Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation.

# Past Stochastic Analysis Seminar

We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention.

We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.

SPDEs with Lévy noise can be used to model chemical, physical or biological phenomena which contain uncertainties. When discretising these SPDEs in order to solve them numerically the problem might be of large order. The goal is to save computational time by replacing large scale systems by systems of low order capturing the main information of the full model. In this talk, we therefore discuss balancing related MOR techniques. We summarise already existing results and discuss recent achievements.

It is well-known that the stochastic heat equation on R^n has a Hölder continuous function-valued solution in the case n=1, and that in dimensions 2 and above the solution is not function-valued but is forced to take values in some wider space of distributions. So what happens if the space has, in some sense, a dimension in between 1 and 2? We turn to the theory of fractals in order to answer this question. It has been shown (Kigami, 2001) that there exists a class of self-similar sets on which natural Laplacians can be defined, and so an analogue to the stochastic heat equation can be posed. In this talk we cover the following questions: Is the solution to this equation function-valued? If so, is it Hölder continuous? To answer the latter we must first prove an analogue of Kolmogorov's celebrated continuity theorem for the self-similar sets that we are working on. Joint work with Ben Hambly.

We give an explicit scheme to reconstruct any C^1 curve from its signature. It is implementable and comes with detailed stability properties. The key of the inversion scheme is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales. Joint work with Terry Lyons.

The Regularity Structures introduced by Martin Hairer allow us to describe the solution of a singular SPDEs by a Taylor expansion with new monomials. We present the two Hopf Algebras used in this theory for defining the structure group and the renormalisation group. We will point out the importance of recursive formulae with twisted antipodes.

Signature of a path provides a top down summary of the path as a driving signal. There have been substantial recent progress in reconstructing paths from its signature, (Lyons-Xu 2016, Geng 2016). In this talk, we focus on obtaining certain quantitative features of paths from their signatures. Hambly-Lyons' showed that the normalized limit of signature gives the length of a C^3 path. The result was recently extended by Lyons-Xu to C^1 paths. The extension of this result to bounded variation paths remains open. We will discuss this open problem.

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions and whose diffusivity has a sub-Riemannian structure. In the case where the endpoints agree, we discuss the convergence of the suitably rescaled fluctuations to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

Malliavin calculus provides a framework to differentiate functionals defined on a Gaussian probability space with respect to the underlying noise. This allows to develop analysis on path space with infinite-dimensional generalisations of Fourier analysis, Sobolev spaces, etc from R^d. In this talk, we attempt to build a Lipschitz à la E. M. Stein (as opposed to Sobolev) function theory on rough path space. This framework allows to pathwise differentiate functionals on rough paths with respect to the underlying rough path. Time permitting, we show how to obtain Feynman-Kac-type representations for solutions to some high-order (>2) linear parabolic equations on R^d.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1} = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1} = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$. Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$. Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems \begin{align*} \dot x = \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y = g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.