论文标题
具有跳跃和随机系数的随机微分方程的最佳控制：随机汉密尔顿 - 雅各比 - 贝尔曼方程
Optimal controls of stochastic differential equations with jumps and random coefficients: Stochastic Hamilton-Jacobi-Bellman equations with jumps
论文作者
论文摘要
在本文中，我们研究以下的非线性向后积分偏微分方程，并跳跃\ begin {equation*} \ left \ {\ strapt {split} -d v（t，x）=＆\ displayStyle \ displayStyle \ indyle \ infor_ { V(t,x),\int_E \left(\mathcal I V(t,e,x,u)+Ψ(t,x+g(t,e,x,u))\right)l(t,e)ν(de)) \\ &+\displaystyle\int_{E}\big[\mathcal I V(t,e,x,u)-\displaystyle （g（t，e，x，u），d v（t，x））\ big]ν（d e）+\ int_ {e} \ big [\ mathcal iψ（t，e，e，e，x，x，x，x，x，x，x，u）\ big]ν（d e） e，x）\tildeμ（d e，dt），\\ v（t，x）=＆\ h（x），\ end {split} \ right。 \ end {equation*}其中$ \tildeμ$是一种泊松随机的martingale量度，$ w $是布朗运动，而$ \ nathcal i $是稍后要指定的非本地运算符。函数$ h $是给定的随机映射，是由相应的非马克维亚最佳控制问题引起的。该方程式显示为随机的汉密尔顿 - 雅各比 - 贝尔曼方程，它以递归实用程序成本功能来表征最佳控制问题的价值函数。方程式的解决方案是一个可预测的随机字段$（V，φ，ψ）$的三重线。我们表明，在某些规律性假设下的值函数是对随机HJB方程的解决方案。该方程式的经典解决方案是值函数，并提供最佳控制。有了关于系数的一些其他假设，通过重新铸造后向后的随机部分积分差分方程来显示出Sobolev空间意义上的存在和独特性，并以poisson跳跃的Hilbert Space中的向后随机演化方程式跳跃。
In this paper, we study the following nonlinear backward stochastic integral partial differential equation with jumps \begin{equation*} \left\{ \begin{split} -d V(t,x) =&\displaystyle\inf_{u\in U}\bigg\{H(t,x,u, DV(t,x),D Φ(t,x), D^2 V(t,x),\int_E \left(\mathcal I V(t,e,x,u)+Ψ(t,x+g(t,e,x,u))\right)l(t,e)ν(de)) \\ &+\displaystyle\int_{E}\big[\mathcal I V(t,e,x,u)-\displaystyle (g(t, e,x,u), D V(t,x))\big]ν(d e)+\int_{E}\big[\mathcal I Ψ(t,e,x,u)\big]ν(d e)\bigg\}dt\\ &-Φ(t,x)dW(t)-\displaystyle\int_{E} Ψ(t, e,x)\tildeμ(d e,dt),\\ V(T,x)=& \ h(x), \end{split} \right. \end{equation*} where $\tilde μ$ is a Poisson random martingale measure, $W$ is a Brownian motion, and $\mathcal I$ is a non-local operator to be specified later. The function $H$ is a given random mapping, which arises from a corresponding non-Markovian optimal control problem. This equation appears as the stochastic Hamilton-Jacobi-Bellman equation, which characterizes the value function of the optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields $(V,Φ,Ψ)$. We show that the value function, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and gives the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the backward stochastic partial integral differential equation with jumps as a backward stochastic evolution equation in Hilbert spaces with Poisson jumps.
