Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and therefore allow for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals.
The coefficients of these power series are solutions of Riccati and linear ordinary differential equations (ODEs) with values in the extended tensor algebra, respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes that is universal (in a sense specified in the introduction) within Itô-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmet…
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universitat de Barcelona（巴塞罗那大学）
(Conditional) Generative Adversarial Networks (GANs) have found great success in recent years, due to their ability to approximate (conditional) distributions over extremely high dimensional spaces. However, they are highly unstable and computationally expensive to train, especially in the time series setting. Recently, it has been proposed the use of a key object in rough path theory, called the signature of a path, which is able to convert the min-max formulation given by the (conditional) GAN framework into a classical minimization problem. However, this method is extremely expensive in ter…
我们对相当一般的状态空间提出了一个完整的矩爆炸理论。这包括仿射变换公式的有效性方面的最小解的广义里卡蒂微分方程系统的表征。同时，我们刻画了这类半正定过程，并证明了WishartSDEs的弱解和强解的存在性.作为应用，我们回答了M. L. Eaton关于非中心Wishart分布的最大参数域。本文的最后一章包括三个关于仿射模型的独立工作，如指数仿射过程的鞅性质的刻画，半正定锥上过程的跳行为的研究，以及加权希尔伯特空间中多元仿射跳扩散的转移密度及其逼近论的存在性结果。
There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general Itô processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz con…
