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- The fractional volatility model : no-arbitrage, leverage and completenessPublication . Mendes, Rui Vilela; Oliveira, Maria João; Rodrigues, A. M.When the volatility process is driven by fractional noise one obtains a model which is consistent with the empirical market data. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behaviors. Here, the no-arbitrage and completeness properties of the models are rigorously studied.
- Self-repelling fractional Brownian motion : a generalized Edwards model for chain polymersPublication . Bornales, Jinky; Oliveira, Maria João; Streit, LudwigWe present an extension of the Edwards model for conformations of individual chain molecules in solvents in terms of fractional Brownian motion, and discuss the excluded volume effect on the end-to-end length of such trajectories or molecules.
- Polymer measure: varadhan's renormalization revisitedPublication . Bock, Wolfgang; Oliveira, Maria João; Silva, José Luís da; Streit, LudwigThrough chaos decomposition we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion.
- Markov evolutions and hierarchical equations in the continuum. I: one-component systemsPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoGeneral birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.
- Glauber dynamics in the continuum via generating functionals evolutionPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoWe construct the time evolution for states of Glauber dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, leading to a local (in time) solution which, under certain initial conditions, might be extended to a global one. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.
- Markov evolutions and hierarchical equations in the continuum. II: multicomponent systemsPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoGeneral birth-and-death as well as hopping stochastic dynamics of infinite multicomponent particle systems in the continuum are considered. We derive the corresponding evolution equations for quasi-observables and correlation functions. We also present sufficient conditions that allow us to consider these equations on suitable Banach spaces.
- An infinite dimensional umbral calculusPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Lytvynov, Eugene; Oliveira, Maria JoãoThe aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.
- Kawasaki dynamics in the continuum via generating functionals evolutionPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoWe construct the time evolution of Kawasaki dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.
- Stirling operators in spatial combinatoricsPublication . Finkelshtein, Dmitri; Kondratiev, Yuri; Lytvynov, Eugene; Oliveira, Maria João
- Stochastic and infinite dimensional analysisPublication . Bernido, Christopher C.; Carpio-Bernido, Maria Victoria; Grothaus, Martin; Kuna, Tobias; Oliveira, Maria João; Silva, José Luís daThis volume presents a collection of papers covering applications from a wide range of systems with infinitely many degrees of freedom studied using techniques from stochastic and infinite dimensional analysis, e.g. Feynman path integrals, the statistical mechanics of polymer chains, complex networks, and quantum field theory. Systems of infinitely many degrees of freedom create their particular mathematical challenges which have been addressed by different mathematical theories, namely in the theories of stochastic processes, Malliavin calculus, and especially white noise analysis. These proceedings are inspired by a conference held on the occasion of Prof. Ludwig Streit’s 75th birthday and celebrate his pioneering and ongoing work in these fields.