Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow
Anna Amirdjanova Jie Xiong
Discrete & Continuous Dynamical Systems - B 2006, 6(4): 651-666 doi: 10.3934/dcdsb.2006.6.651
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
Mathematical Control & Related Fields 2018, 8(2): 451-473 doi: 10.3934/mcrf.2018018

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
Robust optimal investment and reinsurance of an insurer under Jump-diffusion models
Xin Zhang Hui Meng Jie Xiong Yang Shen
Mathematical Control & Related Fields 2018, 8(0): 1-18 doi: 10.3934/mcrf.2019003

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

keywords: Robust control proportional reinsurance stochastic differential game HJBI equation jump-diffusion model

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