论文标题
通过轻度Kolmogorov方程的各向同性$α-$稳定过程驱动的高维半线性SDE的概率计算
Probability computation for high-dimensional semilinear SDEs driven by isotropic $α-$stable processes via mild Kolmogorov equations
论文作者
论文摘要
研究了由添加剂lévy噪声驱动的半连续性，$ n- $ dimensional随机微分方程（SDE）。具体而言，给定$α\ in \ left（\ frac {1} {2}，1 \右）$，利息是由由$2α-$ stable驱动的SDE上的，旋转不变的过程是由布朗尼运动从下属获得的。与其解决方案相关的时间相关的马尔可夫过渡半群与以轻度积分形式向后方程相关的原始连接是通过逐有技术建立的。这样的链接是迭代方法的起点，它允许随着几个参数的变化，具有单批Monte Carlo模拟与SDE相关的概率近似，从而在标准的Monte Carlo方法中具有引人注目的计算优势。该方法还涉及对高维间差kolmogorov向后方程的数值计算的数值计算。然后将其提供的方案及其提供的一阶近似值应用于两个非线性矢量字段，并显示出在尺寸$ n = 100 $中提供令人满意的结果。
Semilinear, $N-$dimensional stochastic differential equations (SDEs) driven by additive Lévy noise are investigated. Specifically, given $α\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by $2α-$stable, rotation-invariant processes obtained by subordination of a Brownian motion. An original connection between the time-dependent Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild integral form is established via regularization-by-noise techniques. Such a link is the starting point for an iterative method which allows to approximate probabilities related to the SDEs with a single batch of Monte Carlo simulations as several parameters change, bringing a compelling computational advantage over the standard Monte Carlo approach. This method also pertains to the numerical computation of solutions to high-dimensional integro-differential Kolmogorov backward equations. The scheme, and in particular the first order approximation it provides, is then applied for two nonlinear vector fields and shown to offer satisfactory results in dimension $N=100$.
