Optimal harvesting from interacting populations in a stochastic environment (Q5949618)

Let the sizes of \(n\) populations \(X=(X_1,X_2,\ldots,X_n)\) be described by the multidimensional stochastic differential equation NEWLINE\[NEWLINE dX(t)=b\big(X(t)\big) dt+\sigma\big(X(t)\big) dB(t),\quad s\leq t\leq T,\quad X(s)=x, NEWLINE\]NEWLINE driven by an \(m\)-dimensional Brownian motion \(B\). Let \(S\subset \mathbb{R}^{n+1}\) be a survival set, and let the time of extinction be defined by NEWLINE\[NEWLINE T(\omega)=\inf\{t>0:(t,X(t))\notin S\}. NEWLINE\]NEWLINE A harvesting strategy is an adapted right-continuous \(n\)-dimensional stochastic process \(\gamma\) with nondecreasing coordinates. If a harvesting strategy \(\gamma\) is applied, then the corresponding population vector \(X^\gamma\) is assumed to satisfy the stochastic equation NEWLINE\[NEWLINE dX^\gamma(t)=b(X^\gamma(t)) dt +\sigma(X^\gamma(t)) dB(t)-d\gamma(t),\quad s\leq t\leq T,\quad X^\gamma(s-)=x.\tag{1} NEWLINE\]NEWLINE The problem is to find the maximal total expected discounted utility NEWLINE\[NEWLINE \Phi(s,x)=\sup {\mathbb E}\int_s^T f(t)\cdot d\gamma(t) NEWLINE\]NEWLINE where \(f\) is a nonrandom \(n\)-dimensional continuous function (the vector of prices per unit), and the supremum is taken over all harvesting strategies \(\gamma\) such that the solution \(X^\gamma\) of (1) does not explode before time \(T\) and \(X^\gamma(T)\in \overline S\). The author obtains a verification theorem giving sufficient conditions so that a given function actually coincides with the value function \(\Phi\) and, if it does, telling us how to find an optimal strategy. The result leads to the conjecture of the one-at-a-time principle that it is almost surely never optimal to harvest from more than one population at a time.