Abstract

Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class $\Sigma (H)$. The second part is devoted to the construction of solutions for the homogeneous skew Brownian motion equation and for the inhomogeneous skew Brownian motion equation. To do this, our ingredients are the techniques and results developed in the first part that we apply on some stochastic processes borrowed from the theory of stochastic calculus for signed measures. Our methods are inspired by those used by Bouhadou and Ouknine in [2013]. Moreover, their solution of the inhomogeneous skew Brownian motion equation is a particular case of those we propose in this paper.

摘要

本论文的贡献由两部分组成。在第一个中，我们为带符号测度的随机微积分理论做出了贡献。例如，我们提供了一些结果，允许刻画鞅和布朗运动的特征，这两个运动都是在有符号度量下定义的。我们还证明了一致可积的​​鞅（根据有符号测度定义）可以表示为相对鞅，我们为类的研究提供了一些新的结果$\Sigma (H)$. 第二部分致力于齐次偏斜布朗运动方程和非齐次偏斜布朗运动方程解的构造。为此，我们的成分是在第一部分中开发的技术和结果，我们将这些技术和结果应用于一些从随机微积分理论借来的有符号测度的随机过程。我们的方法受到 Bouhadou 和 Ouknine 在 [2013] 中使用的方法的启发。此外，他们对非齐次偏斜布朗运动方程的解是我们在本文中提出的特例。