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Optimal Control of Stochastic Partial Differential Equations and Partial (Noisy) Observation Control

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Applied Stochastic Control of Jump Diffusions

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Abstract

Suppose the density Y(tx) of a fish population at time \(t \in [0,T]\) and at the point \(x \in D \subset \mathbb {R}^n\) (where D is a given open set) is modeled by a stochastic partial differential equation (SPDE for short) of the form

$$\begin{aligned} \begin{aligned} {\mathrm {d}}Y(t, x) =&\left[ \frac{1}{2} \Delta Y(t,x) + \alpha Y(t,x) - u(t,x) \right] {\mathrm {d}}t \\&+ \beta Y(t,x) {\mathrm {d}}B(t) + Y(t^-,x) \int _{\mathbb {R}} \zeta \tilde{N} ({\mathrm {d}}t, {\mathrm {d}}\zeta ) ; \ (t, x) \in (0,T) \times D, \end{aligned} \end{aligned}$$

where we assume that \(\zeta \ge -1 + \varepsilon \) a.s. \(\nu ({\mathrm {d}}\zeta )\) for some constant \(\varepsilon > 0\). The boundary conditions are:

$$\begin{aligned} Y(0,x) = \xi (x) ; \ x \in D \end{aligned}$$
$$\begin{aligned} Y(t,x) = \eta (t,x) ; \ (t, x) \in [0,T) \times \partial D. \end{aligned}$$

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Correspondence to Bernt Øksendal .

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Øksendal, B., Sulem, A. (2019). Optimal Control of Stochastic Partial Differential Equations and Partial (Noisy) Observation Control. In: Applied Stochastic Control of Jump Diffusions. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-02781-0_13

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