Rare-Event Simulation Techniques for Structured Fisheries Models

Abstract

Depletion of fish stocks has a disastrous effect on the natural environment. One of the main goals in fisheries management is therefore to prevent the collapse of fish stocks. A key consideration is the impact of different harvest strategies and environmental noise on the likelihood of future stock collapse, or quasi-extinction. Motivated by this problem, we consider several rare-event simulation techniques to increase the speed and accuracy of projections of stock collapse. We consider both a one-dimensional structured fisheries model and a multidimensional age-structured fisheries model in applying our techniques. We observe that the way in which catch is modelled has a substantial impact on the likelihood of quasi-extinction. We reach the overall conclusion that the use of rare-event simulation techniques for structured fisheries models is worthwhile when the probability of quasi-extinction is on the order of $10^{-2}$ or smaller.

Appendix. Auxiliary Results

This section contains the mathematical results underpinning the approximation discussed in Sect. 3.3. We start by providing a formal definition of the large-deviations principle for a set S that describes a rare event. Building on this, we show in a very general setting that we can approximate the corresponding optimal path by selecting a path from what we call maximizing sets (Lemma A.1) and Lemma A.2). Finally, relying on these results, we outline a discretization scheme for stochastic recursions (like the one of Model I) and show how an approximate optimal path can be selected in this case.

1.1 A.1 The Large-Deviations Principle

Let $\lbrace X_{\epsilon }, \epsilon > 0 \rbrace$ be random elements in $\mathbb {R}^{d}$ that satisfy the large-deviations principle (LDP) with rate function $I :\mathbb {R}^{d} \rightarrow [0, \infty ]$ as $\epsilon \rightarrow 0$. We assume here that I is continuous and has compact level sets in order to avoid topological complications. We focus on closed sets S, for which the infimum in (4) is attained on S.

The LDP (4) characterizes the logarithmic asymptotics of $X_{\epsilon }$ being in S via the cost minimization problem (5). An ‘optimal path’ is any $x \in S$ that minimizes the cost, but usually there is a unique optimal path and we focus on this scenario in the remainder of this discussion. We are interested in the optimal path for several reasons. First, it minimizes the cost and thus characterizes the logarithmic asymptotics of $\mathbb {P} (X_{\epsilon } \in S)$. Second, the process $X_{\epsilon }$ conditioned on being in S gets closer and closer to the optimal path as $\epsilon \rightarrow 0$. If S is a disastrous event with very low probability, we may interpret the optimal path as the way in which disaster is likely to happen. Thus, knowing the optimal path may help us to prepare for and possibly mitigate the disastrous event. Third, we may use the optimal path to increase the speed and accuracy of simulations related to the event S via importance sampling.

1.2 A.2 Approximating the Optimal Path

Now suppose that we decompose the set S into finitely many disjoint subsets that become smaller and smaller as $\epsilon$ becomes smaller. Intuitively, one would think that the logarithmic asymptotics of $\mathbb {P} (X_{\epsilon } \in S)$ is determined by a subset of S that contains an optimal path. The next lemma formalizes this idea and states conditions under which it holds.

Lemma A.1

If for every $\epsilon > 0$ the sets $S_{\epsilon , 1}, \dotsc , S_{\epsilon , m_{\epsilon }}$ form a partition of S and if $\epsilon \log m_{\epsilon }$ converges to 0, then

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \epsilon \log \mathbb {P} (X_{\epsilon } \in S) = \lim _{\epsilon \rightarrow 0} \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}). \end{aligned}$$

(8)

Proof

The elementary inequalities

$$\begin{aligned} \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) \le \mathbb {P} (X_{\epsilon } \in S) \le m_{\epsilon } \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) \end{aligned}$$

are a direct consequence of the sets $S_{\epsilon , 1}, \dotsc , S_{\epsilon , m_{\epsilon }}$ being a partition of S. These inequalities imply that

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0} \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) \le \lim _{\epsilon \rightarrow 0} \epsilon \log \mathbb {P} (X_{\epsilon } \in S) \end{aligned}$$

and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \epsilon \log \mathbb {P} (X_{\epsilon } \in S)&\le \liminf _{\epsilon \rightarrow 0} \epsilon \log m_{\epsilon } \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) \\&= \liminf _{\epsilon \rightarrow 0} ( \epsilon \log m_{\epsilon } + \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) ) \\&= \liminf _{\epsilon \rightarrow 0} \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}). \end{aligned}$$

Consequently, $\lim _{\epsilon \rightarrow 0} \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i})$ exists and satisfies (8).$\square$

The previous lemma suggests that we may approximate solutions to the variational problem (5) using sets that are a maximizer of $\max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i})$ (which we call maximizing sets). The next lemma makes this idea precise.

The key condition in the next lemma is that S is compact and that the sets $S_{\epsilon , i}$ become small as $\epsilon \rightarrow 0$. This condition ensures that the maximizing sets give useful information about a solution to the variational problem (5). At a high level, the lemma states that the maximizing sets get arbitrarily close to an optimal path in S both in terms of distance and in terms of the cost associated with the worst path in a maximizing set. From a more practical point of view, this means that we may find an approximation to the solution of the variational problem (5) by determining maximizing sets.

Lemma A.2

Suppose that the conditions of Lemma A.1 hold. Let S be a compact set and define $S^{-} = \lbrace x \in S \, \vert I(x) = \inf _{y \in S} I(y) \rbrace$. Assume that the sets $S_{\epsilon , 1}, \dotsc , S_{\epsilon , m_{\epsilon }}$ are all contained in closed balls with diameter $\delta _{\epsilon }$, where $\delta _{\epsilon } \rightarrow 0$ as $\epsilon \rightarrow 0$. If $S_{\epsilon }^{*} \in \lbrace S_{\epsilon , 1}, \dotsc , S_{\epsilon , m_{\epsilon }} \rbrace$ satisfies $\mathbb {P} (S_{\epsilon }^{*}) = \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i})$, then

$$\begin{aligned} \sup _{x \in S_{\epsilon }^{*}} d(x, S^{-}) \rightarrow 0 \end{aligned}$$

(9)

and

$$\begin{aligned} \sup _{x \in S_{\epsilon }^{*}} I (x) \rightarrow \inf _{x \in S} I (x) = I_{S} \end{aligned}$$

(10)

as $\epsilon \rightarrow 0$.

Proof

We prove the first statement by contradiction. Suppose that (9) does not hold. Since the partitioning sets have a vanishing diameter, this means that there exists a sequence $\epsilon _{k} \rightarrow 0$ such that $d(S_{\epsilon _{k}}^{*}, S^{-}) \ge \eta > 0$. Then the closure C of $\cup _{k = 1}^{\infty } S_{\epsilon _{k}}^{*}$ is contained in S but has a distance of at least $\eta$ to $S^{-}$. Now using that $X_{\epsilon }$ satisfies the LDP with rate function I, we obtain

$$\begin{aligned} - I_{S}&= \lim _{\epsilon \rightarrow 0} \epsilon \log \max _{i = 1, \dotsc , m_{\epsilon }} \mathbb {P} (X_{\epsilon } \in S_{\epsilon , i}) \\&= \lim _{k \rightarrow \infty } \epsilon _{k} \log \mathbb {P} (X_{\epsilon _{k}} \in S_{\epsilon _{k}}^{*}) \\&\le \lim _{k \rightarrow \infty } \epsilon _{k} \log \mathbb {P} (X_{\epsilon _{k}} \in \text {cl} \cup _{k = 1}^{\infty } S_{\epsilon _{k}}^{*}) \\&= - I_{\text {cl} \cup _{k = 1}^{\infty } S_{\epsilon _{k}}^{*}} \\&< - I_{S}, \end{aligned}$$

which is a contradiction.

To prove the second statement, we note that the level sets of I are closed and that S is compact, so $S^{-}$ is compact. The convergence in (10) is a direct consequence of $S^{-}$ being compact and I being continuous.$\square$

The previous lemma considers the case in which S is compact and I has compact level sets, which is the most relevant setting for our applications. However, more general versions of the lemma can be formulated. It suffices that I has compact level sets (although the requirements for the partitioning sets have to be adapted) or that S is compact. If I does not have compact level sets and S is not compact, then the conclusions of the lemma may not hold.

1.3 A.3 Discretization Schemes

In practice it is often convenient to work with a discretized version $\tilde{X}_{\epsilon }$ of a Markov chain $X_{\epsilon }$. We show that intuitive discretization schemes ensure that the discretized process is sufficiently close to the original process in terms of their logarithmic asymptotics for threshold probabilities. Thus we do not lose any relevant information by focussing on the discretized process. In particular, we can use the discretized process to approximate the optimal path of the original process, which we can then use to define a change of measure for the original process.

We focus on a one-dimensional stochastic recursion defined for $t \in \lbrace 1, \dotsc , T \rbrace$ via

$$\begin{aligned} X_{\epsilon } (t + 1)&= g (X_{\epsilon } (t)) e^{\xi _{\epsilon } (t)}, \end{aligned}$$

where $X_{\epsilon } (0)$ is given and the $\xi _{\epsilon } (1), \xi _{\epsilon } (2), \dotsc$ form a sequence of independent random variables all having a normal distribution with mean 0 and variance $\epsilon$. We assume that the function g is a strictly increasing, concave function on $[0, \infty )$, having two fixed points $\rho _{1} < \rho _{2}$ in [0, K), where K can be interpreted as the (normalized) carrying capacity. We are given a threshold $\delta$ that is larger than 0 but smaller than the largest fixed point $\rho _{2}$. For simplicity, we assume that the initial value $X_{\epsilon }(0)$ concentrates on a closed set in $(\delta , \rho _{2})$ and does not depend on $\epsilon$.

We now analyze how this affects the large-deviations behavior of $X_{\epsilon }$. Let E be the event that the process starts in a point in $(\delta , \rho _{2})$ and crosses the threshold $\delta$ at least once before or at time T, so $E = \lbrace (x_0, x_1, \dotsc , x_T) \, \vert \, x_0 \in (\delta , \rho _{2}) \text { and } x_t \le \delta \text { for some } t \le T \rbrace$. As argued before, $X_{\epsilon }$ satisfies the LDP as $\epsilon \rightarrow 0$ and the optimal path associated with E is found by solving the cost optimization problem

$$\begin{aligned} \min&\quad \big ( \log x_{1} - \log g (x_{0}) \big )^{2} + \cdots + \big ( \log x_{T} - \log g (x_{T - 1}) \big )^{2}, \\ \text {s.t.}&\quad (x_{1}, \dotsc , x_{T}) \in E. \end{aligned}$$

Although this is a complicated optimization problem, one property is easy to see: no solution can have $x_{t}$ larger than the fixed point $\rho _2$ for some t. Indeed, replacing each $x_{t} > \rho _{2}$ by $\rho _{2}$ in a given path in E leads to a lower cost (since $g(x_{t}) < x_{t}$ for $x_{t} > \rho _{2}$), while still having the defining property that it crosses the threshold $\delta$.

The previous implies that no optimal path toward $\delta$ can cross the fixed point $\rho _2$, so in particular

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \epsilon \log \mathbb {P} (X_{\epsilon } \in E) = \lim _{\epsilon \rightarrow 0} \epsilon \log \mathbb {P} (X_{\epsilon } \in E \cap [0, K]^{T}). \end{aligned}$$

This enables us to deal with the unbounded state space of $X_{\epsilon }$ by defining a discretized version of $X_{\epsilon }$ that has a bounded state space [0, K] but retains the logarithmic asymptotics of $X_{\epsilon }$ for all events of interest (i.e., events in which $X_{\epsilon }$ reaches low levels).

For fixed $\epsilon > 0$, we define a grid on [0, K] by taking the coarsest equidistant grid $x_{0}, \dotsc , x_{M}$ that includes 0 and K and has grid distance at most $K / m_{\epsilon }$ for a positive integer $m_{\epsilon }$. We also define the $(M + 1) \times (M + 1)$ transition matrix P via

$$\begin{aligned} P_{ij} = \mathbb {P} \big ( X_{\epsilon }(t + 1) \in [x_{j}, x_{j + 1}) \, \big \vert \, X_{\epsilon }(t) \in [x_{i}, x_{i + 1}) \big ), \end{aligned}$$

where we interpret $x_{M + 1} = \infty$. Then the discretized process $\tilde{X}_{\epsilon }$ is a Markov chain with transition matrix P. Its initial value $\tilde{X}_{\epsilon }(0)$ is determined by rounding down $X_{\epsilon }(0)$ to the nearest grid point. If $x_{i_0}, x_{i_1}, \dotsc , x_{i_T}$ is a sequence of grid points not including K, then this definition guarantees that

$$\begin{aligned} P \left( \tilde{X}_{\epsilon } (0) = x_{i_0}, \tilde{X}_{\epsilon } (1) = x_{i_1}, \dotsc , \tilde{X}_{\epsilon } (T) = x_{i_T} \right) \end{aligned}$$

(11)

equals

$$\begin{aligned} P \left( X_{\epsilon } (0) \in \left[ x_{i_0}, x_{i_0 + 1} \right) , X_{\epsilon } (1) \in \left[ x_{i_1}, x_{i_1 + 1} \right) , \dotsc , X_{\epsilon } (T) \in \left[ x_{i_T}, x_{i_T + 1} \right) \right) . \end{aligned}$$

(12)

With these results at hand, we see that

$$\begin{aligned} \mathbb {P} (X_{\epsilon } \in E \cap [0, K]^{T}) = \mathbb {P} (\tilde{X}_{\epsilon } \in E). \end{aligned}$$

In view of Lemma A.2 and the equality of (11) and (12), it now suffices to decrease the grid size and determine a maximizing set for $\mathbb {P} (\tilde{X}_{\epsilon } \in E)$ with the partition based on the grid. This gives an approximate optimal path for $\tilde{X}_{\epsilon }$ and in turn for the original process ${X}_{\epsilon }$.