Optimal Consumption and Investment Choice with Bounded Memory and Recursive Preferences in a Multi-Asset Setting

1. Introduction

The problem of optimal consumption and investment has long been one of the central topics in mathematical finance and stochastic control theory. Since the pioneering work of Merton, a vast literature has investigated how rational investors should dynamically allocate wealth between consumption and financial assets in order to maximise their lifetime utility. In the classical framework, investors continuously rebalance their portfolios between a risk-free asset and one or several risky assets while simultaneously determining an optimal consumption stream. Such models have provided fundamental insights into portfolio allocation, intertemporal decision-making, and equilibrium asset pricing. Nevertheless, despite their mathematical tractability and economic relevance, classical expected utility models remain limited in their ability to capture realistic investor behaviour and preference dynamics.

One of the major shortcomings of the standard expected utility framework lies in the fact that it links risk aversion and intertemporal substitution through a single parameter. In particular, within the constant relative risk aversion (CRRA) specification, the coefficient of relative risk aversion is the reciprocal of the elasticity of intertemporal substitution (EIS). Economically, this implies that an investor’s willingness to smooth consumption across uncertain states of nature is inseparable from the willingness to substitute consumption over time. However, empirical studies suggest that these two aspects of preferences are fundamentally distinct and should not necessarily move together. To overcome this restriction, Epstein and Zin (1989) introduced the recursive utility framework now commonly referred to as Epstein-Zin utility. This utility specification allows risk aversion and intertemporal substitution to be disentangled, thereby offering greater flexibility for modelling long-term investment decisions and asset pricing dynamics. Since then, Epstein-Zin preferences have become an important tool in dynamic portfolio optimisation; see, for example, Kraft et al. (2017); Xing (2017); Herdegen et al. (2023); Feng et al. (2025), and the references therein.

Alongside the development of recursive utility models, another important direction in modern financial modelling concerns the incorporation of behavioural effects, learning mechanisms, and memory dependence into portfolio choice problems. In practice, investors rarely make decisions solely on the basis of current market information. Instead, past performance, historical gains and losses, and previous investment experiences often influence future decisions. Empirical evidence from behavioural finance suggests that investors exhibit path-dependent behaviours such as momentum trading, trend-following, overreaction, and contrarian investment strategies. These observations challenge the classical Markovian paradigm in which only the current state variable matters for decision-making.

To account for such features, several recent studies have introduced delay equations, learning effects, and bounded-memory structures into stochastic control and portfolio optimisation problems. For instance, Chang et al. (2011) considered a stochastic portfolio optimisation model with bounded memory, where investment decisions depend on delayed wealth dynamics. More recently, Zhou and Yongjun (2024) investigated a consumption and portfolio optimisation problem with information learning under stochastic interest rates, while Wang et al. (2025) studied utility maximisation problems incorporating life insurance purchase, information learning and behavioural effects. See also Zhu and Zhang (2025) for the use of bounded memory in the context of robust Nash equilibrium for defined contribution pension games. These contributions and the references therein demonstrate that delayed information and learning mechanisms can substantially alter optimal investment behaviour and generate economically meaningful patterns consistent with observed investor behaviour. However, to the best of our knowledge, no research has ever considered the bounded memory effect when investors have recursive preferences.

Motivated by these developments, the present paper studies an optimal consumption and investment problem in an incomplete financial market where investors exhibit bounded memory and information-learning behaviour while evaluating outcomes through Epstein-Zin recursive preferences. More precisely, we consider a market consisting of a money market account and several risky assets, where the risky assets dynamics are influenced not only by standard market factors but also by delayed information variables linked to past wealth performance. In contrast to standard delay models based solely on historical wealth levels, our framework introduces an integrated delayed wealth variable defined through exponentially weighted gains and losses relative to a benchmark wealth level. This formulation provides a more realistic representation of bounded rationality and behavioural learning, since investors are often more sensitive to relative performance and recent experiences than to absolute wealth levels accumulated far in the past.

The model incorporates two distinct channels of historical information. The first is an integrated delay information variable that captures the exponentially weighted deviation of past wealth from the initial benchmark over a fixed memory horizon. This component reflects gradual learning and bounded-memory effects, giving greater importance to more recent observations while preserving the influence of historical performance. The second is a pointwise delayed wealth variable, corresponding to a lagged observation of wealth at a specific past time. Economically, this variable can be interpreted as a reference point or a delayed signal used by investors to update beliefs and revise strategies. Together, these two components provide a richer and more flexible representation of information learning and behavioural feedback effects in portfolio selection.

From a mathematical perspective, incorporating memory effects into dynamic portfolio optimisation increases the complexity of the problem. The resulting wealth dynamics become non-Markovian and involve delay terms that interact with recursive utility preferences. To address these difficulties, we formulate the optimisation problem through a system of forward-backward stochastic differential equations (FBSDEs). By introducing a suitable ansatz for the value process, we derive explicit representations for the optimal consumption and investment strategies, as well as the associated value function. The explicit solvability of the model enables a detailed analysis of how preference parameters and memory effects jointly shape optimal decision-making.

The main contributions of this paper can be summarised as follows.

1.

We formulate and solve an optimal consumption and investment problem with bounded memory under Epstein-Zin recursive utility. The latter framework naturally generalises the classical CRRA utility case as a particular specification of the preference parameters; see Remark 1.

2.

We introduce a multi-asset incomplete-market model in which each risky asset is influenced by two distinct information-learning mechanisms: an integrated delayed wealth variable reflecting exponentially weighted past performance and a pointwise delayed wealth state representing lagged wealth information. Compared with existing bounded-memory models, our formulation focuses on exponentially weighted gains and losses relative to a benchmark wealth level rather than on absolute historical wealth or on pure average gains and losses.

3.

Using an FBSDE approach together with an explicit ansatz for the value function, we derive closed-form expressions for the optimal consumption and portfolio strategies, as well as the associated value function; see Theorem 1. These explicit formulas make it possible to perform a detailed sensitivity analysis with respect to the memory horizon, learning parameters, risk aversion coefficient, and elasticity of intertemporal substitution.

4.

We provide an economic interpretation of the resulting optimal strategies and show how behavioural learning effects can generate different investment patterns across risky assets. In particular, the analysis highlights the interaction between delayed information, recursive preferences, and dynamic portfolio allocation.

The remainder of the paper is organised as follows. Section 2 introduces the financial market model, the bounded-memory structure, and the associated wealth dynamics. Section 3 formulates the Epstein-Zin optimisation problem and presents the main theoretical results, including the explicit solution of the associated forward-backward stochastic differential equation system. Section 4 investigates the effects of the model parameters on the optimal investment and consumption policies through a sensitivity analysis. Finally, the paper concludes with a summary of the main findings and a discussion of possible directions for future research.

2. Model and Problem Formulation

2.1. Probability Setting and Wealth Process of the Investors

We consider a filtered probability space $(\mathsf{\Omega },\mathcal{F},{\left({\mathcal{F}}_t\right)}_{0\le t\le T},\mathbb{P})$ generated by a n-dimensional Brownian motion $W=(W^1,\dots ,W^n)$. The filtration ${\left({\mathcal{F}}_t\right)}_{0\le t\le T}$ is assumed to satisfy the usual conditions of completeness and right-continuity, so that we can take càdlàg versions for semi-martingales. We define some known spaces of stochastic processes.

$\left(i\right)$

Let $\mathcal{C}$ be the set of non-negative progressively measurable processes on $[0,T]\times \mathsf{\Omega }$.

$\left(ii\right)$

Let ${\mathcal{H}}^2\left(\mathbb{R}\right)$ denote the space of càdlàg adapted $\mathbb{R}$-valued processes ${\left(Y_t\right)}_{0\le t\le T}$ such that $\mathbb{E}\left[{\int }_0^T\right|Y_t{|}^2\mathrm{d}t]<\infty$.

$\left(iii\right)$

Let ${\mathbb{H}}^2\left({\mathbb{R}}^n\right)$ denote the space of predictable ${\mathbb{R}}^n$-valued processes ${\left(Z_t\right)}_{0\le t\le T}$ such that $\mathbb{E}[{\int }_0^T\parallel Z_t{\parallel }^2\mathrm{d}t]<\infty$.

We consider a dynamic financial environment with k risky assets (think of stock, bond, mutual fund) and a money market account. We assume that investors in such a financial market consider historical investment data when choosing their current investment strategies. In particular, they take into account the performance of past strategies. Let h be the information learning horizon, $X_t$ be the level of wealth of the investor at the current time t, and $X_0$ its initial wealth. To accurately evaluate the profits and losses of historical performance, we consider a refinement of the information variables proposed by Chang et al. (2011); Zhou and Yongjun (2024), as follows

$$\begin{array}{cc}\hfill X_t^{int}& :={\int }_{-h}^0{e}^{\alpha s}\left(X_{t+s}-X_0\right)\mathrm{d}s\hfill \end{array}$$

(1)

Here, $\alpha \ge 0$ and $h>0$ represent the average parameter and the delay parameter, respectively, and $X_t^{int}$ represents the integrated delayed information of the investor’s wealth process on the past horizon $[t-h,t]$. $X_t^{int}$ captures the past performance relative to the benchmark $X_0>0$ while ensuring that only gains and losses, not arbitrary wealth levels (as in Chang et al. (2011)), drive the dynamics. The exponential weighting introduces a realistic decay in memory (contrary to Zhou and Yongjun (2024)), giving greater importance to recent results while still retaining a smooth influence of the past. We assume that the investor is endowed with an initial wealth $X_0$ at time $-h$ and does not start investing until time 0; that is, $X_t=X_0$ for all $t\in [-h,0]$. In that case, $X_0^{int}=0$.

Using Equation (1), the dynamics of $X_t^{int}$ with respect to t is derived as follows

$$\begin{array}{cc}\hfill \mathrm{d}{X}_t^{int}& ={\displaystyle \frac{\partial }{\partial t}}\left({\int }_{-h}^0{e}^{\alpha s}\left(X_{t+s}-X_0\right)\mathrm{d}s\right)\mathrm{d}t\hfill \\ \hfill & ={\displaystyle \frac{\partial }{\partial t}}\left({\int }_{t-h}^t{e}^{\alpha (\tau -t)}\left(X_{\tau }-X_0\right)\mathrm{d}\tau \right)\mathrm{d}t\mathrm{for}\tau =t+s\hfill \\ \hfill & ={\displaystyle \frac{\partial }{\partial t}}\left(e^{-\alpha t}{\int }_{t-h}^t{e}^{\alpha \tau }\left(X_{\tau }-X_0\right)\mathrm{d}\tau \right)\mathrm{d}t\hfill \\ \hfill & =\left(-\alpha X_t^{int}+e^{-\alpha t}\left(e^{\alpha t}\left(X_t-X_0\right)-e^{\alpha (t-h)}\left(X_{t-h}-X_0\right)\right)\right)\mathrm{d}t\hfill \\ \hfill & =\left(-\alpha X_t^{int}+X_t-e^{-\alpha h}{X}_{t-h}-\left(1-e^{-\alpha h}\right)X_0\right)\mathrm{d}t.\hfill \end{array}$$

(2)

Following Chang et al. (2011), we assume that the investment dynamics of the stock exhibits an information learning effect, influenced not only by the integrated delayed information variable $X_t^{int}$ over the time period $[t-h,t]$, but also by the pointwise delayed information state

$$\begin{array}{c}\hfill X_t^{point}:=X_{t-h}\mathrm{at}\mathrm{time}t-h.\end{array}$$

(3)

Moreover, as in Zhou and Yongjun (2024), we assume that investors choose, from the money market account, a consumption rate $c_t$ at time $t\in [0,T]$. Hence, the investment dynamics for the money market account and the k-vector of risky assets are governed by the stochastic differential equations

$$\begin{array}{cc}\hfill \mathrm{d}{\pi }_t^M& =\left(r{\pi }_t^M-c_t\right)\mathrm{d}t,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{d}{\pi }_t^R& =\mathrm{diag}\left({\pi }_t^R\right)\left(\left(r+\mu \right)\mathrm{d}t+\sigma \mathrm{d}{W}_t\right)+\left(X_t^{int}{\beta }_1+X_t^{point}{\beta }_2\right)\mathrm{d}t,\hfill \end{array}$$

(5)

where ${\pi }_t^M\in \mathbb{R}$ and ${\pi }_{t,i}^R\in \mathbb{R},i=1,\dots ,k$, denote the amounts invested in the money market account and the i-th risky asset, respectively, $r\ge 0$ denote the interest rate earned by the money market account, $\mu \in {\mathbb{R}}^k$ and $\sigma \in {\mathbb{R}}^{k\times n}$ denote the risk premium vector and the volatility matrix of the risky assets, respectively, and ${\beta }_1\in {\mathbb{R}}^k$ and ${\beta }_2\in {\mathbb{R}}^k$ denote the information learning parameters for $X_t^{int}$ and $X_t^{point}$, respectively, with the components ${\beta }_{1,i}$ and ${\beta }_{2,i},i=1,\dots ,k$, associated with the i-th risky asset. Although both $X_t^{int}$ and $X_t^{point}$ reflect aspects of the historical dynamics of wealth, they serve complementary behavioural roles in the model. More precisely, $X_t^{int}$ describes the average deviation of wealth from the initial benchmark over a recent period, reflecting bounded memory or trend-based learning, while $X_t^{point}$ is constructed as a pointwise time snapshot of past wealth, serving as a proxy for updating immediate belief. As in Zhou and Yongjun (2024), it should be noted that when ${\beta }_1$ is positive, the investor exhibits momentum behaviour and tends to follow past winners. On the other hand, when ${\beta }_1$ is negative, the investor exhibits contrarian behaviour and tends to follow past losers; we refer the reader to Zhou and Yongjun (2024) Figure 1) for a graphical view of the information learning mechanism.

Assumption 1. 

The volatility matrix $\sigma \in {\mathbb{R}}^{k\times n}$ satisfies $\sigma {\sigma }^{\top }$ is invertible, with ${\sigma }^{\top }$ being the transpose of the matrix σ.

Using Equations (4) and (5), the wealth process X of investors evolves according to the stochastic differential equation

$$\begin{array}{cc}\hfill \mathrm{d}{X}_t& =\mathrm{d}\left({\pi }_t^M+{\displaystyle \sum _{i=1}^k}{\pi }_{t,i}^R\right)\hfill \\ \hfill & =\left(r{X}_t+{\displaystyle \sum _{j=1}^k}{\beta }_{1,j}{X}_t^{int}+{\displaystyle \sum _{j=1}^k}{\beta }_{2,j}{X}_t^{point}\right)\mathrm{d}t+{\left({\pi }_t^R\right)}^{\top }\mu \mathrm{d}t+{\left({\pi }_t^R\right)}^{\top }\sigma \mathrm{d}{W}_t-c_t\mathrm{d}t\hfill \\ \hfill & =\left(r{X}_t+{\displaystyle \sum _{j=1}^k}{\beta }_{1,j}{X}_t^{int}+{\displaystyle \sum _{j=1}^k}{\beta }_{2,j}{X}_t^{point}\right)\mathrm{d}t+{\pi }_t^{\top }\eta \mathrm{d}t+{\pi }_t^{\top }\mathrm{d}{W}_t-c_t\mathrm{d}t,\hfill \end{array}$$

(6)

where ${\pi }_t^{\top }:={\left({\pi }_t^R\right)}^{\top }\sigma ,0\le t\le T$, and $\eta :={\sigma }^{\top }{\left(\sigma {\sigma }^{\top }\right)}^{-1}\mu$. Due to Assumption A1, the risk premium $\eta$ is well-defined. Note that working with $\pi$ does not always ensure that we can come back to ${\pi }^R$ (because the matrix $\sigma$ is not necessarily symmetric). To guaranty the reversed relation, ${\pi }_t\left(\omega \right)$ must take values in ${\sigma }^{\top }\mathbb{R}$ for $t\in [0,T],\omega \in \mathsf{\Omega }$.

3. Main Results

3.1. The Epstein-Zin Utility Maximisation with Bounded Memory

To capture investors’ preferences, we consider the Epstein-Zin stochastic differential utility. We specify the utility via an aggregator f and a bequest utility function g, defined as

$$\begin{array}{cc}\hfill f(c,v)& :=\delta e^{-\delta t}{\displaystyle \frac{c^{1-\frac{1}{\psi }}}{1-\frac{1}{\psi }}}{\left((1-\gamma )v\right)}^{1-{\displaystyle \frac{1}{\theta }}}\mathrm{and}g\left(c\right):=e^{-\delta \theta T}{\displaystyle \frac{c^{1-\gamma }}{1-\gamma }}\mathrm{with}\theta :={\displaystyle \frac{1-\gamma }{1-\frac{1}{\psi }}},\hfill \end{array}$$

(7)

where c represents a positive consumption rate, $\delta >0$ the discounting rate, $\psi >0,\psi \ne 1$ the elasticity of intertemporal substitution coefficient (EIS) and $\gamma >0,\gamma \ne 1$ the relative risk aversion. Hence, the Epstein-Zin utility process ${\left(V_t^c\right)}_{t\in [0,T]}$ is given by

$$\begin{array}{cc}\hfill V_t^c& =\mathbb{E}\left[g\left(c_T\right)+{\int }_t^{T}f(c_s,V_s^c)\mathrm{d}s|{\mathcal{F}}_t\right]\mathrm{for}t\in [0,T].\hfill \end{array}$$

(8)

We consider the parameters’ specification

$$\begin{array}{c}\hfill \mathrm{either}\gamma \psi \ge 1,\psi >1\mathrm{or}\gamma \psi \le 1,\psi <1.\end{array}$$

(9)

We give a remark on the relation between the stochastic differential utility defined by (7)-(8) and the classical stochastic differential utility introduced by Duffie and Epstein (1992) (see also Epstein and Zin (1989)).

Remark 1. 

(i) 

Equations (7)-(8) provide a reformulation of the standard Epstein-Zin utility, denoted here by $V^{\mathrm{EZ},c}$. The standard utility has a terminal value given by $g_{\mathrm{EZ}}\left(c\right)={\displaystyle \frac{c^{1-\gamma }}{1-\gamma }}$, and its generator is defined in the difference form: $f_{\mathrm{EZ}}(c,v)=\delta {\displaystyle \frac{c^{1-\frac{1}{\psi }}}{1-\frac{1}{\psi }}}{\left((1-\gamma )v\right)}^{1-{\displaystyle \frac{1}{\theta }}}-\delta \theta v$ (see, e.g., Xing (2017, Eq.(2.1)-(2.2))). Note that, $V^{\mathrm{EZ},c}$ corresponds to the standard Epstein-Zin utility if and only if $V^c:={\left(e^{-\delta \theta t}{V}_t^{\mathrm{EZ},c}\right)}_{t\in [0,T]}$ coincides with the utility process defined by Equations (7)-(8). In particular, this implies $V_0^{\mathrm{EZ},c}=V_0^c$.

(ii) 

With the function f parameterised as above, one readily recognises the CRRA utility as a special case corresponding to the parameter configuration $\gamma ={\displaystyle \frac{1}{\psi }}$ (in that case $\theta =1$).

We consider the following set of admissible consumption streams:

$$\begin{array}{c}\hfill {\mathcal{C}}_a:=\left\{c\in \mathcal{C}\mid V^c\mathrm{is}\mathrm{of}\mathrm{class}\left(\mathrm{D}\right)\mathrm{and}(1-\gamma )V^c>0\right\}.\end{array}$$

(10)

(See, for example, [Def. 1.46 on p.11 Jacod and Shiryaev (2013) for the notion of class (D)). The set ${\mathcal{C}}_a$ defined by Equation (10) aligns with those considered in Matoussi and Xing (2018), as well as Herdegen et al. (2023). All known sufficient conditions for the existence of Epstein–Zin utility over a finite time horizon are summarised in [Prop. 2.1 Matoussi and Xing (2018), which, in particular, ensures that ${\mathcal{C}}_a\ne \varnothing$.

Hence, a pair $(c,\pi )$ of consumption-investment strategy is admissible if $c\in {\mathcal{C}}_a$, ${\pi }_t\left(\omega \right)\in {\sigma }^{\top }{\mathbb{R}}^k$ for $t\in [0,T]$, $\omega \in \mathsf{\Omega }$, and Equation (6) admits a unique strong solution. Investors aim to maximise their utility $V_0^c$ in the class of admissible strategies. Following, for example, Feng et al. (2025); Aurand et al. (2023) and Xing (2017), we later restrict admissible consumption-investment strategies to a permissible set $\mathcal{P}$ (see Equation (25)), and solve the stochastic optimisation problem of investors on this permissible set. Moreover, following Chang et al. (2011), we consider a situation in which investors derive utility from the perceived wealth. That is, we incorporate an additional term that reflects historical investment performance. That way, we allow the model to take into account non-financial drivers of satisfaction and reflect learning behaviours (meaning, investors care not just about wealth outcomes, but also how they are achieved). In this sense, it bridges the gap between standard financial modelling and behavioural modelling. Thus, the goal of investors is to find some strategy $(c^{*},{\pi }^{*})\in \mathcal{P}$ that achieves the optimal value

$$\begin{array}{cc}\hfill V_0^{*}& =\underset{(c,\pi )\in \mathcal{P}}{sup}\mathbb{E}\left[g(X_T+{\beta }_3{X}_T^{int})+{\int }_t^{T}f(c_s,V_s^c)\mathrm{d}s\right],\hfill \end{array}$$

(11)

where $V^c$ is the unique solution to Equation (8), with $c_T=X_T+{\beta }_3{X}_T^{int},{\beta }_3\in \mathbb{R}$.

3.2. The Ansatz

Motivated by the decomposition of the Epstein-Zin utility in Kuissi-Kamdem (2025) (compare with Aurand et al. (2023); Xing (2017)), we speculate that the optimal utility process takes the form

$$\begin{array}{c}\hfill V_t^{*}=e^{-\delta \theta t}{\displaystyle \frac{{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{1-\gamma }}{1-\gamma }}\mathrm{for}t\in [0,T],\end{array}$$

(12)

where $X^{int}$ and X are solutions to the SDEs (2) and (6), respectively, and $(Y,Z)$ is the solution to the BSDE

$$\begin{array}{cc}\hfill \mathrm{d}{Y}_t& =-\mathcal{G}\left(t,X_t,X_t^{int},X_t^{point},Y_t,Z_t\right)\mathrm{d}t+Z_t^{\top }\mathrm{d}{W}_t,Y_T=0,\hfill \end{array}$$

(13)

for some generator $\mathcal{G}$ to be determined. From Equations (13) and (8), we deduce that the process

$$\begin{array}{cc}\hfill L_t^{c,\pi }& :=e^{-\delta \theta t}{\displaystyle \frac{{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{1-\gamma }}{1-\gamma }}\hfill \\ \hfill & \phantom{xx}+{\int }_0^{t}f\left(c_s,e^{-\delta \theta s}{\displaystyle \frac{{(X_s+{\beta }_3{X}_s^{int}+e^{(r+{\beta }_3)s}{Y}_s)}^{1-\gamma }}{1-\gamma }}\right)\mathrm{d}s,t\in [0,T],\hfill \end{array}$$

(14)

should be a local supermartingale for any $(c,\pi )\in \mathcal{P}$ and a local martingale for an optimal strategy $(c^{*},{\pi }^{*})$. Itô’s formula applied to $L^{c,\pi ,\xi }$ gives

$$\begin{array}{cc}\hfill \mathrm{d}{L}_t^{c,\pi }& =(-c_t+\delta {\displaystyle \frac{c_t^{1-\frac{1}{\psi }}}{1-\frac{1}{\psi }}}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{{\displaystyle \frac{1}{\psi }}}+{\displaystyle \frac{1}{2\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right){\parallel \eta \parallel }^2\hfill \\ \hfill & \phantom{Xxx}-e^{(r+{\beta }_3)t}{Z}_t^{\top }\eta +(r+{\beta }_3)X_t+\left({\displaystyle \sum _{j=1}^k}{\beta }_{1,j}-\alpha {\beta }_3\right)X_t^{int}+\left({\displaystyle \sum _{j=1}^k}{\beta }_{2,j}-{\beta }_3{e}^{-\alpha h}\right)X_t^{point}\hfill \\ \hfill & \phantom{Xxx}+(r+{\beta }_3)e^{(r+{\beta }_3)t}{Y}_t-\left(1-e^{-\alpha h}\right){\beta }_3{X}_0-{\displaystyle \frac{\delta \theta }{1-\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\hfill \\ \hfill & \phantom{Xxx}-e^{(r+{\beta }_3)t}\mathcal{G}(t,X_t,X_t^{int},X_t^{point},Y_t,Z_t))e^{-\delta \theta t}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{-\gamma }\mathrm{d}t\hfill \\ \hfill & \phantom{X}-{\displaystyle \frac{\gamma }{2}}{e}^{-\delta \theta t}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{-\gamma -1}\hfill \\ \hfill & \phantom{XX}\times \parallel {\pi }_t+\left(e^{(r+{\beta }_3)t}{Z}_t-{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\eta \right){\parallel }^2\mathrm{d}t\hfill \\ \hfill & \phantom{X}+e^{-\delta \theta t}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{-\gamma }{\left({\pi }_t+e^{(r+{\beta }_3)t}{Z}_t\right)}^{\top }\mathrm{d}{W}_t.\hfill \end{array}$$

(15)

Hence, expecting the drift in Equation (15) to be non-positive for any $(c,\pi )\in \mathcal{P}$ and zero for an optimal strategy $(c^{*},{\pi }^{*})$, we obtain

$$\begin{array}{cc}\hfill & \mathcal{G}(t,X_t,X_t^{int},X_t^{point},Y_t,Z_t)\hfill \\ \hfill & =-Z_t^{\top }\eta +{\displaystyle \frac{1}{2\gamma }}{e}^{-(r+{\beta }_3)t}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right){\parallel \eta \parallel }^2\hfill \\ \hfill & \phantom{xx}-{\displaystyle \frac{\delta \theta }{1-\gamma }}{e}^{-(r+{\beta }_3)t}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)-e^{-(r+{\beta }_3)t}\left(1-e^{-\alpha h}\right){\beta }_3{X}_0\hfill \\ \hfill & \phantom{xx}+e^{-(r+{\beta }_3)t}((r+{\beta }_3)X_t+\left({\displaystyle \sum _{j=1}^k}{\beta }_{1,j}-\alpha {\beta }_3\right)X_t^{int}\hfill \\ \hfill & \phantom{XXXXXXXxx}+\left({\displaystyle \sum _{j=1}^k}{\beta }_{2,j}-{\beta }_3{e}^{-\alpha h}\right)X_t^{point}+(r+{\beta }_3)e^{(r+{\beta }_3)t}{Y}_t)\hfill \\ \hfill & \phantom{xx}+e^{-(r+{\beta }_3)t}\underset{c}{max}\left\{-c_t+\delta {\displaystyle \frac{c_t^{1-\frac{1}{\psi }}}{1-\frac{1}{\psi }}}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{{\displaystyle \frac{1}{\psi }}}\right\}\hfill \\ \hfill & \phantom{xx}+\underset{\pi }{max}\{-{\displaystyle \frac{\gamma }{2}}{e}^{-(r+{\beta }_3)t}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{-1}\hfill \\ \hfill & \phantom{XXXXXXXx}\times \parallel {\pi }_t+\left(e^{(r+{\beta }_3)t}{Z}_t-{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\eta \right){\parallel }^2\}.\hfill \end{array}$$

(16)

The optimisers to the two optimisation problems in Equation (16) are

$$\begin{array}{c}\hfill c_t^{*}={\delta }^{\psi }\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\mathrm{and}{\pi }_t^{*}=-e^{(r+{\beta }_3)t}{Z}_t+{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\eta .\end{array}$$

(17)

Substituting $(c^{*},{\pi }^{*})$ into (16) and (6) we obtain

$$\begin{array}{cc}\hfill & \mathcal{G}(t,X_t,X_t^{int},X_t^{point},Y_t,Z_t)\hfill \\ \hfill & =-Z_t^{\top }\eta +{\displaystyle \frac{1}{2\gamma }}{e}^{-(r+{\beta }_3)t}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right){\parallel \eta \parallel }^2\hfill \\ \hfill & \phantom{xx}+\left({\displaystyle \frac{{\delta }^{\psi }}{\psi -1}}-{\displaystyle \frac{\delta \theta }{1-\gamma }}\right)e^{-(r+{\beta }_3)t}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\hfill \\ \hfill & \phantom{xx}+e^{-(r+{\beta }_3)t}((r+{\beta }_3)X_t+\left({\displaystyle \sum _{j=1}^k}{\beta }_{1,j}-\alpha {\beta }_3\right)X_t^{int}+\left({\displaystyle \sum _{j=1}^k}{\beta }_{2,j}-{\beta }_3{e}^{-\alpha h}\right)X_t^{point}\hfill \\ \hfill & \phantom{XXXXXXXxx}+(r+{\beta }_3)e^{(r+{\beta }_3)t}{Y}_t)-e^{-(r+{\beta }_3)t}\left(1-e^{-\alpha h}\right){\beta }_3{X}_0\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill \mathrm{d}{X}_t& =(\left(r{X}_t+{\displaystyle \sum _{j=1}^k}{\beta }_{1,j}{X}_t^{int}+{\displaystyle \sum _{j=1}^k}{\beta }_{2,j}{X}_t^{point}\right)-{\delta }^{\psi }\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)-e^{(r+{\beta }_3)t}{Z}_t^{\top }\eta \hfill \\ \hfill & \phantom{Xxx}+{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right){\parallel \eta \parallel }^2)\mathrm{d}t\hfill \\ \hfill & \phantom{X}+{\left(-e^{(r+{\beta }_3)t}{Z}_t+{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\eta \right)}^{\top }\mathrm{d}{W}_t.\hfill \end{array}$$

(19)

For the heuristic derivation above to make sense, we should first prove existence of a solution to the linear forward-backward stochastic differential equation (1), (19) and (13), with the generator of the backward stochastic differential equation given by Equation (16). For easier reference, we summarise the latter FBSDE as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_t=(\left(r{X}_t+{\displaystyle \sum _{j=1}^k}{\beta }_{1,j}{X}_t^{int}+{\displaystyle \sum _{j=1}^k}{\beta }_{2,j}{X}_t^{point}\right)-{\delta }^{\psi }\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\hfill \\ \phantom{XXXxx}-e^{(r+{\beta }_3)t}{Z}_t^{\top }\eta +{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right){\parallel \eta \parallel }^2)\mathrm{d}t\hfill \\ \phantom{XXxx}+{\left(-e^{(r+{\beta }_3)t}{Z}_t+{\displaystyle \frac{1}{\gamma }}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\eta \right)}^{\top }\mathrm{d}{W}_t\hfill \\ \mathrm{d}{X}_t^{int}=\left(-\alpha X_t^{int}+X_t-e^{-\alpha h}{X}_t^{point}-\left(1-e^{-\alpha h}\right)X_0\right)\mathrm{d}t\hfill \\ \mathrm{d}{Y}_t=-(-Z_t^{\top }\eta +\left({\displaystyle \frac{{\delta }^{\psi }}{\psi -1}}-{\displaystyle \frac{\delta \theta }{1-\gamma }}+{\displaystyle \frac{1}{2\gamma }}{\parallel \eta \parallel }^2\right)e^{-(r+{\beta }_3)t}\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)\hfill \\ \phantom{XXxx}+e^{-(r+{\beta }_3)t}((r+{\beta }_3)X_t+\left({\displaystyle \sum _{j=1}^k}{\beta }_{1,j}-\alpha {\beta }_3\right)X_t^{int}+\left({\displaystyle \sum _{j=1}^k}{\beta }_{2,j}-{\beta }_3{e}^{-\alpha h}\right)X_t^{point}\hfill \\ \phantom{XXXXXXXXxxx}+(r+{\beta }_3)e^{(r+{\beta }_3)t}{Y}_t)-e^{-(r+{\beta }_3)t}\left(1-e^{-\alpha h}\right){\beta }_3{X}_0)\mathrm{d}t+Z_t^{\top }\mathrm{d}{W}_t\hfill \\ X_0>0,X_0^{int}=0,Y_T=0.\hfill \end{array}\right.\end{array}$$

(20)

To obtain an explicit solution to the FBSDE (20) and to the optimal control problem (Section 3.1), we consider the following

Assumption A2. 

The interest rate r as well as the parameters ${\beta }_1=({\beta }_{1,1},\dots ,{\beta }_{1,k}),{\beta }_2=({\beta }_{2,1},\dots ,{\beta }_{2,k}),{\beta }_3,\alpha$ and h satisfy

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=1}^k}{\beta }_{1,j}-\alpha {\beta }_3=(r+{\beta }_3){\beta }_{3}and{\displaystyle \sum _{j=1}^k}{\beta }_{2,j}={\beta }_3{e}^{-\alpha h}.\end{array}$$

(21)

Remark 2. 

We want to point out that Assumption A2 is consistent with a similar condition for the particular case of constant relative risk aversion (CRRA) utility (corresponding to $\gamma ={\displaystyle \frac{1}{\psi }}$) as in Chang et al. (2011) (for $k=1$) and Zhou and Yongjun (2024) and Wang et al. (2025) (for $k=1$ and $\alpha =0$).

We define the process ${\left({\phi }_t\right)}_{t\in [0,T]}$ by

$$\begin{array}{cc}\hfill {\phi }_t& :=exp\left(\left({\displaystyle \frac{\left(-{\delta }^{\psi }+\delta \right)\psi }{\psi -1}}+{\displaystyle \frac{\gamma -1}{2{\gamma }^2}}{\parallel \eta \parallel }^2\right)t+{\displaystyle \frac{1}{\gamma }}{\eta }^{\top }{W}_t\right),\hfill \end{array}$$

(22)

and the constants

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\kappa }_1:={\displaystyle \frac{{\delta }^{\psi }-\delta \psi }{\psi -1}}+{\displaystyle \frac{1}{2\gamma }}{\parallel \eta \parallel }^2+r+{\beta }_3\hfill \\ {\kappa }_2:=-{\kappa }_1-{\delta }^{\psi }\hfill \\ \tilde{x}:={\displaystyle \frac{{\kappa }_1+{\delta }^{\psi }}{{\kappa }_1{e}^{{\kappa }_{2}T}+{\delta }^{\psi }}}\left(1-{\displaystyle \frac{{\beta }_3}{r+{\beta }_3}}\left(1-e^{-(r+{\beta }_3)T}\right)\left(1-e^{-\alpha h}\right)\right)X_0.\hfill \end{array}\right.\end{array}$$

(23)

Now, we can obtain the following result on a solution to the linear FBSDE (2), (19), (13) and (16).

Proposition 1. 

Suppose Assumptions A1 and A2 hold. Then a solution $(X,X^{int},Y,Z)\in {\mathcal{H}}^2\left(\mathbb{R}\right)\times {\mathcal{H}}^2\left(\mathbb{R}\right)\times {\mathcal{H}}^2\left(\mathbb{R}\right)\times {\mathbb{H}}^2\left({\mathbb{R}}^n\right)$ to the FBSDE (2), (19), (13) and (16) is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{X}_t+{\beta }_3{X}_t^{int}=\tilde{x}{\phi }_t-e^{(r+{\beta }_3)t}{Y}_t\hfill \\ Y_t={\displaystyle \frac{1}{r+{\beta }_3}}\left(e^{-(r+{\beta }_3)T}-e^{-(r+{\beta }_3)t}\right)\left(1-e^{-\alpha h}\right){\beta }_3{X}_0\hfill \\ \phantom{XXx}+{\displaystyle \frac{\tilde{x}{\kappa }_1}{{\kappa }_2}}\left(e^{{\kappa }_{2}T}-e^{{\kappa }_{2}t}\right)exp\left({\displaystyle \frac{2\gamma -1}{2{\gamma }^2}}{\parallel \eta \parallel }^{2}t+{\displaystyle \frac{1}{\gamma }}{\eta }^{\top }{W}_t\right)\hfill \\ Z_t={\displaystyle \frac{\tilde{x}{\kappa }_1}{\gamma {\kappa }_2}}\left(e^{{\kappa }_{2}T}-e^{{\kappa }_{2}t}\right)exp\left({\displaystyle \frac{2\gamma -1}{2{\gamma }^2}}{\parallel \eta \parallel }^{2}t+{\displaystyle \frac{1}{\gamma }}{\eta }^{\top }{W}_t\right)\eta ,0\le t\le T.\hfill \end{array}\right.\end{array}$$

(24)

Proof. 

See Appendix A. □

3.3. Solution to the Stochastic Optimisation Problem

With the strategy $(c^{*},{\pi }^{*})$, given by Equation (17), well defined (due to Proposition 1), it remains to prove its optimality among a suitable set of strategies. We consider the permissible set

$$\begin{array}{cc}\hfill \mathcal{P}:=\left\{\right(c,\pi )\mathrm{admissible}\mid & \mathrm{the}\mathrm{process}{\left(X_t+{\beta }_3{X}_t^{int}+e^{(r+{\beta }_3)t}{Y}_t\right)}^{1-\gamma },t\in [0,T],\hfill \\ \hfill & \mathrm{is}\mathrm{of}\mathrm{class}\left(\mathrm{D}\right)\}.\hfill \end{array}$$

(25)

As already mentioned in Section 3.1, restricting the set of admissible strategies to a set of permissible ones has also been considered in, for example, Feng et al. (2025); Aurand et al. (2023) and Xing (2017). We refer the reader to Aurand et al. (2023, Rmk. 3.6) for details on the motivation for such restriction. In the present paper, for the motivation for our definition of $\mathcal{P}$, we use arguments similar to those of Aurand et al. (2023, Rmk. 3.6).

We are now ready to give an explicit expression for the solution to the stochastic optimisation problem (11).

Theorem 1. 

Suppose Assumptions A1 and A2 hold. Let $\tilde{x}$ be defined as in Equation (23). Then the optimal consumption $c^{*}$ and investment ${\pi }^{*}$ strategies given by

$$\begin{array}{c}\hfill c_t^{*}={\delta }^{\psi }\tilde{x}{\phi }_t\mathrm{and}{\pi }_t^{*}={\displaystyle \frac{1}{\gamma }}\tilde{x}\left(-{\displaystyle \frac{{\kappa }_1}{{\kappa }_2}}\left(e^{{\kappa }_2(T-t)}-1\right)+1\right){\phi }_t\eta \end{array}$$

(26)

solve the control problem (11), and their associated value function is given by

$$\begin{array}{cc}\hfill V_0^{*}& ={\displaystyle \frac{{\tilde{x}}^{1-\gamma }}{1-\gamma }}.\hfill \end{array}$$

(27)

In particular,

$$\begin{array}{c}\hfill {\pi }_t^{R,*}={\displaystyle \frac{1}{\gamma }}\tilde{x}\left(-{\displaystyle \frac{{\kappa }_1}{{\kappa }_2}}\left(e^{{\kappa }_2(T-t)}-1\right)+1\right){\phi }_t{\left(\sigma {\sigma }^{\top }\right)}^{-1}\mu \end{array}$$

(28)

Proof. 

See Appendix B □

4. Sensitivity Analysis

To illustrate the general results from the previous sections, we analyse the sensitivity of the value function, and the optimal consumption and investment strategies with respect to some parameters. For simplicity, we fix $k=2$ (two risky assets) and $n=2$ (two independent Brownian motions). Following Escobar-Anel et al. (2022), unless otherwise stated, the basic model parameters are summarised in Table 1.

In this setting, the matrix $\sigma$, and the vectors $\eta$ and ${\left(\sigma {\sigma }^{\top }\right)}^{-1}\mu$ are given by

$$\begin{array}{cc}\hfill & \sigma =\left(\begin{array}{cc}{\sigma }_1& 0\\ {\sigma }_2\rho & {\sigma }_2\sqrt{1-{\rho }^2}\end{array}\right),\eta ={\displaystyle \frac{1}{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }^2\right)}}\left(\begin{array}{c}{\sigma }_1{\sigma }_2^2{\mu }_1\left(1-{\rho }^2\right)\\ {\sigma }_2\sqrt{1-{\rho }^2}\left({\sigma }_1^2{\mu }_2-{\sigma }_1{\sigma }_2{\mu }_1\rho \right)\end{array}\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \mathrm{and}{\left(\sigma {\sigma }^{\top }\right)}^{-1}\mu ={\displaystyle \frac{1}{{\sigma }_1^2{\sigma }_2^2\left(1-{\rho }^2\right)}}\left(\begin{array}{c}{\sigma }_2^2{\mu }_1-{\sigma }_1{\sigma }_2{\mu }_2\rho \\ {\sigma }_1^2{\mu }_2-{\sigma }_1{\sigma }_2{\mu }_1\rho \end{array}\right),\hfill \end{array}$$

(30)

where ${\sigma }_1\ne 0,{\sigma }_2\ne 0$, and $\rho \in (-1,1)$ is the correlation coefficient between the two risky assets. Recall the definition of ${\kappa }_1,{\kappa }_2$ and $\tilde{x}$ in Equation (23). From Equations (26) and (28), we deduce that the optimal consumption and investment strategies are given by

$$\begin{array}{cc}\hfill & c_t^{*}={\delta }^{\psi }\tilde{x}{\phi }_t,{\pi }_{1,t}^{R,*}={\displaystyle \frac{{\sigma }_2^2{\mu }_1-{\sigma }_1{\sigma }_2{\mu }_2\rho }{\gamma {\sigma }_1^2{\sigma }_2^2\left(1-{\rho }^2\right)}}\tilde{x}\left(-{\displaystyle \frac{{\kappa }_1}{{\kappa }_2}}\left(e^{{\kappa }_2(T-t)}-1\right)+1\right){\phi }_t\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \mathrm{and}{\pi }_{2,t}^{R,*}={\displaystyle \frac{{\sigma }_1^2{\mu }_2-{\sigma }_1{\sigma }_2{\mu }_1\rho }{\gamma {\sigma }_1^2{\sigma }_2^2\left(1-{\rho }^2\right)}}\tilde{x}\left(-{\displaystyle \frac{{\kappa }_1}{{\kappa }_2}}\left(e^{{\kappa }_2(T-t)}-1\right)+1\right){\phi }_t.\hfill \end{array}$$

(32)

Figure 1 illustrates the evolution of the optimal investment and consumption strategies over time. The figure shows that these strategies are monotonic functions of time. In particular, Figure 1a,b indicate that when the weight parameter associated with the pointwise delayed wealth, ${\beta }_3$, is negative, the optimal investment in risky asset 1 decreases over time, whereas the optimal investment in risky asset 2 increases over time. By contrast, when ${\beta }_3$ is non-negative, the opposite pattern is observed: the investment in risky asset 1 increases over time while the investment in risky asset 2 decreases over time.

Furthermore, Figure 1c,d show that optimal consumption increases over time regardless of the sign of ${\beta }_3$. This suggests that investors progressively raise their consumption levels as they approach the terminal horizon, independently of the influence of delayed wealth effects.

Figure 2 illustrates the influence of the average parameter $\alpha$ on the optimal investment and consumption strategies. As shown in Figure 2c, for a fixed value of $\alpha$, optimal consumption increases over time. Moreover, for any given time t, higher values of the average parameter $\alpha$ lead to higher consumption levels. A similar behaviour can be observed in Figure 2a for the investment in risky asset 1, where larger values of $\alpha$ are associated with greater investment allocations.

In contrast, Figure 2b reveals an inverse relationship for risky asset 2: the allocation to this asset decreases as the average parameter $\alpha$ increases.

Figure 3 exhibit trends similar to those observed in Figure 2, with the average parameter $\alpha$ replaced by the delay parameter h.

Figure 4 shows that the risk aversion coefficient $\gamma$ influences the optimal investment and consumption strategies in a qualitatively similar pattern to the average parameter $\alpha$.

Figure 5 illustrates the influence of the elasticity of intertemporal substitution (EIS) parameter $\psi$ on the optimal strategies. The figure highlights three main features. First, the allocation to risky asset 1 increases both over time, for a fixed value of $\psi$, and with higher values of $\psi$ at any given time t. Second, the investment in risky asset 2 exhibits the opposite behaviour, decreasing with both time and the EIS parameter. Third, the optimal consumption strategy rises over time while declining as $\psi$ increases.

5. Conclusions

This paper develops a consumption–investment framework that integrates bounded memory and information learning within an Epstein-Zin utility setting. By incorporating both integrated and pointwise delayed wealth variables, the model captures realistic behavioural features such as trend-following and contrarian strategies.

Using martingale optimality principle with forward-backward stochastic differential equations, we obtain explicit solutions for optimal consumption and portfolio allocation under structural assumptions. These solutions reveal how memory effects and learning parameters directly influence investment behaviour and welfare.

The sensitivity analysis reveals that the optimal consumption and investment strategies are strongly influenced by investors’ learning behaviour, memory effects, and preference parameters. In particular, the weight assigned to delayed wealth information, represented by ${\beta }_3$, significantly affects portfolio allocation dynamics, generating opposite investment patterns across the two risky assets depending on whether past performance enters negatively or positively into investors’ perceived wealth.

The analysis further shows that the memory-related parameters, namely the average parameter $\alpha$ and the delay horizon h, play an important role in shaping optimal decisions. Higher values of these parameters generally increase allocations to risky asset 1 and consumption levels, while reducing the proportion invested in risky asset 2. Similar effects are observed for the risk aversion coefficient $\gamma$, highlighting the close interaction between investors’ preferences and learning mechanisms.

Finally, the elasticity of intertemporal substitution parameter $\psi$ exhibits differentiated effects on investment and consumption behaviour. Larger values of $\psi$ increase the allocation to risky asset 1 but decrease both the allocation to risky asset 2 and the consumption rate. These findings demonstrate that incorporating information learning and bounded memory alters optimal consumption-investment decisions and provides a richer behavioural interpretation of dynamic portfolio choice in incomplete markets. Future research could extend this framework to include stochastic interest rates, market frictions, or heterogeneous agents, further enhancing its applicability.