On small time asymptotics for rough differential equations driven by fractional Brownian motions

We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of Peter Laurenc

Pathwise McKean-Vlasov Theory with Additive Noise

Abstract. We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 10, 40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, c\ue0dl\ue0g noise, and reflecting boundaries. Last not least, we generalize Dawson-G\ue4rtner large deviations and the central limit theorem to a non-Brownian noise setting

Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces II

The strong convergence of Wong-Zakai approximations of the solution to the reflecting stochastic differential equations was studied in [2]. We continue the study and prove the strong convergence under weaker assumptions on the domain.Comment: To appear in "Stochastic Analysis and Applications 2014-In Honour of Terry Lyons", Springer Proceedings in Mathematics and Statistic

Flows driven by Banach space-valued rough paths

We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.Comment: 8 page

Investigation on Dabigatran Etexilate and Worsening of Renal Function in Patients with Atrial fibrillation : the IDEA Study

BACKGROUND AND OBJECTIVES: Warfarin-related nephropathy is an unexplained acute kidney injury, and may occur in patients with supratherapeutic INR, in the absence of overt bleeding. Similar findings have been observed in rats treated with dabigatran etexilate. We conducted a prospective study in dabigatran etexilate-treated patients to assess the incidence of dabigatran-related nephropathy and to investigate the possible correlation between dabigatran plasma concentration (DPC) and worsening renal function. METHOD: One hundred and seven patients treated long term with dabigatran etexilate for non-valvular atrial fibrillation (NVAF) were followed up for 90 days. DPC, serum creatinine (SCr) and serum cystatin C were prospectively measured. Ninety five patients had complete follow-up data and were evaluable for primary endpoint. RESULTS: Eleven patients had supratherapeutic DPC, defined as DPC higher than 200 ng/ml at study enrolment, but at the end of follow-up no patient showed a persistent increase in SCr. No patients experienced acute kidney injury. CONCLUSIONS: Our study shows that no persistent renal detrimental effect is associated with dabigatran treatment. An increase in SCr during dabigatran treatment is reversible and it seems to be unrelated to dabigatran itself

Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201

Continuous Equilibrium in Affine and Information-Based Capital Asset Pricing Models

We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents' optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents' risk aversion on the implied volatility of simultaneously-traded European-style options.Comment: 24 pages, 4 figure

From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index α∈(1/8,1/4)\alpha\in(1/8,1/4)

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area

Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadha

Planar digital nanoliter dispensing system based on thermocapillary actuation

We provide guidelines for the design and operation of a planar digital nanodispensing system based on thermocapillary actuation. Thin metallic microheaters embedded within a chemically patterned glass substrate are electronically activated to generate and control 2D surface temperature distributions which either arrest or trigger liquid flow and droplet formation on demand. This flow control is a consequence of the variation of a liquid’s surface tension with temperature, which is used to draw liquid toward cooler regions of the supporting substrate. A liquid sample consisting of several microliters is placed on a flat rectangular supply cell defined by chemical patterning. Thermocapillary switches are then activated to extract a slender fluid filament from the cell and to divide the filament into an array of droplets whose position and volume are digitally controlled. Experimental results for the power required to extract a filament and to divide it into two or more droplets as a function of geometric and operating parameters are in excellent agreement with hydrodynamic simulations. The capability to dispense ultralow volumes onto a 2D substrate extends the functionality of microfluidic devices based on thermocapillary actuation previously shown effective in routing and mixing nanoliter liquid samples on glass or silicon substrates