## Publications

80 results found

Crisan D, Holm DD, Leahy J-M,
et al., 2022, Variational principles for fluid dynamics on rough paths, *arXiv*

In this paper, we introduce a new framework for parametrization schemes (PS)in GFD. Using the theory of controlled rough paths, we derive a class of roughgeophysical fluid dynamics (RGFD) models as critical points of rough actionfunctionals. These RGFD models characterize Lagrangian trajectories in fluiddynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps.Three constrained variational approaches are formulated for the derivation ofthese models. The first is the Clebsch formulation, in which the constraintsare imposed as rough advection laws. The second is the Hamilton-Pontryaginformulation, in which the constraints are imposed as right-invariant roughvector fields. The third is the Euler--Poincar\'e formulation in which thevariations are constrained. These variational principles lead directly to theLie--Poisson Hamiltonian formulation of fluid dynamics on geometric roughpaths. The GRP framework preserves the geometric structure of fluid dynamicsobtained by using Lie group reduction to pass from Lagrangian to Eulerianvariational principles, thereby yielding a rough formulation of the Kelvincirculation theorem. The rough-path variational approach includes non-Markovianperturbations of the Lagrangian fluid trajectories. In particular, memoryeffects can be introduced through this formulation through a judicious choiceof the rough path (e.g. a realization of a fractional Brownian motion). In thespecial case when the rough path is a realization of a semimartingale, werecover the SGFD models in Holm (2015). However, by eliminating the need forstochastic variational tools, we retain a pathwise interpretation of theLagrangian trajectories. In contrast, the Lagrangian trajectories in thestochastic framework are described by stochastic integrals which do not have apathwise interpretation. Thus, the rough path formulation restores thisproperty.

Lang O, Crisan D, 2022, Well-posedness for a stochastic 2D Euler equation with transport noise, *Stochastics and Partial Differential Equations: Analysis and Computations*, ISSN: 2194-0401

We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.

Crisan D, Holm DD, Street OD, 2021, Wave-current interaction on a free surface, *STUDIES IN APPLIED MATHEMATICS*, Vol: 147, Pages: 1277-1338, ISSN: 0022-2526

Beskos A, Crisan D, Jasra A,
et al., 2021, Score-based parameter estimation for a class of continuous-time state space models, *SIAM Journal on Scientific Computing*, ISSN: 1064-8275

We consider the problem of parameter estimation for a class ofcontinuous-time state space models. In particular, we explore the case of apartially observed diffusion, with data also arriving according to a diffusionprocess. Based upon a standard identity of the score function, we consider twoparticle filter based methodologies to estimate the score function. Bothmethods rely on an online estimation algorithm for the score function of$\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. Thefirst approach employs a simple Euler discretization and standard particlesmoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time,where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step.The second approach is new and based upon a novel diffusion bridgeconstruction. It yields a new backward type Feynman-Kac formula incontinuous-time for the score function and is presented along with a particlemethod for its approximation. Considering a time-discretization, the cost is$\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs,we then consider multilevel methodologies for the score function. We illustrateour parameter estimation method via stochastic gradient approaches in severalnumerical examples.

Cass T, Crisan D, Dobson P,
et al., 2021, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, *Electronic Journal of Probability*, Vol: 26, Pages: 1-72, ISSN: 1083-6489

We study the long time behaviour of a large class of diffusion processes on RN, generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hörmander Condition (HC). Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock. We demonstrate the importance of the class of UFG processes in several respects: i) we show that UFG processes constitute a family of SDEs which exhibit, in general, multiple invariant measures (i.e. they are in general non-ergodic) and for which one is able to describe a systematic procedure to study the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic HC are UFG processes, this paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic.

Street OD, Crisan D, 2021, Semi-martingale driven variational principles, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, ISSN: 1364-5021

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler-Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

Crisan D, Dobson P, Ottobre M, 2021, Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups, *Transactions of the American Mathematical Society*, Vol: 374, Pages: 3289-3330, ISSN: 0002-9947

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii).Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

Crisan D, Lang O, 2021, Local well-posedness for the great lake equation with transport noise, *Revue Roumaine de Mathematiques Pures et Appliquees*, Vol: 66, Pages: 131-155, ISSN: 0035-3965

This work is a continuation of the authors' work in [11]. In the equation satisfied by an incompressible uid with stochastic transport is analysed. Here we lift the incompressibility constraint. Instead we assume a weighted incompressibility condition. This condition is inspired by a physical model for a uid in a basin with a free upper surface and a spatially varying bottom topography (see [23]). Moreover, we assume a different form of the vorticity to stream function operator that generalizes the standard Biot-Savart operator which appears in the Euler equation. These two properties are exhibited in the physical model called the great lake equation. For this reason we refer to the model analysed here as the stochastic great lake equation. Just as in [11], the deterministic model is perturbed with transport type noise. The new vorticity to stream function operator generalizes the curl operator and it is shown to have good regularity properties. We also show that the initial smoothness of the solution is preserved. The arguments are based on constructing a family of viscous solutions which is proved to be relatively compact and to converge to a truncated version of the original equation. Finally, we show that the truncation can be removed up to a positive stopping time.

Akyildiz ÖD, Crisan D, Míguez J, 2020, Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization, *Statistics and Computing*, Vol: 30, Pages: 1645-1663, ISSN: 0960-3174

We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.

Cotter C, Crisan D, Holm DD, et al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Cotter C, Crisan D, Holm D, et al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

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Crisan D, López-Yela A, Miguez J, 2020, Stable approximation schemes for optimal filters, *SIAM-ASA Journal on Uncertainty Quantification*, Vol: 8, Pages: 483-509

A stable filter has the property that it asymptotically "forgets"initial perturbations. As a result of this property, it is possible to construct approximations of such filters whose errors remain small in time, in other words approximations that are uniformly convergent in the time variable. As uniform approximations are ideal from a practical perspective, finding criteria for filter stability has been the subject of many papers. In this paper, we seek to construct approximate filters that stay close to a given (possibly) unstable filter. Such filters are obtained through a general truncation scheme and, under certain constraints, are stable. The construction enables us to give a characterization of the topological properties of the set of optimal filters. In particular, we introduce a natural topology on this set, under which the subset of stable filters is dense.

Crisan D, Ortiz-Latorre S, 2019, A high order time discretization of the solution of the non-linear filtering problem, *Stochastics and Partial Differential Equations: Analysis and Computations*, ISSN: 2194-0401

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the m-power of the mesh of the partition for arbitrary m∈N. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.

Cotter C, Crisan D, Holm DD, et al., 2019, A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation, Publisher: arXiv

In this work, we apply a particle filter with three additional procedures(model reduction, tempering and jittering) to a damped and forcedincompressible 2D Euler dynamics defined on a simply connected bounded domain.We show that using the combined algorithm, we are able to successfullyassimilate data from a reference system state (the ``truth") modelled by ahighly resolved numerical solution of the flow that has roughly $3.1\times10^6$degrees of freedom for $10$ eddy turnover times, using modest computationalhardware. The model reduction is performed through the introduction of a stochasticadvection by Lie transport (SALT) model as the signal on a coarser resolution.The SALT approach was introduced as a general theory using a geometricmechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows onthe numerical implementation for SALT presented by Cotter et al, SIAMMultiscale Model. Sim. (2019) for the flow in consideration. The modelreduction is substantial: The reduced SALT model has $4.9\times 10^4$ degreesof freedom. Forecast reliability and estimated asymptotic behaviour of the particlefilter are also presented.

Crisan D, Flandoli F, Holm DD, 2019, Solution properties of a 3D stochastic euler fluid equation, *Journal of Nonlinear Science*, Vol: 29, Pages: 813-870, ISSN: 0938-8974

We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

Paulin D, Jasra A, Crisan DO,
et al., 2019, Optimization based methods for partially observed chaotic systems, *Foundations of Computational Mathematics*, Vol: 19, Pages: 485-559, ISSN: 1615-3375

In this paper we consider filtering and smoothing of partially observed chaotic dynamical systems that are discretely observed, with an additive Gaussian noise in the observation. These models are found in a wide variety of real applications and include the Lorenz 96’ model. In the context of a fixed observation interval T, observation time step h and Gaussian observation variance σ2Z, we show under assumptions that the filter and smoother are well approximated by a Gaussian with high probability when h and σ2Zh are sufficiently small. Based on this result we show that the maximum a posteriori (MAP) estimators are asymptotically optimal in mean square error as σ2Zh tends to 0. Given these results, we provide a batch algorithm for the smoother and filter, based on Newton’s method, to obtain the MAP. In particular, we show that if the initial point is close enough to the MAP, then Newton’s method converges to it at a fast rate. We also provide a method for computing such an initial point. These results contribute to the theoretical understanding of widely used 4D-Var data assimilation method. Our approach is illustrated numerically on the Lorenz 96’ model with state vector up to 1 million dimensions, with code running in the order of minutes. To our knowledge the results in this paper are the first of their type for this class of models.

Chassagneux JF, Crisan D, Delarue F, 2019, Numerical method for FBSDEs of McKean-Vlasov type, *Annals of Applied Probability*, Vol: 29, Pages: 1640-1684, ISSN: 1050-5164

© Institute of Mathematical Statistics, 2019 This paper is dedicated to the presentation and the analysis of a numerical scheme for forward-backward SDEs of the McKean-Vlasov type, or equivalently for solutions to PDEs on the Wasserstein space. Because of the mean field structure of the equation, earlier methods for classical forward-backward systems fail. The scheme is based on a variation of the method of continuation. The principle is to implement recursively local Picard iterations on small time intervals. We establish a bound for the rate of convergence under the assumption that the decoupling field of the forward-backward SDE (or equivalently the solution of the PDE) satisfies mild regularity conditions. We also provide numerical illustrations.

Barre J, Crisan DO, Goudon T, 2019, Two-dimensional pseudo-gravity model: particles motion in a non-potential singular force field, *Transactions of the American Mathematical Society*, Vol: 371, Pages: 2923-2962, ISSN: 0002-9947

We analyze a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self-consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation. We establish the existence of solutions under a suitable smallness condition on the total mass or, equivalently, for a sufficiently large diffusion coefficient. When a symmetry assumption is fulfilled, the solutions satisfy strengthened estimates (exponential moments). We also investigate the convergence of the $ N$-particles description towards the PDE system in the mean field regime.

Cotter CJ, Crisan D, Holm DD,
et al., 2019, Numerically modelling stochastic lie transport in fluid dynamics, *SIAM Journal on Scientific Computing*, Vol: 17, Pages: 192-232, ISSN: 1064-8275

We present a numerical investigation of stochastic transport in ideal fluids.According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principlesof transformation theory and multi-time homogenisation, respectively, imply aphysically meaningful, data-driven approach for decomposing the fluid transportvelocity into its drift and stochastic parts, for a certain class of fluidflows. In the current paper, we develop new methodology to implement thisvelocity decomposition and then numerically integrate the resulting stochasticpartial differential equation using a finite element discretisation forincompressible 2D Euler fluid flows. The new methodology tested here is foundto be suitable for coarse graining in this case. Specifically, we performuncertainty quantification tests of the velocity decomposition of Cotter et al.(2017), by comparing ensembles of coarse-grid realisations of solutions of theresulting stochastic partial differential equation with the "true solutions" ofthe deterministic fluid partial differential equation, computed on a refinedgrid. The time discretization used for approximating the solution of thestochastic partial differential equation is shown to be consistent. We includecomprehensive numerical tests that confirm the non-Gaussianity of the streamfunction, velocity and vorticity fields in the case of incompressible 2D Eulerfluid flows.

Crisan D, McMurray E, 2018, Cubature on Wiener space for McKean-Vlasov SDEs with smooth scalar interaction, *Annals of Applied Probability*, Vol: 29, Pages: 130-177, ISSN: 1050-5164

We present two cubature on Wiener space algorithms for the numerical solutionof McKean-Vlasov SDEs with smooth scalar interaction. The analysis hinges onsharp gradient to time-inhomogeneous parabolic PDEs bounds. These bounds may beof independent interest. They extend the classical results of Kusuoka \&Stroock. Both algorithms are tested through two numerical examples.

Crisan D, Míguez J, Ríos-Muñoz G, 2018, On the performance of parallelisation schemes for particle filtering, *Eurasip Journal on Advances in Signal Processing*, Vol: 2018, ISSN: 1687-6172

© 2018, The Author(s). Considerable effort has been recently devoted to the design of schemes for the parallel implementation of sequential Monte Carlo (SMC) methods for dynamical systems, also widely known as particle filters (PFs). In this paper, we present a brief survey of recent techniques, with an emphasis on the availability of analytical results regarding their performance. Most parallelisation methods can be interpreted as running an ensemble of lower-cost PFs, and the differences between schemes depend on the degree of interaction among the members of the ensemble. We also provide some insights on the use of the simplest scheme for the parallelisation of SMC methods, which consists in splitting the computational budget into M non-interacting PFs with N particles each and then obtaining the desired estimators by averaging over the M independent outcomes of the filters. This approach minimises the parallelisation overhead yet still displays desirable theoretical properties. We analyse the mean square error (MSE) of estimators of moments of the optimal filtering distribution and show the effect of the parallelisation scheme on the approximation error rates. Following these results, we propose a time–error index to compare schemes with different degrees of parallelisation. Finally, we provide two numerical examples involving stochastic versions of the Lorenz 63 and Lorenz 96 systems. In both cases, we show that the ensemble of non-interacting PFs can attain the approximation accuracy of a centralised PF (with the same total number of particles) in just a fraction of its running time using a standard multicore computer.

Crisan DO, Miguez J, 2018, Nested particle filters for online parameter estimation in discrete-time state-space Markov models, *Bernoulli*, Vol: 24, Pages: 3039-3086, ISSN: 1350-7265

We address the problem of approximating the posterior probability distribution of the fixedparameters of a state-space dynamical system using a sequential Monte Carlo method. Theproposed approach relies on a nested structure that employs two layers of particle filters toapproximate the posterior probability measure of the static parameters and the dynamic statevariables of the system of interest, in a vein similar to the recent “sequential Monte Carlosquare” (SMC2) algorithm. However, unlike the SMC2scheme, the proposed technique operatesin a purely recursive manner. In particular, the computational complexity of the recursive stepsof the method introduced herein is constant over time. We analyse the approximation of integralsof real bounded functions with respect to the posterior distribution of the system parameterscomputed via the proposed scheme. As a result, we prove, under regularity assumptions, that theapproximation errors vanish asymptotically inLp(p≥1) with convergence rate proportional to1√N+1√M, whereNis the number of Monte Carlo samples in the parameter space andN×Mis the number of samples in the state space. This result also holds for the approximation of thejoint posterior distribution of the parameters and the state variables. We discuss the relationshipbetween the SMC2algorithm and the new recursive method and present a simple example inorder to illustrate some of the theoretical findings with computer simulations.Keywords:particle filtering, parameter estimation, model inference, state space models, recursivealgorithms, Monte Carlo, error bounds.

Crisan D, Holm DD, 2018, Wave breaking for the Stochastic Camassa-Holm equation, *Physica D: Nonlinear Phenomena*, Vol: 376-377, Pages: 138-143, ISSN: 0167-2789

We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.

Crisan DO, Kurtz T, Janjigian C, 2018, Particle representations for stochastic partial differential equations with boundary conditions, *Electronic Journal of Probability*, Vol: 23, Pages: 1-29, ISSN: 1083-6489

In this article, we study weighted particle representations for a class of stochastic partial differential equations (SPDE) with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations. The locations are given by independent, stationary reflecting diffusions in abounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise W which is the stochastic input for the SPDE. The weights interact through V, the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on theboundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure V is the unique solution for each of the nonlinear stochastic par-tial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14, 15]

Paulin D, Jasra A, Crisan D,
et al., 2018, On concentration properties of partially observed chaotic systems, *Advances in Applied Probability*, Vol: 50, Pages: 440-479, ISSN: 0001-8678

In this paper we present results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to ∞. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant multiplied by the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant multiplied by the standard deviation of the noise. To some extent, applications to the three-dimensional Lorenz 63 model and to the Lorenz 96 model of arbitrarily large dimension are considered.

Crisan DO, McMurray E, 2018, Smoothing properties of McKean-Vlasov SDEs, *Probability Theory and Related Fields*, Vol: 171, Pages: 97-148, ISSN: 1432-2064

In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean–Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean–Vlasov SDEs.

Crisan D, Moral PD, Houssineau J,
et al., 2017, Unbiased multi-index Monte Carlo, *Stochastic Analysis and Applications*, Vol: 36, Pages: 257-273, ISSN: 0736-2994

We introduce a new class of Monte Carlo-based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte Carlo (MIMC) that has the added advantage of reducing the computational effort, relative to i.i.d. sampling from the most precise discretization, for a given level of error. We cover extensions of results regarding variance and optimality criteria for the new approach. We apply the methodology to the problem of computing an unbiased mollified version of the solution of a partial differential equation with random coefficients. A second application concerns the Bayesian inference (the smoothing problem) of an infinite-dimensional signal modeled by the solution of a stochastic partial differential equation that is observed on a discrete space grid and at discrete times. Both applications are complemented by numerical simulations.

Crisan DO, Miguez J, 2017, Uniform convergence over time of a nested particle filtering scheme forrecursive parameter estimation in state–space Markov models, *Advances in Applied Probability*, Vol: 49, Pages: 1170-1200, ISSN: 1475-6064

We analyse the performance of a recursive Monte Carlo method forthe Bayesian estimation of the staticparameters of a discrete–time state–space Markov model. The algorithm employs two layers of particlefilters to approximate the posterior probability distribution of the model parameters. In particular, thefirst layer yields an empirical distribution of samples on the parameter space, while the filters in the secondlayer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. Thisapproach relates the novel algorithm to the recent sequential Monte Carlo square (SMC2) method, whichprovides anon-recursivesolution to the same problem. In this paper, we investigate the approximationof integrals of real bounded functions with respect to the posterior distribution of the system parameters.Under assumptions related to the compactness of the parametersupport and the stability and continuityof the sequence of posterior distributions for the state–space model, we prove that theLpnorms of theapproximation errors vanish asymptotically (as the number of Monte Carlo samples generated by thealgorithm increases) and uniformly over time. We also prove that, under the same assumptions, theproposed scheme can asymptotically identify the parameter valuesfor a class of models. We conclude thepaper with a numerical example that illustrates the uniform convergence results by exploring the accuracyand stability of the proposed algorithm operating with long sequences of observations.

Beskos A, Crisan D, Jasra A,
et al., 2017, A stable particle filter for a class of high-dimensional state-space models, *Advances in Applied Probability*, Vol: 49, Pages: 24-48, ISSN: 0001-8678

We consider the numerical approximation of the filtering problem in high dimensions, thatis, when the hidden state lies in Rd with large d. For low-dimensional problems, one of themost popular numerical procedures for consistent inference is the class of approximationstermed particle filters or sequential Monte Carlo methods. However, in high dimensions,standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponentialin d for the algorithm to be stable in an appropriate sense. We develop a new particlefilter, called the space–time particle filter, for a specific family of state-space models indiscrete time. This new class of particle filters provides consistent Monte Carlo estimatesfor any fixed d, as do standard particle filters. Moreover, when there is a spatial mixingelement in the dimension of the state vector, the space–time particle filter will scale muchbetter with d than the standard filter for a class of filtering problems. We illustrate thisanalytically for a model of a simple independent and identically distributed structure anda model of an L-Markovian structure (L ≥ 1, L independent of d) in the d-dimensionalspace direction, when we show that the algorithm exhibits certain stability propertiesas d increases at a cost O(nN d2), where n is the time parameter and N is the numberof Monte Carlo samples, which are fixed and independent of d. Our theoretical resultsare also supported by numerical simulations on practical models of complex structures.The results suggest that it is indeed possible to tackle some high-dimensional filteringproblems using the space–time particle filter that standard particle filters cannot handle.

Crisan DO, Ottobre M, 2016, Pointwise gradient bounds for degenerate semigroups (of UFG type), *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 472, ISSN: 1364-5021

In this paper we consider diffusion semigroups generated by second order differentialoperators of degenerate type. The operators that we consider do not, in general, satisfy theH¨ormander condition and are not hypoelliptic. In particular, instead of working under the H¨ormanderparadigm, we consider the so-called UFG condition, introduced by Kusuoka and Strook in theeighties. The UFG condition is weaker than the uniform H¨ormander condition, the smoothing effecttaking place only in certain directions (rather than in every direction, as it is the case when theH¨ormander condition is assumed). Under the UFG condition, Kusuoka and Strook deduced sharpsmall time asymptotic bounds for the derivatives of the semigroup in the directions where smoothingoccurs. In this paper, we study the large time asymptotics for the gradients of the diffusionsemigroup in the same set of directions and under the same UFG condition. In particular, we identifyconditions under which the derivatives of the diffusion semigroup in the smoothing directionsdecay exponentially in time. This paper constitutes therefore a stepping stone in the analysis ofthe long time behaviour of diffusions which do not satisfy the H¨ormander condition

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