Let

`y`

be measured stock size, subject to measurement error from true stock`x`

`q`

be 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