Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
https://vmsta.org/https://vmsta.org/Latest articles of Modern Stochastics: Theory and Applications
http://vmsta.org/journal/VMSTA/feeds/latest
enFri, 01 Dec 2023 22:47:03 +0200<![CDATA[A quantitative functional central limit theorem for shallow neural networks]]>
https://vmsta.org/journal/VMSTA/article/280
https://vmsta.org/journal/VMSTA/article/280We prove a quantitative functional central limit theorem for one-hidden-layer neural networks with generic activation function. Our rates of convergence depend heavily on the smoothness of the activation function, and they range from logarithmic for nondifferentiable nonlinearities such as the ReLu to $\sqrt{n}$ for highly regular activations. Our main tools are based on functional versions of the Stein–Malliavin method; in particular, we rely on a quantitative functional central limit theorem which has been recently established by Bourguin and Campese [Electron. J. Probab. 25 (2020), 150].
PDFXML]]>We prove a quantitative functional central limit theorem for one-hidden-layer neural networks with generic activation function. Our rates of convergence depend heavily on the smoothness of the activation function, and they range from logarithmic for nondifferentiable nonlinearities such as the ReLu to $\sqrt{n}$ for highly regular activations. Our main tools are based on functional versions of the Stein–Malliavin method; in particular, we rely on a quantitative functional central limit theorem which has been recently established by Bourguin and Campese [Electron. J. Probab. 25 (2020), 150].
PDFXML]]>Valentina Cammarota,Domenico Marinucci,Michele Salvi,Stefano VigognaTue, 28 Nov 2023 00:00:00 +0200<![CDATA[Author index]]>
https://vmsta.org/journal/VMSTA/article/277
https://vmsta.org/journal/VMSTA/article/277PDF XML]]>PDF XML]]>Fri, 13 Oct 2023 00:00:00 +0300<![CDATA[Keywords index]]>
https://vmsta.org/journal/VMSTA/article/278
https://vmsta.org/journal/VMSTA/article/278PDF XML]]>PDF XML]]>Fri, 13 Oct 2023 00:00:00 +0300<![CDATA[2010 Mathematics Subject Classification index]]>
https://vmsta.org/journal/VMSTA/article/279
https://vmsta.org/journal/VMSTA/article/279PDF XML]]>PDF XML]]>Fri, 13 Oct 2023 00:00:00 +0300<![CDATA[BDG inequalities and their applications for model-free continuous price paths with instant enforcement]]>
https://vmsta.org/journal/VMSTA/article/276
https://vmsta.org/journal/VMSTA/article/276Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of “typical” price paths. In this paper an outer measure on the space $[0,+\infty )\times \Omega $ is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô’s isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement.
PDFXML]]>Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of “typical” price paths. In this paper an outer measure on the space $[0,+\infty )\times \Omega $ is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô’s isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement.
PDFXML]]>Rafał Marcin ŁochowskiWed, 04 Oct 2023 00:00:00 +0300<![CDATA[Variance Gamma (nonlocal) equations]]>
https://vmsta.org/journal/VMSTA/article/275
https://vmsta.org/journal/VMSTA/article/275Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed.
PDFXML]]>Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed.
PDFXML]]>Fausto ColantoniMon, 25 Sep 2023 00:00:00 +0300<![CDATA[Critical branching processes in a sparse random environment]]>
https://vmsta.org/journal/VMSTA/article/274
https://vmsta.org/journal/VMSTA/article/274We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.
PDFXML]]>We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.
PDFXML]]>Dariusz Buraczewski,Congzao Dong,Alexander Iksanov,Alexander MarynychTue, 29 Aug 2023 00:00:00 +0300<![CDATA[Perpetual cancellable American options with convertible features]]>
https://vmsta.org/journal/VMSTA/article/273
https://vmsta.org/journal/VMSTA/article/273The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.
PDFXML]]>The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.
PDFXML]]>Tsvetelin ZaevskiFri, 04 Aug 2023 00:00:00 +0300<![CDATA[Multi-mixed fractional Brownian motions and Ornstein–Uhlenbeck processes]]>
https://vmsta.org/journal/VMSTA/article/272
https://vmsta.org/journal/VMSTA/article/272The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.
PDFXML]]>The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.
PDFXML]]>Hamidreza Maleki Almani,Tommi SottinenMon, 26 Jun 2023 00:00:00 +0300<![CDATA[On geometric recurrence for time-inhomogeneous autoregression]]>
https://vmsta.org/journal/VMSTA/article/271
https://vmsta.org/journal/VMSTA/article/271The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${X_{n+1}}={\alpha _{n}}{X_{n}}+{\varepsilon _{n}}$, where ${\alpha _{n}}$ are constants, and ${\varepsilon _{n}}$ are independent random variables. Conditions on ${\alpha _{n}}$ and distributions of ${\varepsilon _{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${X^{(1)}}$ and ${X^{(2)}}$ assuming that both ${\alpha _{n}^{(i)}}$, $i\in \{1,2\}$, and distributions of ${\varepsilon _{n}^{(i)}}$, $i\in \{1,2\}$, are close enough.
PDFXML]]>The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${X_{n+1}}={\alpha _{n}}{X_{n}}+{\varepsilon _{n}}$, where ${\alpha _{n}}$ are constants, and ${\varepsilon _{n}}$ are independent random variables. Conditions on ${\alpha _{n}}$ and distributions of ${\varepsilon _{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${X^{(1)}}$ and ${X^{(2)}}$ assuming that both ${\alpha _{n}^{(i)}}$, $i\in \{1,2\}$, and distributions of ${\varepsilon _{n}^{(i)}}$, $i\in \{1,2\}$, are close enough.
PDFXML]]>Vitaliy GolomoziyWed, 03 May 2023 00:00:00 +0300