DCDS-B
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
MCRF
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
MCRF
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
MCRF
A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance
Jie Xiong Shuaiqi Zhang Yi Zhuang
Mathematical Control & Related Fields 2018, 8(0): 1-20 doi: 10.3934/mcrf.2019013

In this article, we study a class of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding Nash equilibrium control is required to be adapted to the filtration generated by the observation process. To find the Nash equilibrium point, we establish the maximum principle as a necessary condition and derive the verification theorem as a sufficient condition. Applying the theoretical results and stochastic filtering theory, we obtain the explicit investment strategy of a partial information financial problem.

keywords: Forward-backward stochastic differential equation stochastic differential game maximum principle equilibrium point stochastic filtering

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