| Abstract: |
| We establish a stochastic maximum principle for partially observed optimal control problem, in which the state dynamic is driven by an $\alpha-$stable process ($1<\alpha<2$) and the observation process contains both Brownian and jump noises. By employing the separation principle, the original control problem is transformed into an infinite-dimensional setting, where the unnormalized conditional density of the state satisfies a fractional Zakai equation involving a fractional Laplacian and Poisson jumps. Under suitable assumptions, the well-posedness of the weak solution to the associated fractional forward-backward stochastic partial differential equation (FBSPDE) is proved within a Gelfand triple framework. The main result provides a necessary condition for optimality in the form of a Hamiltonian inequality, thereby extending earlier work to allow the control to enter the diffusion coefficient and the observation to include jumps. |
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