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### Open Access Journals

DCDS-B

We consider the optimal harvesting and planting control problem to maximize the expected total net benefits in the stochastic logistic population model. The variational inequality associated with this problem is given by the degenerate form of elliptic type with quadratic coefficients. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a solution to the variational inequality. The optimal harvesting and planting policy is characterized in terms of two thresholds for the variational inequality.

DCDS-B

We study the variational
inequality for a 1-dimensional linear-quadratic control
problem with discretionary stopping.
We establish the existence of a unique
strong solution via stochastic analysis and
the viscosity solution technique.
Finally, the optimal policy is shown to exist from the
optimality conditions.

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