Related papers: From Markov Processes to Semimartingales
We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic…
In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on…
Semi-Markov processes play an important role in the effective description of partially accessible systems in stochastic thermodynamics. They occur, for instance, in coarse-graining procedures such as state lumping and when analyzing waiting…
Mean-field models approximate large stochastic systems by simpler differential equations that are supposed to approximate the mean of the larger system. It is generally assumed that as the stochastic systems get larger (i.e., more people or…
This paper contributes to the study of stochastic processes of the class $(\Sigma)$. First, we extend the notion of the above-mentioned class to c\`adl\`ag semi-martingales, whose finite variational part is considered c\`adl\`ag instead of…
The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at…
The non-Markovianity of the stochastic process called the quantum semi-Markov (QSM) process is studied using a recently proposed quantification of memory based on the deviation from semigroup evolution, that provides a unified description…
This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov…
The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive n-th order…
In this paper we consider (probability-)measure valued processes, which we call MVMs, which have a natural martingale structure. Following previous work of Eldan and Cox-K\"allblad, these processes are known to have a close connection to…
Building upon the rule-algebraic stochastic mechanics framework, we present new results on the relationship of stochastic rewriting systems described in terms of continuous-time Markov chains, their embedded discrete-time Markov chains and…
Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\geq0}$ be the associated Markov process. For $u\in D(\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\tilde{u}(X_{t})-\tilde{u}(X_{0})$ and…
Various and ubiquitous information systems are being used in monitoring, exchanging, and collecting information. These systems are generating massive amount of event sequence logs that may help us understand underlying phenomenon. By…
In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter $\varepsilon$. Understanding the asymptotic behavior of such processes is needed in order to study the…
In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite…
Given a semi-Markov law, using an additional parameter, we consider a family of stochastic flows corresponding to that law. Then we suitably select a particular flow, for which we obtain expressions of the meeting and merging probabilities…
We introduce the concept evolutionary semigroups on path spaces, generalizing the notion of transition semigroups to possibly non-Markovian stochastic processes. We study the basic properties of evolutionary semigroups and, in particular,…
We define dynamic treatment regimes and associated potential outcomes for data described by marked point processes (MPPs). These definitions motivate MPP analogues of the commonly used consistency, exchangeability, and positivity conditions…
A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative…
Complex systems may often be characterized by their hierarchical dynamics. In this paper do we present a method and an operational algorithm that automatically infer this property in a broad range of systems; discrete stochastic processes.…