# Optimal Harvest under a Gilpin-Ayala Model Driven by the Hawkes Process

Nyassoke Titi Gaston Clément, Sadefo Kamdem Jules, Fono Louis Aimé

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Nyassoke Titi Gaston Clément, Sadefo Kamdem Jules, Fono Louis Aimé. Optimal Harvest under a Gilpin-Ayala Model Driven by the Hawkes Process. Results in Applied Mathematics, 2025, 26, pp.100564. 10.1016/j.rinam.2025.100564. hal-04987283

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Submitted on 20 Mar 2025

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# Optimal harvest under a Gilpin–Ayala model driven by the Hawkes process

Nyassoke Titi Gaston Clément<sup>a,∗</sup>, Sadefo Kamdem Jules<sup>b,c</sup>, Fono Louis Aimé<sup>d</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

Keywords:

Stochastic Gilpin–Ayala

Self-exciting Hawkes process

Sufficient maximum principle

Optimal effort

## ABSTRACT

This paper analyzes the optimal effort for a risk-averse fisherman where the biomass process follows a Hawkes jump-diffusion process with Gilpin–Ayala drift. The main feature of the Hawkes process is to capture the phenomenon of clustering. The price process is of the mean-reverting type. We prove a sufficient maximum principle for the optimal control of a stochastic system consisting of an SDE driven by the Hawkes process and, by the concavity of the Hamiltonian, we obtain the optimal effort of the fisherman for a risk-averse investor.

## 1. Introduction

Fisheries management refers to activities that protect fishery resources so that sustainable exploitation is possible. The fishing effort is a measure of fishing intensity, e.g. the number of days on the boat per year or (in a rod-and-reel fishery) the number of hours on the boat per year [1]. Good fisheries management often requires modeling the evolution of fish biomass. [2] first proposes the representation of fish growth under harvesting by the equation:

$$ \frac{dX_t}{X_t} = rX_t \left(1 - \frac{X_t}{K}\right) - qE_t X_t, \quad (1) $$

where $X_t$ is the stock biomass, $r$ is the intrinsic rate of growth, $K$ is the carrying capacity, $q$ is a catch-ability parameter describing the efficiency of the fishing gear, and $E_t$ is the fishing effort. Notice that $h_t = qE_tX_t$ is the harvest. Considering $E$ to be a constant in the Schaefer model, Eq. (1) gives a nonzero equilibrium point at $\bar{x} = K \left(1 - \frac{qE}{r}\right)$ where $E < r/q$. If $E > r/q$ then $\bar{x} < 0$ and the population is driven to extinction. The effort maximizing the sustainable yield (or the equilibrium harvest) $\bar{h}$, defined as $\bar{h} = qE\bar{x}$, is obtained as $\bar{E} = r/2q$.

One of the main concerns of fisheries management is to determine the optimal fishing effort considering the growth dynamics of a particular fish species in a disturbed and changing environment, e.g., dry season or flood season [3–5] and natural disasters [6], which ensures sustainability and profits for the fisherman who is considered a risk-averse investor [7–9].

Sudden environmental shocks such as storms, volcanic eruptions, etc. influence the authors when modeling the dynamics of a biomass. Most of them use the Levy jump-diffusion model to account for the jumps caused by these sudden events. The Levy jump-diffusion model contains three different types of distributions, namely a normal distribution for the Wiener process, a Poisson distribution for the jump events, and a specific distribution for the jump sizes. The use of a Poisson distribution means that

<footnote><sup>a</sup> Laboratoire de Mathématiques, Université de Bertoua, Cameroon</footnote>

<footnote><sup>b</sup> MRE UR 209, Université de Montpellier, France</footnote>

<footnote><sup>c</sup> Faculté D'Economie, Université de Montpellier, France</footnote>

<footnote><sup>d</sup> Laboratoire de Mathématiques, Université de Douala, Cameroon</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: titigaston@yahoo.fr (Nyassoke T.G.C.), jules.sadefo-kamdem@umontpellier.fr (Sadefo K.J.), lfono2000@yahoo.fr (Fono L.A.).</footnote>
the following assumptions are made: the only changes in the process are unit jumps and the time intervals between jumps are independent. However, these assumptions are not entirely realistic: the occurrence of certain sudden events can cause the occurrence of other events, such as a contagion effect. In the case of an earthquake, for example, Hawkes processes were introduced by Alan Hawkes in 1971 [10] to model the effects of aftershocks. Similar behavior to the earthquake is observed by [11], who studies how clusters of crime patterns emerge in cases of violent crime, or by [12,13] for the time-dependent behavior of the infinite-server system. Hence, the study of Hawkes processes initiated by [14] and the references therein has found extensive applications in fields such as finance, DNA modeling, and neuroscience.

Notice that Hawkes processes can be defined using two different approaches, namely the intensity-based approach and the clustering approach. We only consider the former.

In this paper, we apply the Hawkes process to the management of renewable resources, in particular, to fisheries. Following the work of [15], we replace the Poisson process with the Hawkes process. More precisely, we consider a Gilpin–Ayala growth model for fishery resources with a self-exciting Hawkes process. The Gilpin–Ayala process, a generalization of the logistic growth model, offers superior flexibility to capture biomass dynamics, particularly under extreme events, while the Hawkes process reflects the clustering and contagion effects of environmental shocks. Additionally, the inclusion of a stochastic mean-reverting fish price process further enhances the applicability of the model as it captures the volatility and unpredictability of market conditions faced by fishermen. This dual-layered stochastic framework allows for a precise determination of optimal fishing effort, balancing resource conservation with economic profitability.

Moreover, we derived sufficient optimality conditions for the stochastic optimal control problem where the investor has to deal with uncertainties from various sources, as described previously. Note that many authors ([16–18]) have specified the necessary conditions for optimality that can be used in our context. [17] proved that the maximum principle differs from the classical principle in that the first-order adjoint equation turns out to be a linear mean-field backward SDE.

The remainder of the paper is organized as follows. In Section 2, we first present the Hawkes jump Gilpin–Ayala process and mean-reverting stochastic price process, then we set the Optimal control problem to be solve. In Section 3, we use the maximum principle approach to provide the sufficient condition for the optimality of controls in our stochastic system. Section 4 concludes the paper.

## 2. The model

### 2.1. Hawkes process

We first give a summary of the mathematical definitions and properties of the Hawkes process, which will be used throughout the remaining sections.

Let $N_t$ denote the number of arrivals (events) over time $t$. Then $N_t$ is a counting process. Denote by $(\tau_1, \tau_2, \dots)$ the corresponding point process of the arrival times of the events. We consider a counting process $(N_t)_{t \ge 0}$ with the history of arrivals up to time $t$, $(\mathcal{H}_t)_{t \ge 0}$, that satisfies the following conditions:

$$ \mathbb{P}\{N_{t+h} - N_t = k \mid \mathcal{H}_t\} = \begin{cases} \lambda_t h + o(h), & k=1 \\ o(h), & k>1 \\ 1 - \lambda_t h + o(h), & k=0 \end{cases} \qquad (2) $$

where $\lambda$ is the conditional intensity function and is given by

$$ \lambda_t = \lim_{h \to 0} \frac{E(N_{t+h} - N_t \mid \mathcal{H}_t)}{h} $$

The intensity function, if it exists, uniquely determines the finite-dimensional distributions of the point process, and thus can be used to characterize the process [19].

**Definition 2.1 ([20])**. Consider $(N_t)_{t \ge 0}$, a counting process with history $(\mathcal{H}_t)_{t \ge 0}$ that satisfies the condition in (2) and has conditional intensity function of the form

$$ \lambda_t = \lambda_\infty + \int_0^t \mu(t-s) dN_s = \lambda_\infty + \sum_{\tau_i < t} \mu(t-\tau_i), \qquad (3) $$

where $\lambda_\infty > 0$ is called background intensity and $\mu: [0, \infty) \to (0, \infty)$ is called the excitation function. Assume that $\mu(\cdot) \neq 0$ to avoid the trivial case of a homogeneous Poisson process. The process $N_t$ is called a one-dimensional Hawkes process. In particular, if the excitation function is an exponentially decreasing function, then $N_t$ is called a self-exciting Hawkes process.

To ensure the stationarity of the univariate Hawkes process, the excitation function must fulfill the condition

$$ \int_0^\infty \mu(s) ds < 1 \qquad (4) $$

For the self-exciting Hawkes process, a common exponentially decaying excitation function is

$$ \mu(t) = \beta e^{-\alpha t} \qquad (5) $$
where $\alpha, \beta > 0$. In this case, the form of the conditional intensity function (3) becomes

$$ \lambda_t = \lambda_\infty + \int_0^t \beta e^{-\alpha(t-s)} dN_s = \lambda_\infty + \beta \sum_{t_i < t} e^{-\alpha(t-t_i)} \quad (6) $$

with the stationarity condition $\beta < \alpha$. Applying Ito's product rule to (6), we obtain the following stochastic differential equation (SDE)

$$ \begin{aligned} d\lambda_t &= \alpha(\lambda_\infty - \lambda_t)dt + \beta dN_t; & \lambda_0 \neq \lambda_\infty \\ d\lambda_t &= \alpha(\lambda_\infty - \lambda_t)dt + \int_{\mathbb{R}\setminus\{0\}} \beta N(dt, dz); & \end{aligned} \quad (7) $$

$$ d\lambda_t = \left[ \alpha(\lambda_\infty - \lambda_t) + \int_{\mathbb{R}\setminus\{0\}} \beta\lambda_t v(dz) \right] dt + \int_{\mathbb{R}\setminus\{0\}} \beta\tilde{N}(dt, dz); $$

$\tilde{N}(dt, dz) = N(dt, dz) - \lambda_t v(dz)dt$ denotes the compensated Poisson measure on $[0, T] \times \mathbb{R}\setminus\{0\}$ (see [21]). $v(dz)$ is the Lévy Measure for the jump size, i.e. a $\sigma$-finite Borel measure on $\mathbb{R}\setminus\{0\}$ with the property $\int_{\mathbb{R}\setminus\{0\}} \min(1, z^2)v(dz) < +\infty$.

## 2.2. Problem formulation

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \ge 0}, \mathbb{P})$ be a complete probability space with a filtration $\{\mathcal{F}_t\}_{t \ge 0}$ satisfying the usual conditions (i.e. it is increasing and right-continuous while $\mathcal{F}_0$ contains all P-null sets).

The dynamics of the population size $X_t$ and the unit price of fish $Y_t$, at time $t$, are as follows

$$ \left\{ \begin{aligned} dX_t &= X_{t-} \left[ r - aX_{t-}^* - qE(t) \right] dt + \sigma X_{t-} dW_t + \int_{\mathbb{R}\setminus\{0\}} \eta(t, X_{t-}, \lambda, z) \tilde{N}(dt, dz) \\ dY_t &= \theta(\tilde{Y}_0 - Y_t) dt + \sigma_Y Y_t dW_t^Y \\ d\lambda_t &= \left[ \alpha(\lambda_\infty - \lambda_t) + \int_{\mathbb{R}\setminus\{0\}} \beta\lambda_t v(dz) \right] dt + \int_{\mathbb{R}\setminus\{0\}} \beta\tilde{N}(dt, dz) \end{aligned} \right. \quad (8) $$

where $\tilde{Y}_0$ is the long-run mean, $\theta$ is a mean-reverting speed parameter, (the price process $Y_t$ is a Ornstein-Uhlenbeck process with the level-dependent volatility process, $r$ is the intrinsic growth rate and, $E$ is the fishing effort exerted on the population, $q > 0$ is the catchability coefficient, $\kappa > 0$ denotes the Gilpin-Ayala parameter, $a = r/K > 0$, and $K$ is the environmental carrying capacity, $\sigma$ and $\sigma_Y$ are population and price volatility respectively. The standard Brownian motions $W_t$ and $W_t^Y$, $t \ge 0$, defined on $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \ge 0}, \mathbb{P})$ are supposed to be independent. And $\eta: [0, T] \times \mathbb{R} \times (0, \infty) \times \mathbb{R} \to \mathbb{R}$ is a continuous function with respect to $v$ and is measurable satisfying the following assumption: there exist $\rho: \mathbb{R} \to \mathbb{R}_+$ with $\int_{\mathbb{R}\setminus\{0\}} \rho^2(z)v_t(dz) < +\infty$, such that for all $t, s \in [0, T]$, $x, x' \in \mathbb{R}$ and $\lambda, \lambda' \in (0, \infty)$, $\|\eta(t, x, \lambda, z) - \eta(t, x', \lambda', z)\| < \rho(z)\|(x, \lambda) - (x', \lambda')\|$, $\|\eta(t, x, \lambda, z)\| < \rho(z)(1 + \|(x, \lambda)\|)$. The reader is referred to [22] for more precise properties of random measures.

We use the quadratic cost function

$$ c(E_t) = c_1 E_t + c_2 E_t^2 $$

where $c_1, c_2 > 0$ are constants, and define the profit function as

$$ \pi(X_t, Y_t, E_t) = qE_t X_t Y_t - (c_1 E_t + c_2 E_t^2) \quad (9) $$

We use the power utility function $u(x) = (x^{1-\gamma})/(1-\gamma)$, where $\gamma$ is the constant relative risk aversion coefficient (CRRA) and $\gamma \in (0, 1)$, to characterize our risk adverse fisherman.

Let $X_s^{t,x}, Y_s^{t,y}$ and $\lambda_s^{t,x}, \lambda_s^{t,y}$ denote the solutions of the SDE (8) starting at $s = t$ at points $x, y$ and $\lambda$. The problem is then to find the optimal control effort $E_t$ that maximizes the present value $J(t, x, y, \lambda, E_t)$ defined as follows:

$$ J(t, x, y, \lambda, E_t) = \mathbb{E}_{x, y, \lambda} \left[ \int_t^T e^{-\zeta(s-t)} \frac{\pi(X_s^{t,x}, Y_s^{t,y}, E_s)^{1-\gamma}}{1-\gamma} ds + e^{-\zeta(T-t)} V(X_T, Y_T, \lambda_T) \right]. \quad (10) $$

Here $\mathbb{E}_{x, y, \lambda}$ is the conditional expectation given $X_t^{t,x} = x, Y_t^{t,y} = y$ and $\lambda_t^{t,x} = \lambda$, where $T$ is the finite time horizon $\zeta > 0$ is a discount factor (see [15]).

We say that the control process $E_s$ is admissible if the following two conditions are satisfied:

1. equations for the state, price and jump intensity processes of System (8) have a unique strong solution $(X_s, Y_s, \lambda_s)$;

$$ 2. \quad \mathbb{E}_{x, y, \lambda} \left( \int_t^T \| e^{-\zeta(s-t)} \frac{\pi(X_s^{t,x}, Y_s^{t,y}, E_s)^{1-\gamma}}{1-\gamma} \| ds + \| e^{-\zeta(T-t)} V(X_T, Y_T, \lambda_T) \| \right) < \infty. $$

It is assumed that effort is limited (see [23]), so that $A$, the set of admissible controls, is bounded. The fisherman seeks to maximize the total expected discounted utility of both profit and terminal biomass. Therefore, the value function can be written as follows:

$$ v(t, x, y, \lambda) = \sup_{E_t \in A} J(t, x, y, \lambda, E_t). \quad (11) $$

Notice that we have allowed the boundary condition and the final condition in the following form $v(t, 0, y, \lambda) = 0$ and $v(T, x, y, \lambda) = \Lambda^{\gamma} \frac{x^{1-\gamma}}{1-\gamma} \Lambda > 0$, respectively.
# 3. Stochastic maximum principle

## 3.1. Lemma

For the processes $X_t$, $Y_t$, and $\lambda_t$ we make the following assumptions on the coefficients: There exist constants $C_X^1 < \infty$, $C_Y^1 < \infty$ and $C_\lambda^1 < 1$ such that for all $h, t \in [0, T]$, $x, y, \lambda, x', y', \lambda' \in \mathbb{R}_+$:

### A1 At most linear growth:

$$ 
\begin{aligned}
& E\|X_h^{t,x}\|^2 \le C_X^1(1 + \|x\|^2 + \|\lambda\|^2), \quad E\|Y_h^{t,y}\|^2 \le C_Y^1(1 + \|y\|^2 + \|\lambda\|^2), \\
& E\|\lambda_h^{t,\lambda}\|^2 \le C_\lambda^1(1 + \|\lambda\|^2).
\end{aligned} 
\quad (12) $$

### A2 Lipschitz continuity:

$$ 
\begin{aligned}
& E\|X_h^{t,x} - X_h^{t,\lambda'}\|^2 \le C_X^1(\|x - x'\|^2 + \|\lambda - \lambda'\|^2), \\
& E\|Y_h^{t,y} - Y_h^{t,\lambda'}\|^2 \le C_Y^1(\|y - y'\|^2 + \|\lambda - \lambda'\|^2), \\
& E\|\lambda_h^{t,\lambda} - \lambda_h^{t,\lambda'}\|^2 \le C_\lambda^1(\|\lambda - \lambda'\|^2).
\end{aligned} 
\quad (13) $$

The reader is referred to [15], for the sake of proof.

## 3.2. The Hamiltonian and adjoint processes

We have adapted the stochastic calculus to account for the specific properties of Hawkes processes, then the Hamiltonian of the problem (11) is the function $\mathcal{H} : [0, T] \times \mathbb{R}_+ \times \mathbb{R}_+ \times \mathbb{R}_+ \times A \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}_1 \times \mathbb{R}_2 \to \mathbb{R}$ defined by

$$ 
\begin{aligned}
\mathcal{H}(s, x, y, \lambda, E_s, p_X, p_Y, p_\lambda, q_X, q_Y, q_\lambda, \theta_X, \theta_\lambda) = & \\
& l(s, x, y, \lambda, E_s) + f(s, x, E_s)p_X + \theta(\bar{Y}_0 - y)p_Y + \left[ \alpha(\lambda_\infty - \lambda_s) + \int_{\mathbb{R}\setminus\{0\}} \beta\lambda_s v(dz) \right] p_\lambda \\
& + \sigma x q_X + \sigma_y y q_Y + \int_{\mathbb{R}\setminus\{0\}} \theta_X(s, z)\eta(s, x, \lambda, z)\lambda_s v(dz) \\
& + \int_{\mathbb{R}\setminus\{0\}} \beta\theta_\lambda(s, z)\lambda_s v(dz)
\end{aligned} 
\quad (14) $$

where $l(s, x, y, \lambda, E_s) = e^{-\zeta(s-t)} \frac{\pi(x, y, E_s)^{1-\gamma}}{1-\gamma}$, and $f(s, x, E_s) = [r - ax^\kappa - qE_s] x$.

Here $\mathbb{R}_1$ and $\mathbb{R}_2$ are sets of all functions $\theta_X : [0, T] \times \mathbb{R}\setminus\{0\} \to \mathbb{R}$, $\theta_\lambda : [0, T] \times \mathbb{R}\setminus\{0\} \to \mathbb{R}$, respectively, for which the integrals in (14) converge, that is,

$$ \mathbb{R}_1 = \{\theta_X; \int_{\mathbb{R}\setminus\{0\}} \|\theta_X(s, z)\eta(s, X_s, \lambda, z)\|\lambda_s v(dz) < \infty\}, $$

$$ \mathbb{R}_2 = \{\theta_\lambda; \int_{\mathbb{R}\setminus\{0\}} \beta\theta_\lambda(s, z)\lambda_s v(dz) < \infty\}. $$

Suppose that the Hamiltonian defined in (14) is sufficiently differentiable (at least $C^1$) with respect to the variables $x, y, \lambda, E$. The first-order adjoint equations involved in the stochastic maximum principle for our control problem are linear backward SDE of mean-field type (see [24–26])

$$ 
\begin{aligned}
dp_X &= -\left[\frac{\partial H(t)}{\partial x}\right] ds + q_X dW_s + \int_{\mathbb{R}\setminus\{0\}} \theta_X(s, z) \tilde{N}(ds, dz) \\
&= -[(r - (\kappa + 1)ax^\kappa - qE_s)p_X + \sigma q_X - qE_s y]ds + q_X dW_s \\
&\quad + \int_{\mathbb{R}\setminus\{0\}} \theta_X(s, z) \tilde{N}(ds, dz)
\end{aligned} 
\quad (15) $$

$$ p_X(T) = e^{-\zeta(T-t)} \frac{\partial V(X_T, Y_T, \lambda_T)}{\partial x} = -e^{-\zeta(T-t)} \Lambda^\gamma X_T^{-\gamma} $$

$$ 
\begin{aligned}
dp_Y &= -\left[\frac{\partial H(t)}{\partial y}\right] ds + q_Y dW_s^Y \\
&= -[-\theta_Y p_Y - q E_s x + \sigma_Y q_Y] ds + q_Y dW_s^Y
\end{aligned} 
\quad (16) $$

$$ p_Y(T) = e^{-\zeta(T-t)} \frac{\partial V(X_T, Y_T, \lambda_T)}{\partial y} = 0 $$
$$ 
\begin{aligned}
dp_{\lambda} &= - \left[ \frac{\partial H(t)}{\partial \lambda} \right] ds + \int_{\mathbb{R}\setminus\{0\}} \theta_{\lambda}(s, z) \tilde{N}(ds, dz) \\
&= - \left[ -\alpha\lambda + \int_{\mathbb{R}\setminus\{0\}} \beta\nu(dz) \right] p_{\lambda} ds + \int_{\mathbb{R}\setminus\{0\}} \theta_{\lambda}(s, z) \tilde{N}(ds, dz)
\end{aligned} 
\quad (17) $$

$$ p_{\lambda}(T) = e^{-\zeta(T-t)} \frac{\partial V(X_T, Y_T, \lambda_T)}{\partial \lambda} = 0, \quad s \in [0, T]. $$

### 3.3. The sufficient maximum principle

**Theorem 3.1.** Let $E_t^* \in A$ be an arbitrary admissible control with the corresponding solutions $(X_s^*, \hat{Y}_s, \hat{\lambda}_s)$, and $(p_X(s), q_X(s), \theta(s, z))$, $(p_Y(s), q_Y(s), \theta(s, z))$ and $(p_\lambda(s), \theta(s, z))$, respectively, of (8), (15)-(17), where $X_s^* = X_{E_t^*}$.

Set

$$ H^*(s) = \mathcal{H}(s, X_s^*, \hat{Y}_s, \hat{\lambda}_s, E_t^*, p_X, p_Y, p_\lambda, q_X, q_Y, q_\lambda, \theta_X, \theta_\lambda). $$

Suppose that the following conditions are satisfied:

#### 1. The function

$$ (s, x, y, \lambda, E_s) \to H(s, x, y, \lambda, E_s), $$

$$ \text{where } H(s, x, y, \lambda, E_s) = \mathcal{H}(s, x, y, \lambda, E_s, p_X, p_Y, p_\lambda, q_X, q_Y, q_\lambda, \theta_X, \theta_\lambda) \text{ is concave for all } s \in [0, T]. $$

#### 2. The maximum condition:

$$ H^*(s) = \max_{E_s \in A} \mathcal{H}(s, X_s^*, \hat{Y}_s, \hat{\lambda}_s, E_s, p_X, p_Y, p_\lambda, q_X, q_Y, q_\lambda, \theta_X, \theta_\lambda), \quad s \in [0, T]. \quad (18) $$

#### 3. Integrability conditions:

$$ \mathbb{E} \left[ \int_t^T (X_s - X_s^*)^2 \left( q_X^2 + \int_{\mathbb{R}\setminus\{0\}} \theta_X^2(s, z) \lambda_s \nu(dz) \right) ds \right] < \infty $$

$$ \mathbb{E} \left[ \int_t^T (Y_s - \hat{Y}_s)^2 \left( q_Y^2 + \int_{\mathbb{R}\setminus\{0\}} \theta_Y^2(s, z) \lambda_s \nu(dz) \right) ds \right] < \infty $$

$$ \mathbb{E} \left[ \int_t^T (\lambda_s - \hat{\lambda}_s)^2 \left( \int_{\mathbb{R}\setminus\{0\}} \theta_\lambda^2(s, z) \lambda_s \nu(dz) \right) ds \right] < \infty $$

$$ \mathbb{E} \left[ \int_t^T p_X^2 \left( \sigma^2(X_s - X_s^*)^2 + \int_{\mathbb{R}\setminus\{0\}} (\eta(s, X_s^*, \hat{\lambda}_s, z) - \eta(s, X_s, \lambda_s, z))^2 \lambda_s \nu(dz) \right) ds \right] < \infty $$

$$ \mathbb{E} \left[ \int_t^T p_Y^2 (\sigma_Y^2 (Y_s - \hat{Y}_s)^2) ds \right] < \infty $$

$$ \mathbb{E} \left[ \int_t^T p_\lambda^2 \int_{\mathbb{R}\setminus\{0\}} \beta^2 (\hat{\lambda}_s - \lambda_s)^2 \nu(dz) ds \right] < \infty $$

Then $E_t^*$ is an optimal control.

**Proof of Theorem 3.1.** The proof is standard and, for the sake of completeness, we provide it here.

Choose $E_t^* \in A$ with corresponding solutions $(X_s^*, \hat{Y}_s, \hat{\lambda}_s)$, and $(p_X(s), q_X(s), \theta(s, z))$, $(p_Y(s), q_Y(s), \theta(s, z))$ and $(p_\lambda(s), \theta(s, z))$. In the following we show that $J(t, x, y, \lambda, E_t^*) - J(t, x, y, \lambda, E_t) \le 0$.

$$ J(t, x, y, \lambda, E_t^*) - J(t, x, y, \lambda, E_t) = I_1 + I_2 $$

$$ 
\begin{aligned}
I_1 &= \mathbb{E}_{X,y,\lambda} \left[ \int_t^T e^{-\zeta(s-t)} \left( \frac{\pi(X_s^*, \hat{Y}_s, E_t^*)^{1-\gamma}}{1-\gamma} - \frac{\pi(X_s, Y_s, E_s)^{1-\gamma}}{1-\gamma} \right) ds \right]. \\
&= \mathbb{E}_{X,y,\lambda} \left[ \int_t^T (l(X_s^*, \hat{Y}_s, \hat{\lambda}_s, E_t^*) - l(X_s, Y_s, \lambda_s, E_s)) ds \right]
\end{aligned}
$$

$$ I_2 = \mathbb{E}_{X,y,\lambda} [e^{-\zeta(T-t)} (V(X_T^*, \hat{Y}_T, \hat{\lambda}_T) - V(X_T, Y_T, \lambda_T))] $$

By the definition of $H$ we have

$$ 
\begin{aligned}
I_1 &= \mathbb{E}_{X,y,\lambda} \left[ \int_t^T \left\{ H^*(s) - H(s) + (f(s, X_s, E_s) - f(s, X_s^*, E_t^*))p_X \right. \right. \\
&\qquad \left. + \theta(\hat{Y}_s - Y_s)p_Y + \left( \alpha(\hat{\lambda}_s - \lambda_s) + \int_{\mathbb{R}\setminus\{0\}} \beta(\lambda_s - \hat{\lambda}_s)\nu(dz) \right) p_\lambda + \sigma(X_s - X_s^*)q_X \right]
\end{aligned}
$$
$$ +\sigma_Y (Y_s - \hat{Y}_s)q_Y + \int_{\mathbb{R}\setminus\{0\}} \theta_X(s, z)(\eta(s, X_s, \lambda_s, z)\lambda_s - \eta(s, X_s^*, \hat{\lambda}_s, z)\hat{\lambda}_s)\nu(dz) \\ +\int_{\mathbb{R}\setminus\{0\}} \beta(\lambda_s - \hat{\lambda}_s)\theta_{\lambda}(s, z)\nu(dz) \Big\} ds \Big]. \quad (19) $$

Since the function

$$ (x, y, \lambda, E_s) \to H(s, x, y, \lambda, E_s), $$

is concave, we have

$$ I_1 \geq \mathbb{E}_{x,y,\lambda} \left[ \int_t^T \left\{ \begin{aligned} & \frac{\partial H^*(s)}{\partial x}(X_s^* - X_s) + \frac{\partial H^*(s)}{\partial y}(\hat{Y}_s - Y_s) + \frac{\partial H^*(s)}{\partial \lambda}(\hat{\lambda}_s - \lambda_s) \\ & + \frac{\partial H^*(s)}{\partial E_s}(E_t^* - E_s) + (f(s, X_s, E_s) - f(s, X_s^*, E_t^*))p_X + \theta(\hat{Y}_s - Y_s)p_Y \\ & + \left(\alpha(\hat{\lambda}_s - \lambda_s) + \int_{\mathbb{R}\setminus\{0\}} \beta(\lambda_s - \hat{\lambda}_s)\nu(dz)\right) p_\lambda \\ & + \sigma(X_s - X_s^*)q_X + \int_{\mathbb{R}\setminus\{0\}} (\eta(s, X_s, \lambda_s, z) - \eta(s, X_s^*, \hat{\lambda}_s, z))\theta_X(s, z)\lambda_s\nu(dz) \end{aligned} \right\} ds \right]. \quad (20) $$

By concavity of $(x, y, \lambda) \to e^{-\zeta(T-t)}V(x, y, \lambda)$, $\forall t$,

$$ \begin{aligned} I_2 &\geq \mathbb{E}_{x,y,\lambda} \left[ e^{-\zeta(T-t)} \frac{\partial V(X_T^*, \hat{Y}_T, \hat{\lambda}_T)}{\partial x} (X_T^* - X_T) \right. \\ &\qquad \left. + e^{-\zeta(T-t)} \frac{\partial V(X_T^*, \hat{Y}_T, \hat{\lambda}_T)}{\partial y} (\hat{Y}_T - Y_T) + e^{-\zeta(T-t)} \frac{\partial V(X_T^*, \hat{Y}_T, \hat{\lambda}_T)}{\partial \lambda} (\hat{\lambda}_T - \lambda_T) \right] \\ &= \mathbb{E}_{x,y,\lambda} \left[ p_X(T)(X_T^* - X_T) + p_Y(T)(\hat{Y}_T - Y_T) + p_\lambda(T)(\hat{\lambda}_T - \lambda_T) \right] \\ &= \mathbb{E}_{x,y,\lambda} \left[ \int_t^T p_X(s)(f(s, X_s^*, E_t^*) - f(s, X_s, E_s))ds + \int_t^T \left[ -\frac{\partial H^*(t)}{\partial x} \right] (X_s^* - X_s)ds \right. \\ &\qquad \left. + \int_t^T \sigma(X_s^* - X_s)q_X ds \right] \\ &\qquad + \int_t^T \left( \int_{\mathbb{R}\setminus\{0\}} (\eta(s, X_s^*, \lambda_s, z) - \eta(s, X_s, \lambda_s, z))\theta_X(s, z)\lambda_s\nu(dz) \right) ds \\ &\qquad + \int_t^T p_Y(s)\theta(Y_s - \hat{Y}_s)ds + \int_t^T \left[ -\frac{\partial H^*(t)}{\partial y} \right] (\hat{Y}_s - Y_s)ds + \int_t^T \sigma_Y(\hat{Y}_s - Y_s)q_Y ds \\ &\qquad \left. + \int_t^T \left( \alpha(\hat{\lambda}_s - \lambda_s) + \int_{\mathbb{R}\setminus\{0\}} \beta(\lambda_s - \hat{\lambda}_s)\nu(dz) \right) p_\lambda(s)ds + \int_t^T \left[ -\frac{\partial H^*(t)}{\partial \lambda} \right] (\hat{\lambda}_s - \lambda_s)ds \right]. \end{aligned} $$

Then,

$$ J(t, x, y, \lambda, E_t^*) - J(t, x, y, \lambda, E_t) = I_1 + I_2 \geq \mathbb{E}_{x,y,\lambda} \left[ \frac{\partial H^*(s)}{\partial E_s} (E_t^* - E_s) \right] $$

From the maximum condition (18), we have

$$ E_t^* = \arg\max_{E_s \in \mathcal{A}} \mathbb{E}[H(t)] = \arg\max_{E_s \in \mathcal{A}} \mathbb{E}[\mathcal{J}(s, X_s^*, \hat{Y}_s, \hat{\lambda}_s, E_s, p_X, p_Y, p_\lambda, q_X, q_Y, q_\lambda, \theta_X, \theta_\lambda)]. $$

Thus,

$$ \frac{\partial \mathbb{E}[H^*(t)]}{\partial E_s} = 0. $$

We can therefore conclude that

$$ J(t, x, y, \lambda, E_t^*) - J(t, x, y, \lambda, E_t) \geq 0. $$

## 4. Conclusion

In this paper, we establish a sufficient stochastic maximum principle for optimal control of the Hawkes-jump Gilpin-Ayala process, a comprehensive framework that integrates advanced modeling features to address real-world complexities in renewable resource management.

By incorporating a fishery-specific utility function that characterizes fishermen's preferences and risk aversion, we not only ensure the relevance of the model for practical decision making, but also align it with sustainable economic goals.

Our results provide actionable information for fisheries management, encouraging policymakers, industry stakeholders, and conservation organizations to adopt this framework. By explicitly considering both ecological and economic uncertainties, the model
facilitates the design of adaptive strategies that ensure long-term sustainability and profitability. Beyond fisheries, the framework can also be extended to other renewable resources, such as forest and water management, where extreme events and market uncertainties are critical factors.

## Ethical approval

This article does not contain any studies with human participants or animals carried out by any of the authors.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

No data was used for the research described in the article.

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