A tree approach to $p$-variation and to integration

Pathwise stochastic integration with finite variation processes uniformly approximating càdlàg processes

For any real-valued stochastic process $X$ with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-…

Quadratic variation, p-variation and integration with applications to stock price modelling

The paper develops a calculus for a class of real-valued functions having a quadratic variation. The main result is a solution of the representation problem for a class of evolutions having a quadratic variation. The result is applied to build up an asset pricing model. Also in the paper there are some results concerning an extension of the class of all semimartingales.

Asymptotic results on Hoppe trees and its variations

A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees. The generalization provides diversity among the nodes, making the model more flexible for applications. We also analyze the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weigh…

Continuous and Tractable models for the Variation of Evolutionary Rates

We propose a continuous model for evolutionary rate variation across sites and over the tree and derive exact transition probabilities under this model. Changes in rate are modelled using the CIR process, a diffusion widely used in financial applications. The model directly extends the standard gamma distributed rates across site model, with one additional parameter governing changes in rate down the tree. The parameters of the model can be estimated directly from two well-known statistics: the index of dispersion and the gamma shape parameter of the rates across sites model. The CIR model can…

A superhedging approach to stochastic integration

Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is develo…

Pathwise integration with respect to paths of finite quadratic variation

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pa…

随机森林中需要多少棵树？一种结合平台搜索与Optuna集成的重新审视方法

How Many Trees in a Random Forest? A Revisited Approach with Plateau Search and Optuna Integration

提出一种基于三元组平台搜索的算法，通过监控袋外分数的相对变化自动确定随机森林的树数量，避免预设搜索范围，并提供了理论分析和实验验证。

Stochastic integration with respect to additive functionals of zero quadratic variation

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an Itô formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.

关于圆映射的连续统实树及其图像

On continuum real trees of circle maps and their graphs

本文通过填充研究给出Picard定理的简短直接证明，建立变差与变差指数的等价概念，并将连续统树的维数理论与其轮廓函数图像的维数理论联系起来。

Some notes on trees and paths

These notes cover background material on trees which are used in the paper `On uniqueness of the signature of a path of variation and the reduced path group'.