# Multiple Uncertainty

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.)

See github log