<p style='text-indent:20px;'>We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.</p>

中文翻译：

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具有空间参数的局部鞅驱动系统的随机最大值原理

<p style='text-indent:20px;'>考虑带空间参数的局部鞅驱动的随机微分方程动力系统的随机最优控制问题。假设控制域的凸性，我们得到了随机极大值原理作为最优控制的必要条件，并证明了它在适当条件下的充分性。还讨论了该设置下的随机线性二次问题。</p>