论文标题
随机Volterra方程及其Euler方案的凸点顺序
Convex ordering for stochastic Volterra equations and their Euler schemes
论文作者
论文摘要
在本文中，我们有兴趣在连续$ \ r^d $可估算的路径和单调凸订单时，将解决方案与随机伏尔泰方程进行比较凸订单的随机Volterra方程。即使通常这些解决方案既不是半木星也不是马尔可夫的过程，我们也能够在其系数上表现出可以进行比较的条件。我们的方法在于首先比较他们的欧拉计划，然后随着时间步长消失而采取限制。我们认为两种类型的Euler方案取决于Volterra内核的离散方式。确保第一个方案比较比较的条件比第二个方案稍弱，这是收敛的另一种方式。此外，我们扩展了存在的起始值所需的集成性，并且收敛导致文献中只能假设有限的一阶矩，这是凸订单的自然框架。
In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous $\R^d$-valued paths and for the monotonic convex order when $d=1$. Even if in general these solutions are neither semi-martingales nor Markov processes, we are able to exhibit conditions on their coefficients enabling the comparison. Our approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. We consider two types of Euler schemes depending on the way the Volterra kernels are discretized. The conditions ensuring the comparison are slightly weaker for the first scheme than for the second one and this is the other way round for convergence. Moreover, we extend the integrability needed on the starting values in the existence and convergence results in the literature to be able to only assume finite first order moments, which is the natural framework for convex ordering.
