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We consider the stochastic optimal control problem for the dynamical systemof the stochastic differential equation driven by a local martingale with aspatial parameter. Assuming the convexity of the control domain, we obtain thestochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions. The stochasticlinear quadratic problem in this setting is also discussed.

Stochastic maximum principle for systems driven by local martingales with spatial parameters