(\underline{\mu }>r\) implies \(\pi ^*>0\), i.e. the investor holds a long position of the stock. The worst-case drift is therefore its lower bound. Likewise, \(\overline{\mu }<r\) implies \(\pi ^*<0\), and therefore, the worst-case drift takes its upper bound. If \(\underline{\mu }\le r\le \overline{\mu }\), then \(\pi ^*\equiv 0\), so the estimation of the drift is irrelevant in this situation.

From Table 1, we categorize five different optimal portfolio strategies \(\pi ^*\) according to various scenarios.

(i) *Borrow-to-buy strategy* When \(\beta ^1\ge 1\), the investor will borrow \((\min \{\beta ^1,\overline{\pi }\}-1)\) units of her wealth with borrowing rate *R* to invest in the stock, and the optimal portfolio is \(\min \{\beta ^1,\overline{\pi }\}\). The reason is that in this situation, \(\underline{\mu }\ge R+(1-p)\overline{\sigma }^2\), i.e. the stock’s return even with the worst estimation of the drift is still higher than the borrowing cost. Hence, the stock’s high risk premium attracts the investor to borrow to invest as much as possible to approach the optimal strategy without constraint.

(ii) *Full-position strategy* When \(\beta ^1\le 1\le \beta ^2\), the investor will simply invest all her wealth in the stock with no additional borrowing or lending. In this case, since \(\underline{\mu }\le R+(1-p)\overline{\sigma }^2\), there exists a possibility that the stock’s return may not be good enough to compensate for the borrowing cost. As a result, the investor would prefer not to borrow. On the other hand, since \(\underline{\mu }\ge r+(1-p)\overline{\sigma }^2\), the stock’s return even in the worst scenario is still better than the return from the bank account, and accordingly, the investor would put all her money in the stock rather than in the bank account.

(iii) *Lend-and-buy strategy* When \(0\le \beta ^2\le 1\), the investor will invest \(\beta ^2\) proportion of her wealth in the stock, and the remaining proportion \((1-\beta ^2)\) in the bank account to earn the interest rate *r*. This is similar to the standard Merton’s strategy with Sharpe ratio \((\underline{\mu }-r)/\overline{\sigma }\).

(iv) *No-trading strategy* When \(\beta ^2\le 0\le \beta ^3\), the investor will put all her money in the bank account. In this case, \(\underline{\mu }\le r\le \overline{\mu }\), so there is a risk that the return from buying the stock is not as good as holding the bank account, and the investor would prefer not to invest in the stock. On the other hand, the best estimation of the drift \(\overline{\mu }\) is still better than the interest rate *r*, so implementing a shortsale strategy may incur a potential loss for the investor. This refrains her from short selling the stock.

(v) *Shortsale strategy* When \(\beta ^3\le 0\), the investor will hold short position in the stock as much as possible, which is \(\max \{\beta _3,\underline{\pi }\}\) units of her wealth in this situation. Consequently, she keeps \((1-\max \{\beta _3,\underline{\pi }\})\) units of her wealth in the bank account in order to earn the interest rate *r*.

The optimal portfolio strategies

\(\beta ^1\ge 1\) | \(\beta ^1\le 1\le \beta ^2\) | \(0\le \beta ^2\le 1\) | \(\beta ^2\le 0\le \beta ^3\) | \(\beta ^3\le 0\) | |
---|---|---|---|---|---|

\(\pi ^*_s\) | \(\min \{\beta ^1,\overline{\pi }\}\) | 1 | \(\beta ^2\) | 0 | \(\max \{\beta ^3,\underline{\pi }\}\) |

In the existing literature, the worst-case parameters are usually bang-bang type, i.e., they take values at the boundaries of the uncertain parameter set. Next, we give an example where the worst-case drift and volatility are an interior point in the uncertain parameter set. In particular, the worst-case volatility may not be its upper bound anymore.

### Assumption 4.3

Assume that \(R=r,\overline{\pi }=+\infty ,\underline{\pi }=-\infty \) and \(\mathbb {B}=\{(\mu ,\sigma ):\mu =\underline{\mu }+\alpha ,\sigma =\underline{\sigma }^2+k\alpha ^q,\alpha \in [\,0,\overline{\alpha }\,]\,\}\), where \(\underline{\mu },\underline{\sigma },k,q,\overline{\alpha }\) are constants satisfying \(\underline{\sigma }\ge 0, k>0,0<q<1,\overline{\alpha }\ge 0\).

The set \(\mathbb {B}\) indicates that the ambiguities about drift and volatility are correlated. A higher return is associated with a larger risk. The limiting case \(q=1\) means that the relationship between the ambiguity about drift and the ambiguity about the volatility square is linear, which is just Example 2.4 in [10]. The other spectrum \(q=0\) means no ambiguity about volatility. Finally, \(0<q<1\) means that the relationship between the ambiguity about drift and the ambiguity about the volatility square is sub-linear.

### Theorem 4.4

Under Assumption 4.3, the worst-case parameters \((\mu ^*,\sigma ^* )\) and the optimal portfolio \(\pi ^*\) are given as follows:

### Proof

*g*, and the conclusion follows from Theorems 3.2 and 3.3. \(\square \)

If \(\underline{\mu }-r\le -\overline{\alpha }\), then the investor will short sell her stock, which is similar to *Shortsale strategy* in Theorem 4.2. Moreover, \(\pi ^*<0\) implies that the worst-case drift \(\mu ^*\) and volatility \(\sigma ^*\) attain their respective upper bounds \(\underline{\mu }+\overline{\alpha }\) and \(\sqrt{\underline{\sigma }^2+k(\overline{\alpha })^q}\).

If \(-\overline{\alpha }<\underline{\mu }-r{< 0}\), then the lower bound of the drift \(\underline{\mu }\le r\) and the upper bound \(\underline{\mu }+\overline{\alpha }>r\). Similar to *No-trading strategy* in Theorem 4.2, the investor may suffer losses if she buys or short sells the stocks, so she will simply invest all her money in the bank account. Moreover, \(\pi ^*=0\) implies that the estimation of the drift and volatility is irrelevant in this situation and, without loss of generality, we let \((\mu ^*,\sigma ^*)=\left( \,r,\sqrt{\underline{\sigma }^2+k(r-\underline{\mu })^q}\,\right) \).

If \(\underline{\mu }-r>0\), then the investor will invest in the stock according to the optimal proportion \(\pi ^*=(\mu ^*-r)/((1-p)(\sigma ^*)^2)>0\). If there is no ambiguity about volatility, the worst-case drift is its lower bound \(\underline{\mu }\) and \(\alpha ^*=0\). Since the correlation between the uncertain drift and uncertain volatility is positive, the worst-case parameter \(\alpha ^*=\alpha _0\), which is an interior point of the interval \([\,0,\overline{\alpha }\,]\) if \(0<\underline{\mu }-r <\widehat{\alpha }\). In particular, the worst-case volatility may not be its upper bound anymore. This is in contrast to Theorem 4.2, where the worst-case parameters take values at the boundaries of the uncertain parameter set.

### 4.2 The Optimal consumption under power utility

In this section, we compute the explicit solutions to ODEs (15) and (19), which in turn allows us to construct the optimal consumption of the maxmin problem (5) [cf. (27) or (29)]. Note that if \(\lambda =0\) in (4) and \(\overline{c}=\underline{c}\), the consumption does not play a role and the optimal consumption strategy is simply \(c^*_t=\overline{c}=\underline{c}\). Hence, we focus on the case \(\lambda >0\) and \(\overline{c}>\underline{c}\) in the rest of the paper. We first present the result for power utility.

### Theorem 4.5

Let \(T>0\) be a large enough number. For the power utility case, the optimal consumption \(c^*_t=c^*(t)\), \(t\in [0,T]\), is a deterministic process, where \(c^*(t)=\widetilde{x}_{c,P}^*(q_P(t))\) with \(\widetilde{x}_{c,P}^*(\cdot )\) given in (27) and \(q_P(\cdot )\) given in Table 8. Moreover, the optimal consumption \(c^*_t\) is summarized in Table 2.^{2}

*K*in the table corresponds to the future investment contributing factor in (15), and has the explicit form

The optimal consumption in the case of \(\underline{c}>0\)

### Proof

Due to its length, the proof is postponed to “Appendix B”. \(\square \)

Table 2 lists all the possible consumption patterns under different parameters. For example, \(\underline{c}I_0^{123}+\widehat{c}(t)I^{12}_{123}+\overline{c}I^4_{12}\) in the first row and the first column (left-top corner) is the optimal consumption when the market parameters satisfy \(\underline{c}<\overline{c}<\lambda ^{1/(1-p)}\) and \(\rho -pK\in (-\infty ,(1-p)\underline{c}\,)\). More specifically, in the time interval \([0,T_{123}]\), the investor will consume at the minimum rate \(\underline{c}\). Then the investor will consume at the optimal rate \(\widehat{c}(t)=\lambda ^{1/(1-p)}{\exp (q_P(t)/(p-1))}\) in the time interval \([T_{123},T_{12}]\), since in this case \(\underline{c}\le \widehat{c}(t)\le \overline{c}\). Finally, in the remaining time interval \([T_{12},T]\), the investor will consume at the maximum rate \(\overline{c}\).

In contrast, in the right-bottom corner, we obtain a reversed consumption pattern when \(\lambda ^{1/(1-p)}<\underline{c}<\overline{c}\) and \(\rho -pK\in ((1-p)\overline{c},+\infty )\). That is, the consumption will be decreasing from the maximum rate \(\overline{c}\) in \([0,T_{321}]\), to \(\widehat{c}(t)\) in \([T_{321},T_{32}]\), and finally to the minimal rate \(\underline{c}\) in \([T_{32},T]\).

In the following, we give some intuitive explanations of different consumption patterns. From the expression of \(f_P\) and \(c^*(t)\), we know that the optimal consumption \(c^*_t=c^*(t)\) achieves the maximum of the concave function \(f_P(q_P(t),\cdot )\) in the interval \([\underline{c},\overline{c}]\). Moreover, note that \(\widehat{c}(t)=\lambda ^{1/(1-p)}\exp (q_P(t)/(p-1))\) as in (27) is the maximum point of \(f_P(q_P(t),\cdot )\) on \(\mathbb {R}_+\). Hence, \(c^*_t=\widehat{c}(t)\) if \(\underline{c}<\widehat{c}(t)<\overline{c}\). Otherwise, \(c^*_t\) will be either \(\underline{c}\) or \(\overline{c}\).

From the proof of Proposition 4.6 below, we know \(q_P(t)\) is monotone in time *t*, so is \(\widehat{c}(t)\). As a result, whether \(\widehat{c}(t)\) stays in \([\underline{c},\overline{c}]\) or not only depends on its values at the two end points \(\widehat{c}(T)\) and \(\widehat{c}(0)\), and their relationship with \(\underline{c}\) and \(\overline{c}\).

In fact, it follows from \(q_P(T)=0\) that \(\widehat{c}(T)=1/\lambda ^{1-p}\). By the continuity of \(\widehat{c}(t)\), when *t* approaches maturity *T*, \(c^*(t)\) will reach its upper bound \(\overline{c}\) if \(\underline{c}<\overline{c}<\lambda ^{1/(1-p)}\); \(c^*(t)\) will be precisely \(\widehat{c}(t)\) if \(\underline{c}<\lambda ^{1/(1-p)}<\overline{c}\); \(c^*(t)\) will reach its lower bound \(\underline{c}\) if \(\lambda ^{1/(1-p)}<\underline{c}<\overline{c}\). The above three situations thus determine the classification of the rows in Table 2.

On the other hand, we have the following asymptotic results for \(\lim \limits _{T\rightarrow +\infty }\widehat{c}(0)\) in Table 3 (see also “Appendix B”, in particular (49)–(52)). By the continuity of \(\widehat{c}(t)\), when *T* is large enough and *t* is near initial time 0, \(c^*(t)=\underline{c}\) if \(\rho -pK\in (-\infty ,(1-p)\underline{c}\,)\); \(c^*(t)=\widehat{c}\) if \(\rho -pK\in ((1-p)\underline{c},(1-p)\overline{c}\,)\); and \(c^*(t)=\overline{c}\) if \(\rho -pK\in ((1-p)\overline{c},+\infty )\). Consequently, the above three situations divide the columns in Table 2.

The limit of \(\widehat{c}(0)\) when \(T\rightarrow +\infty \)

\(\rho -pK\) | \((-\infty ,(1-p)\underline{c}\,)\) | \(\{(1-p)\underline{c}\,\}\) | \(((1-p)\underline{c},(1-p)\overline{c}\,)\) | \(\{(1-p)\overline{c}\,\}\) | \(((1-p)\overline{c},+\infty )\) |
---|---|---|---|---|---|

\(\lim \limits _{T\rightarrow +\infty }\,\widehat{c}(0)\) | \(<\underline{c}\) | \(=\underline{c}\) | \(\in (\,\underline{c},\overline{c}\,)\) | \(=\overline{c}\) | \(>\overline{c}\) |

### Proposition 4.6

*t*, as specified in Table 4. The symbols \(\nearrow ,\searrow \) and \(\perp \) represent nondecreasing, nonincreasing and independent of time

*t*, respectively.

The optimal consumption in time

\(\underline{c}<\overline{c}<\lambda ^{1/(1-p)}\) | \(\nearrow \) | |||||

\(\underline{c}\le \lambda ^{1/(1-p)}\le \overline{c}\) | \(\rho -pK<(1-p)\lambda ^{1/(1-p)}\) | \(\rho - pK=(1-p)\lambda ^{1/(1-p)}\) | \(\rho -pK>(1-p)\lambda ^{1/(1-p)}\) | |||

\(\nearrow \) | \(\perp \) | \(\searrow \) | ||||

\(\lambda ^{1/(1-p)}<\underline{c}<\overline{c}\) | \(\searrow \) |

### Proof

We have shown that \(q_P(t)\) is either nonincreasing or nondecreasing for \(t\in [0,T]\). Thus, it suffices to consider the sign of \(q_P^{\prime }(T)\).

*K*is given in (31). However, Theorem 4.5 implies that \(c^*(t)\equiv \overline{c}\) if \(\rho -pK\ge (1-p)\overline{c}\) in this case, so we only need to consider the situation \(\rho -pK< (1-p)\overline{c}\) for the monotone property of \(c^*(t)\). Together with \(\overline{c}<\lambda ^{1/(1-p)}\), we further obtain that

The other two cases \(\underline{c}\le \lambda ^{1/(1-p)}\le \overline{c}\) and \(\lambda ^{1/(1-p)}<\underline{c}<\overline{c}\) can be treated in a similar way, so their proofs are omitted. \(\square \)

### 4.3 The optimal consumption under logarithm utility

### Theorem 4.7

*T*is a large enough number. For the logarithm utility case, the optimal consumption \(c_t^*=\widetilde{x}_{c,L}^* (q_L(t)),\ t\in [\,0,T\,]\), is a deterministic process, with \(\widetilde{x}_{c,L}^*(\cdot )\) and \(q_L(\cdot )\) given respectively in (29) and (34). Moreover, the optimal consumption \(c^*_t\) is summarized in Table 5.

^{3}

The optimal consumption in the case of \(\underline{c}\ge 0\)

*t*for \(\rho \ge \lambda \), and nondecreasing with respect to

*t*for \(\rho \le \lambda \).

### Proof

First, it is clear that the solution of ODE (19) takes the form (34). From (29), we know that \(\widetilde{x}^*_{c,L}(x_q)\) is nonincreasing with respect to \(x_q\). Moreover, the expression (34) implies that \(q_L(\cdot )\) is nondecreasing with respect to *t* when \(\rho \ge \lambda \), and nonincreasing with respect to *t* when \(\rho \le \lambda \). Then, the monotonicity of \(c^*_t=\widetilde{x}^*_{c,L}(q_L(t))\) follows immediately.

*T*is large enough. Moreover, since \(q_L(\cdot )\) is continuous and strictly increasing with respect to

*t*, there exists unique \((T_1,T_2)\) such that

## 5 The impacts of model uncertainty, portfolio-consumption constraints and borrowing costs

In this section, we investigate the impacts of model uncertainty, portfolio-consumption constraints and borrowing costs on the worst-case parameters \((\mu ^*,\sigma ^*)\) and the optimal portfolio-consumption strategies \((\pi ^*,c^*)\).

### Proposition 5.1

*R*, the constraint set \([\underline{\pi },\overline{\pi }]\times [\underline{c},\overline{c}]\), and the uncertain parameter set \([\underline{\mu },\overline{\mu }]\times [\underline{\sigma }^2,\overline{\sigma }^2]\), as specified in Table 6. The symbols \(\searrow \), \(\nearrow \), \(\perp \) and NM represent nonincreasing, nondecreasing, independent and non-monotone of the corresponding variable. For example, the bottom row and the first column (left-bottom corner) means \(c_s^*\) is nondecreasing in the borrowing rate

*R*.

The comparative statistics

| \(\underline{\pi }\) | \(\overline{\pi }\) | \(\underline{c}\) | \(\overline{c}\) | \(\underline{\mu }\) | \(\overline{\mu }\) | \(\underline{\sigma }\) | \(\overline{\sigma }\) | |
---|---|---|---|---|---|---|---|---|---|

\(\mu ^*_s\) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\perp \) |

\(\sigma ^*_s\) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\nearrow \) |

\(\pi ^*_s\) | \(\searrow \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\perp \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\searrow \) |

\(c^*_s\) | \(\nearrow \) | \(\nearrow \) | \(\searrow \) | \(\nearrow \) | NM | \(\searrow \) | \(\nearrow \) | \(\perp \) | \(\nearrow \) |

Before proceeding to the proof, we provide some intuitive explanations for the above results. The impacts of different parameters on the worst-case parameters \((\mu _s^*,\sigma ^*_s)\) and the optimal portfolio \(\pi _s^*\) are obvious from the results in Theorem 4.2. So we only discuss about their impacts on the optimal consumption \(c^*_s\).

By the expression (27) and \(c^*_s=\widetilde{x}_{c,P}^*(q_P(s))\), the parameters (*R*, \(\underline{\pi }\), \(\overline{\pi }\), \(\underline{\mu }\), \(\overline{\mu }\), \(\underline{\sigma }\), \(\overline{\sigma }\)) will effect the optimal consumption through the channel of the opportunity process \(e^{q_P(s)}\), which is the investor’s extra utilities obtained by optimizing over all the admissible portfolio-consumption strategies (least affected by model uncertainty) in the remaining horizon [*s*, *T*]. A closer look at the ODE (15) for \(q_P(s)\) tells us that those parameters will only enter into the future investment contributing factor \(g(x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})\) in (23). Increasing the borrowing cost *R* will make the future investment contributing factor \(g(x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})\) smaller, so the opportunity process will also become smaller, i.e. the investor will obtain less utilities in the remaining horizon. In turn, her current optimal consumption will go up. Similarly, enlarging the uncertainty parameters interval \([\underline{\mu },\overline{\mu }]\times [\underline{\sigma }^2,\overline{\sigma }^2]\) or shrinking the portfolio constraint interval \([\underline{\pi },\overline{\pi }]\) will also make the future investment contributing factor \(g(x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})\) smaller, and therefore, the current optimal consumption will arise.

The more striking result is probably the impact of the consumption constraint interval \([\underline{c},\overline{c}]\) on the optimal consumption \(c^*_s\). Note that the constraint interval will only effect the consumption contributing factor \(f_P(q_P(s),c^*(s))\) in (22), with \(c^*(s)=\widetilde{x}_{c,P}^*(q_P(s))\). The smaller interval will lead to a smaller consumption contributing factor \(f_P(q_P(s),c^*(s))\) as indicated in (24). In turn, the investor will obtain less utilities in the remaining horizon [*s*, *T*]. This might suggest the current optimal consumption would increase. However, it is not always the case, as there is less chance for the unconstrained optimal consumption \(\widehat{c}(s)\) to stay in the shrinking interval \([\underline{c},\overline{c}]\). If \(\widehat{c}(s)\) reaches the lower bound \(\underline{c}\), then the optimal consumption will further arise as \(\underline{c}\) increases. On the other hand, if \(\widehat{c}(s)\) reaches the upper bound \(\overline{c}\), then the optimal consumption will go down for \(\overline{c}\) becomes smaller, thus offsets the previous increasing impact on the optimal consumption when \(\overline{c}\) is decreasing. *This means the optimal consumption is non-monotone in its upper bound* \(\overline{c}\).

### Proof

*The monotone property of*\(\mu _s^*\) According to Theorem 4.2, the worst-case drift can be rewritten as

(ii) *The monotone property of* \(\sigma _s^*\) The conclusion simply follows from the expression of the worst-case volatility \(\sigma _s^*=\overline{\sigma }\) for \(s\in [0,T]\) in Theorem 4.2.

(iii) *The monotone property of* \(\pi _s^*\) First, the expressions of \(\beta ^1,\beta ^2,\beta ^3\) in Theorem 4.2 imply that they are all nondecreasing in \(\underline{\mu },\overline{\mu }\), nonincreasing in \(R,\overline{\sigma }\) and independent of \(\underline{\sigma },\underline{c},\overline{c}\), so is the optimal portfolio \(\pi _s^*\), as \(\pi _s^*\) is nondecreasing with respect to \(\beta ^1,\beta ^2,\beta ^3\) (cf. Fig. 1).

(iv) *The monotone property of* \(c_s^*\) We first study the impacts of different parameters on the solution \(q_P(t)\) of ODE (15). Note that (*R*, \(\underline{\pi }\), \(\overline{\pi }\), \(\underline{\mu }\), \(\overline{\mu }\), \(\underline{\sigma }\), \(\overline{\sigma }\)) will effect \(q_P(t)\) only through \(g(x_{\pi }^*;x_{\mu }^*,{x_\sigma ^*})=K\), where *K* is given in (31).

It is obvious from the expression (31) that *K* is nonincreasing in *R*. Moreover, since *K* is the maximum value of \(g(x_{\pi };x_{\mu }^*,{x_\sigma ^*})\) over \(x_{\pi }\in [\underline{\pi },\overline{\pi }]\), *K* is nonincreasing in \(\underline{\pi }\) and nondecreasing in \(\overline{\pi }\). On the other hand, *K* is also the minimum value of \(g(x_{\pi }^*;x_{\mu },{x_\sigma })\) over \((x_{\mu },{x_\sigma })\in [\underline{\mu },\overline{\mu }]\times [\underline{\sigma }^2,\overline{\sigma }^2]\). Therefore, *K* is nondecreasing in \(\underline{\mu },\underline{\sigma }\) and nonincreasing in \(\overline{\mu },\overline{\sigma }\). However, the expression of *K* further implies that *K* is independent of \(\underline{\sigma }\).

It then follows from the comparison theorem for ODE (15) that its solution \(q_P(s)\) is nonincreasing in *R*, \(\underline{\pi }\), \(\overline{\mu }\), \(\overline{\sigma }\), nondecreasing in \(\overline{\pi }\) and \(\underline{\mu }\), and independent of \(\underline{\sigma }\). The conclusion about the optimal consumption \(c^*_s\) then follows from Theorem 3.2 together with the expression (27).

In terms of the impacts of \(\underline{c}\) and \(\overline{c}\) on \(c_s^*\), since \(f_P(q_P(s),c^*(s))\), with \(c^*(s)=\widetilde{x}_{c,P}^*(q_P(s))\), is the maximum value of \(f_P(q_P(s),x_c)\) over \(x_c\in [\underline{c},\overline{c}]\), it is nonincreasing in \(\underline{c}\) and nondecreasing in \(\overline{c}\). Following the comparison theorem for ODE (15) and the expression (27) once again, we conclude \(\widehat{c}(s)\) is nondecreasing in \(\underline{c}\) and nonincreasing in \(\overline{c}\).

*t*is close to

*T*, then \(c_1^*(t)=\overline{c}_1>\overline{c}_2=c_2^*(t)\). On the other hand, when

*T*is large enough and

*t*is close to zero, we have

Finally, we present the result for the logarithm utility case. Its proof is omitted as it is similar to the proof for the power utility case.

### Proposition 5.2

*R*, the constraint set \([\underline{\pi },\overline{\pi }]\times [\underline{c},\overline{c}]\), and the uncertain parameter set \([\underline{\mu },\overline{\mu }]\times [\underline{\sigma }^2,\overline{\sigma }^2]\), as showed in Table 7. The symbols \(\searrow \), \(\nearrow \), \(\perp \) represent nonincreasing, nondecreasing and independent of the corresponding variable.

The comparative statistics for log

| \(\underline{\pi }\) | \(\overline{\pi }\) | \(\underline{c}\) | \(\overline{c}\) | \(\underline{\mu }\) | \(\overline{\mu }\) | \(\underline{\sigma }\) | \(\overline{\sigma }\) | |
---|---|---|---|---|---|---|---|---|---|

\(\mu ^*\) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\perp \) |

\(\sigma ^*\) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\nearrow \) |

\(\pi ^*\) | \(\searrow \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\perp \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\searrow \) |

\(c^*\) | \(\perp \) | \(\perp \) | \(\perp \) | \(\nearrow \) | \(\nearrow \) | \(\perp \) | \(\perp \) | \(\perp \) | \(\perp \) |

## Footnotes

- 1.
- 2.
In the case of \(\underline{c}=0\), the results are similar to those in Table 2 except that \(\underline{c}I_0^{123}+\widehat{c}(t)I^{12}_{123}+\overline{c}I^4_{12}\) and \(\underline{c}I_0^{23}+\widehat{c}(t)I^4_{23}\) are replaced by \(\widehat{c}(t)I^{12}_0+\overline{c}I^4_{12}\) and \(\widehat{c}(t)\), respectively. Note that when \(\underline{c}=0\), since \(\lambda >0\), the last two rows about the optimal consumption are then irrelevant.

- 3.
Note that when \(\underline{c}=0\), since \(\lambda >0\) and \(\rho \ge 0\), the first row and the first column about the optimal consumption are then irrelevant.

- 4.
In the case of \(\underline{c}=0\), the results are similar to those in Table 8 except that \(q_{123}\) and \(q_{23}\) are replaced by \(q_{12}\) and \(q_{2}\), respectively. Note that in this case, the forth and fifth rows in Table 8 and \(q_{123}, q_{23}, q_{32}, q_{321}, q^3, T_{123}, T_{23}, T_{32}, T_{321}\) are irrelevant.

## Notes

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