Journal ArticleDOI
Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes
Nelson Vadori,Anatoliy Swishchuk +1 more
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In this article, the authors provide strong law of large numbers and central limit theorem results for inhomogeneous Markov chains with respect to a continuous-time Markov chain model.Abstract:
Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in...read more
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Journal ArticleDOI
General Compound Hawkes Processes in Limit Order Books
Anatoliy Swishchuk,Aiden Huffman +1 more
TL;DR: In this article, a Law of Large Numbers and a Functional Central Limit Theorem (FCLT) are applied to limit order books to study the link between price volatility and order flow, where the volatility in mid price changes is expressed in terms of parameters describing the arrival rates and mid-price process.
Journal ArticleDOI
A Semi-Markovian Modeling of Limit Order Markets
Anatoliy Swishchuk,Nelson Vadori +1 more
TL;DR: Cont and de Larrard as mentioned in this paper extended their framework to arbitrary distributions for book events interarrival times (possibly nonexponential) and both the nature of a new book event and its corresponding inter-rival time depend on the characteristics of the previous book event.
Journal ArticleDOI
Policyholder cluster divergence based differential premium in diabetes insurance
TL;DR: In this article, a differential premium approach based on policyholder cluster divergence was proposed by employing fuzzy c-means algorithm with an extended initial multistate Markov model to formulate the differential premium that matches the policyholder's risk category.
Journal ArticleDOI
General semi-markov model for limit order books
Anatoliy Swishchuk,Tyler Hofmeister,Katharina Cera,Katharina Cera,Julia Schmidt,Julia Schmidt +5 more
TL;DR: In this paper, the authors considered a general semi-Markov model for limit order books with two states that incorporates price changes that are not fixed to one tick, and they introduced an even more general case of the semi Markov model that incorporates an arbitrary number of states for the price changes.
Reference EntryDOI
Limit Theorems for Markov Renewal Processes
TL;DR: In this paper, the authors define Markov renewal type equations and introduce the key renewal theorem, which is used in solving Markov renew-type equations in the limit, and demonstrate how the theorem can be used over an example.
References
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Book
Markov Processes: Characterization and Convergence
TL;DR: In this paper, the authors present a flowchart of generator and Markov Processes, and show that the flowchart can be viewed as a branching process of a generator.
Book
Semi-Markov Processes and Reliability
Nikolaos Limnios,G. Oprisan +1 more
TL;DR: The theory of Markov processes has its origins in the studies by A. A. Markov (1856-1922) of sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian mo- tion.
Journal ArticleDOI
Limit Theorems for Markov Renewal Processes
Ronald Pyke,Ronald A. Schaufele +1 more
TL;DR: In this paper, the authors studied Doeblin Ratio limit laws, the weak and strong laws of large numbers, and the Central Limit theorem for Markov Renewal processes for the special case of a Markov chain.