Let
ybe measured stock size, subject to measurement error from true stockxqbe harvest quota, subject to implementation error from actual harvest,h
Stock next year is also subject to stochastic growth shock \(z_g\), (note that \(f\) will also depend on the harvest, h, unless \(x_t\) is taken as the escapement population, \(x-h\)).
\[ x_{t+1} = z_g f(x_t) \]
In discrete space, let X be a vector representing the probability distribution of having stock \(x\) at time \(t\), restricted to some finite grid of dimension \(n\). Let \(P_g(x | \mu, \sigma_g)\) be the probability density function of the shock \(z_g\) over states \(x\), given parameter \(\sigma\) and mean \(\mu\). We can write the distribution \(X_{t+1}\) as the result of a matrix-vector product, \(X_{t+1} = \mathbb{F} X_t\) where the elements of F are given by
\[ F_{ij} = P_g(x_i, f(x_j), \sigma_g) \]
and \(\mathbb{F}\) is also a function of the harvest level, \(\mathbb{F_h}\).
When we consider measurement uncertainty, we allow the vector \(\vec Y\) to represent the measured stock size, and the true stock size \(\vec X\) is a random variable distributed around it. In the discrete space we may represent this as \(X = \mathbb{M} \vec Y\), where \(M\) is an \(n\) by \(n\) grid representing the role of measurement uncertainty, \(M_{ij} = P_m(x_i, x_j, \sigma_m)\). The state equation now evolves in belief space,
\[ Y_{t+1} = \mathbb{F}_h \mathbb{M} \vec Y_t \]
If there is implementation uncertainty as well and \(f\) is a function of the escapement population (those remaining after a harvest \(h\)), we can impliment this as an additional transition matrix \(I_{ij} = P_i(x_i, x_j-h, \sigma_i)\), and the state equation becomes
\[ Y_{t+1} = \mathbb{F} \mathbb{M} \mathbb{I}_h \vec Y_t \]
Note that the implementation matrix is a function of the harvest level, so that we need a different matrix for each \(h\), whether or not we introduce measurement uncertainty (instead of a different \(\mathbb{F}\) matrix for each harvest as before.)
Results
See github log