Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow
Anna Amirdjanova Jie Xiong
We derive a large deviation principle for a stochastic Navier-Stokes equation for the vorticity of a two-dimensional fluid when the magnitude of the random term tends to zero. The key is the verification of the exponential tightness for the stochastic vorticity.
keywords: vorticity stochastic Navier-Stokes equation exponential tightness. Large deviation principle
A second-order stochastic maximum principle for generalized mean-field singular control problem
Hancheng Guo Jie Xiong

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.

keywords: Stochastic maximum principle mean-field control problem singular control Fréchet derivative range theorem of vector-valued measures

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