论文标题
具有共同噪声的McKean-Vlasov方程的适应性和驯服方案
Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise
论文作者
论文摘要
在本文中，我们首先建立了McKean-Vlasov随机微分方程（McKean-Vlasov SDES）的良好性，具有常见的噪声，可能具有在状态变量中具有超级线性增长的系数。其次，我们为McKean-Vlasov Sdes提供了稳定的时间步变计划。具体而言，我们为与McKean-Vlasov方程相关的相互作用粒子系统提出了一个显式驯服的Euler和Tame Milstein方案。我们证明稳定性和强烈的融合订单$ 1/2 $和$ 1 $。为了获得我们的主要结果，我们采用了瓦斯汀空间中的微积分技术。驯服米尔斯坦方案的强烈收敛的证明只需要在状态和测量成分中连续差异的系数。为了证明我们的理论发现，我们提出了几个数值示例，包括随机$ 3/2 $波动率模型的平均场版本和带有乘法噪声的随机双井动力学。
In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.
