论文标题
部分可观测时空混沌系统的无模型预测
Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analyses in total variation distance)
论文作者
论文摘要
我们处理McKean-Vlasov和Boltzmann类型跳跃方程。这意味着随机方程的系数取决于溶液的定律，并且该方程是由具有强度度量的泊松点测度驱动的，该测量也取决于溶液的定律。在[3]中，Alfonsi和Bally证明，在某些适当的条件下，存在此类方程式的$ x_t $，并且是独一无二的。人们还证明$ x_t $是分析弱方程的概率解释。此外，此方程的Euler方案$ x_t^{\ Mathcal {p}} $将wasserstein距离收敛到$ x_t $。在本文中，在更受限制的假设下，我们表明欧拉方案$ x_t^{\ mathcal {p}} $在总变化距离中收敛到$ x_t $，$ x_t $具有平稳的密度（这是分析弱方程的函数解决方案）。另一方面，鉴于仿真，我们使用截断的欧拉方案$ x^{\ Mathcal {p}，m} _t $，其在任何紧凑的间隔中都有有限的跳跃数量。我们证明$ x^{\ Mathcal {p}，m} _ {t} $也将总变化距离收敛到$ x_t $。最后，我们根据与$ x^{\ Mathcal {p}，m} _t $相关的粒子系统提供了一种算法，以近似$ x_t $的法律密度。获得误差的完整估计。
We deal with Mckean-Vlasov and Boltzmann type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. In [3], Alfonsi and Bally have proved that under some suitable conditions, the solution $X_t$ of such equation exists and is unique. One also proves that $X_t$ is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme $X_t^{\mathcal{P}}$ of this equation converges to $X_t$ in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme $X_t^{\mathcal{P}}$ converges to $X_t$ in total variation distance and $X_t$ has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme $X^{\mathcal{P},M}_t$ which has a finite numbers of jumps in any compact interval. We prove that $X^{\mathcal{P},M}_{t}$ also converges to $X_t$ in total variation distance. Finally, we give an algorithm based on a particle system associated to $X^{\mathcal{P},M}_t$ in order to approximate the density of the law of $X_t$. Complete estimates of the error are obtained.
