arXiv 2312.097792023-12-15math.PR

一维随机微分方程解的凸序

Convex ordering of solutions to one-dimensional SDEs

Benjamin Jourdain, Gilles Pagès

AI总结：

本文研究一维布朗SDE强解的凸性传播与解的凸序，通过欧拉离散格式证明扩散系数空间凸性对多时刻及路径凸泛函凸序的作用，给出可避免凸性套利的可计算近似方法。

摘要：

本文研究空间变量满足Lipschitz条件且关于时间变量一致的一维布朗随机微分方程（SDE）强解的凸性传播，以及两类此类方程解之间的凸序关系。我们证明，这些性质仅在单个时刻对过程的凸函数成立时无需额外假设，但若要推广至两个时刻的凸函数，则需要一个几乎等价于扩散系数空间凸性的假设。在扩散系数满足该空间凸性的条件下，这两个性质甚至对整条路径的凸泛函都成立。对于方向凸泛函，扩散系数的空间凸性则不再是必需的。我们的证明方法是先对欧拉型时间离散格式建立相关结果，再将其推广到极限布朗扩散过程。由此我们得到了一类近似方法，它们通过保持凸性传播和比较性来避免“凸性套利”，且可通过蒙特卡洛模拟计算。

英文摘要：

In this paper, we are interested in the propagation of convexity by the strong solution to a one-dimensional Brownian stochastic differential equation with coefficients Lipschitz in the spatial variable uniformly in the time variable and in the convex ordering between the solutions of two such equations. We prove that while these properties hold without further assumptions for convex functions of the processes at one instant only, an assumption almost amounting to spatial convexity of the diffusion coefficient is needed for the extension to convex functions at two instants. Under this spatial convexity of the diffusion coefficients, the two properties even hold for convex functionals of the whole path. For directionally convex functionals, the spatial convexity of the diffusion coefficient is no longer needed. Our method of proof consists in first establishing the results for time discretization schemes of Euler type and then transferring them to their limiting Brownian diffusions. We thus exhibit approximations which avoid {\em convexity arbitrages} by preserving convexity propagation and comparison and can be computed by Monte Carlo simulation.

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