Page 561 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 561
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
Financial markets with a memory
Yuliya Mishura, myus@univ.kiev.ua
Taras Shevchenko National University of Kyiv, Ukraine
We present general conditions for the weak convergence of a discrete-time additive scheme to
a stochastic process with memory in the space D[0, T ]. Then we investigate the convergence
of the related multiplicative scheme to a process that can be interpreted as an asset price with
memory. As an example, we study an additive scheme that converges to fractional Brownian
motion, which is based on the Cholesky decomposition of its covariance matrix. The second
example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The
multiplicative counterparts for these two schemes are also considered. As an auxiliary result
of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the
Cholesky decomposition of the covariance matrix of a stationary Gaussian process.
On some fractional queues
Enrica Pirozzi, enrica.pirozzi@unina.it
Dipartimento di Matematica e Applicazioni,
Università di Napoli Federico II, Napoli, Italy
Coauthors: Giacomo Ascione, Nikolai Leonenko
We show some recents advances in the study of fractional queueing models such as fractional
M/M/1 queues and fractional Erlang queues M/Ek/1. We also focus on a fractional M/M/1
queue with catastrophes. Starting from fractional M/M/1 queues, we study the transient be-
haviour, in which the time-change plays a key role. An alternative expression for the transient
distribution of the fractional M/M/1 model is provided. The state probabilities for the fractional
queue with catastrophes, the distributions of the busy period for fractional queues without and
with catastrophes and the distribution of the time of the first occurrence of a catastrophe are also
obtained.
Furthermore, we introduce a fractional generalization of the Erlang Queues M/Ek/1. Such
process is obtained through a time-change via inverse stable subordinator of the classical queue
process. The fractional Kolmogorov forward equation for such process is considered, then we
use such equation to obtain an interpretation of this process in the queuing theory context. We
give some results such as the transient state probabilities and some features of this fractional
queue model, the mean queue length, the distribution of the busy periods and some conditional
distributions of the waiting times.
Finally, we also show some results of the study of a fractional M/M/∞ queueing system
constructed as a suitable time–changed birth–death process.
On recent advancement in limit theory for fractional type processes
Mark Podolskij, mpodolskij@math.au.dk
Aarhus University, Denmark
In this talk we review some recent results on limit theorems for fractional type Levy driven
processes. In particular, we will discuss central and non-central limit theorems for statistics of
559
PROCESSES (MS-18)
Financial markets with a memory
Yuliya Mishura, myus@univ.kiev.ua
Taras Shevchenko National University of Kyiv, Ukraine
We present general conditions for the weak convergence of a discrete-time additive scheme to
a stochastic process with memory in the space D[0, T ]. Then we investigate the convergence
of the related multiplicative scheme to a process that can be interpreted as an asset price with
memory. As an example, we study an additive scheme that converges to fractional Brownian
motion, which is based on the Cholesky decomposition of its covariance matrix. The second
example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The
multiplicative counterparts for these two schemes are also considered. As an auxiliary result
of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the
Cholesky decomposition of the covariance matrix of a stationary Gaussian process.
On some fractional queues
Enrica Pirozzi, enrica.pirozzi@unina.it
Dipartimento di Matematica e Applicazioni,
Università di Napoli Federico II, Napoli, Italy
Coauthors: Giacomo Ascione, Nikolai Leonenko
We show some recents advances in the study of fractional queueing models such as fractional
M/M/1 queues and fractional Erlang queues M/Ek/1. We also focus on a fractional M/M/1
queue with catastrophes. Starting from fractional M/M/1 queues, we study the transient be-
haviour, in which the time-change plays a key role. An alternative expression for the transient
distribution of the fractional M/M/1 model is provided. The state probabilities for the fractional
queue with catastrophes, the distributions of the busy period for fractional queues without and
with catastrophes and the distribution of the time of the first occurrence of a catastrophe are also
obtained.
Furthermore, we introduce a fractional generalization of the Erlang Queues M/Ek/1. Such
process is obtained through a time-change via inverse stable subordinator of the classical queue
process. The fractional Kolmogorov forward equation for such process is considered, then we
use such equation to obtain an interpretation of this process in the queuing theory context. We
give some results such as the transient state probabilities and some features of this fractional
queue model, the mean queue length, the distribution of the busy periods and some conditional
distributions of the waiting times.
Finally, we also show some results of the study of a fractional M/M/∞ queueing system
constructed as a suitable time–changed birth–death process.
On recent advancement in limit theory for fractional type processes
Mark Podolskij, mpodolskij@math.au.dk
Aarhus University, Denmark
In this talk we review some recent results on limit theorems for fractional type Levy driven
processes. In particular, we will discuss central and non-central limit theorems for statistics of
559
