Optimal projection filters with information geometry

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo, Hanzon and Le Gland (1998, 1999) [19, 20], where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (2016) [6] where a direct L2 metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong, Brigo and Rossi Ferrucci (2019, 2018) [10, 9], based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges

Rough differential equations driven by signals in Besov spaces

Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in H\"older and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is extended from H\"older to Besov spaces. As an application we solve stochastic differential equations driven by random functions in Besov spaces and Gaussian processes in a pathwise sense.Comment: Former title: "Rough differential equations on Besov spaces", 37 page

Itô Stochastic Differentials

We give an infinitesimal meaning to the symbol dXt for a continuous semimartingale X at an instant in time t. We define a vector space structure on the space of differentials at time t and deduce key properties consistent with the classical Itô integration theory. In particular, we link our notion of a differential with Itô integration via a stochastic version of the Fundamental Theorem of Calculus. Our differentials obey a version of the chain rule, which is a local version of Itô’s lemma. We apply our results to financial mathematics to give a theory of portfolios at an instant in time

Controlled viscosity solutions of fully nonlinear rough PDEs

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers

Projections of SDEs onto Submanifolds

In [AB16] the authors define three projections of $\mathbb R^d$-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, It\^o-vector and It\^o-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient $\mathbb R^d$-coordinates. We use these to show that the It\^o-vector and It\^o-jet projections satisfy respectively a weak and mean-square optimality criterion "for small t": this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in [AB16], which is formulated in terms of weak and strong It\^o-Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality

Focus on stochastic flows and climate statistics

The atmosphere and ocean are examples of dynamical systems that evolve in accordance with the laws of physics. Therefore, climate science is a branch of physics that is just as valid and important as the more traditional branches, which include particle physics, condensed-matter physics, and statistical mechanics. This 'focus on' collection of New Journal of Physics brings together original research articles from leading groups that advance our understanding of the physics of climate. Areas of climate science that can particularly benefit from input by physicists are emphasised. The collection brings together articles on stochastic models, turbulence, quasi-linear approximations, climate statistics, statistical mechanics of atmospheres and oceans, jet formation, and reduced-form climate models. The hope is that the issue will encourage more physicists to think about the climate problem