论文标题
具有弱奇异内核的随机Volterra积分方程的数值方法
Numerical methods for stochastic Volterra integral equations with weakly singular kernels
论文作者
论文摘要
在本文中，我们首先建立了与弱奇异内核的随机燃烧器积分方程的解决方案的存在，独特性和Hölder连续性。然后，我们提出了一个$θ$ -euler-maruyama计划和米尔斯坦方案，以数值求解方程，并在任何$ p \ geq 1 $中以$ l^{p} $ norm中的两种方案获得了强大的收敛速度。对于$θ$ -EULER-MARUYAMA方案，速率为$ \ min \ {1-α，\ frac {1} {2} {2}-β\}〜％（0 <α<1，0 <α<1，0 <β<\ frac {1} {1} {2} {2} {2} $α\ neq \ frac 12 $，其中$（0 <α<1，0 <β<\ frac {1} {2}）$。这些关于收敛速率的结果与带有常规核的随机燃烧器积分方程的相似方案显着不同。获得我们的结果的困难是方程式缺乏ITô公式。为了解决这个困难，我们使用泰勒公式，然后对解决方案满足的方程进行复杂的分析。
In this paper, we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a $θ$-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geq 1$. For the $θ$-Euler-Maruyama scheme the rate is $\min\{1-α,\frac{1}{2}-β\}~ % (0<α<1, 0< β<\frac{1}{2})$ and for the Milstein scheme the rate is $\min\{1-α,1-2β\}$ when $α\neq \frac 12$, where $(0<α<1, 0< β<\frac{1}{2})$. These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of Itô formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies.
