arXiv subjects

Mariusz Niewęgłowski

Publications and source records attributed to Mariusz Niewęgłowski.

7 recordsLinked to original sources

arXiv2026

Loglinear Hawkes processes: existence, nonexplosion, stability and moment bounds

This paper discusses a special class of nonlinear Hawkes processes, where the rate function is the exponential function. We call these processes loglinear Hawkes processes. Such processes are used with success in computational neuroscience and we provide a mathematical background for them. After obtaining existence from a standard Poisson embedding, we give sufficient conditions for explosion and nonexplosion that cover a large class of memory functions. In order to give sufficient stability conditions, we construct an autoregressive integer-valued time series that gives pathwise upper bounds for the bin counts of a loglinear Hawkes process. From here, a Markov chain argument shows that the regeneration time for a loglinear Hawkes process has finite expectation if the memory function is negative enough in a neighborhood of zero. Stability in total variation follows. Finally, we show that both the loglinear Hawkes process under a large class of initial conditions and its stationary version have finite exponential moments.

math.PR↗

arXiv2019

Semimartingales and Shrinkage of Filtration

We consider a complete probability space $(Ω,\mathcal{F},\mathbb{P})$, which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed to satisfy the usual conditions and such that $\mathbb{F} \subset \mathbb{G}$. On this probability space we consider a real valued special $\mathbb{G}$-semimartingale $X$. The purpose of this work is to study the following two problems: A. If $X$ is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale characteristics of $X$ in terms of the $\mathbb{G}$-semimartingale characteristics of $X$. B. If $X$ is not $\mathbb{F}$-adapted, given that the $\mathbb{F}$-optional projection of $X$ is a special semimartingale, compute the $\mathbb{F}$-semimartingale characteristics of $\mathbb{F}$-optional projection of $X$ in terms of the $\mathbb{G}$-canonical decomposition and $\mathbb{G}$-semimartingale characteristics of $X$.

math.PR↗

arXiv2015

Conditional Markov Chains Revisited Part I: Construction and properties

In this paper we continue the study of conditional Markov chains (CMCs) with finite state spaces, that we initiated in Bielecki, Jakubowski and Niewęgłowski (2014a) in an effort to enrich the theory of CMCs that was originated in Bielecki and Rutkowski (2004). We provide an alternative definition of a CMC and an alternative construction of a CMC via a change of probability measure. It turns out that our construction produces CMCs that are also doubly stochastic Markov chains (DSMCs), which allows for study of several properties of CMCs using tools available for DSMCs.

math.PR↗

arXiv2015

Conditional Markov Chains Part II: Consistency and Copulae

In this paper we continue the study of conditional Markov chains (CMCs) with finite state spaces, that we initiated in Bielecki, Jakubowski and Niewęgłowski (2015). Here, we turn our attention to the study of Markov consistency and Markov copulae with regard to CMCs, and thus we follow up on the study of Markov consistency and Markov copulae for ordinary Markov chains that we presented in Bielecki, Jakubowski and Niewęgłowski (2013).

math.PR↗

arXiv2013

Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDE

At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process and compensated Poisson random measure and claims depend on rating. To find a tool to calculate hedging strategy we prove a Feynman-Kac type theorem. This result is of independent interest and has many applications, since it enables to calculate some conditional expectations using related PIDE's. We illustrate our theory on two examples of market. The first is a general exponential Lévy model with stochastic volatility, and the second is a generalization of exponential Lévy model with regime-switching.

q-fin.PR↗

arXiv2013

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions to SDE's. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate Markov property. To prove uniqueness we solve a general martingale problem for \cadlag processes. This result is of independent interest. In the last section we present application of our results considering generalized exponential Levy model.

math.PR↗

arXiv2013

Intricacies of Dependence between Components of Multivariate Markov Chains: Weak Markov Consistency and Markov Copulae

This article continues our study of Markovian consistency and Markov copulae. In particular, we characterize the weak Markovian consistency for finite Markov chains. We discuss some aspects of dependence between the components of a multivariate Markov chain in the context of weak Markovian consistency and strong Markovian consistency. In this connection, we also introduce and discuss the concept of weak Markov copulae.

math.PR↗