## Publications

60 results found

Crisan D, Moral PD, Houssineau J,
et al., 2018, Unbiased multi-index Monte Carlo, *Stochastic Analysis and Applications*, Vol: 36, Pages: 257-273, ISSN: 0736-2994

© 2018 Taylor & Francis Group, LLC. 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.

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

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Crisan D, McMurray E, 2017, Smoothing properties of McKean–Vlasov SDEs, *Probability Theory and Related Fields*, Pages: 1-52, ISSN: 0178-8051

© 2017 The Author(s) 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.

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Crisan D, 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

© 2016 The Author(s) Published by the Royal Society. All rights reserved. In this paper, we consider diffusion semigroups generated by second-order differential operators of degenerate type. The operators that we consider do not, in general, satisfy the Hörmander condition and are not hypoelliptic. In particular, instead of working under the Hörmander paradigm, we consider the so-called UFG (uniformly finitely generated) condition, introduced by Kusuoka and Strook in the 1980s. The UFG condition is weaker than the uniform Hörmander condition, the smoothing effect taking place only in certain directions (rather than in every direction, as it is the case when the Hörmander condition is assumed). Under the UFG condition, Kusuoka and Strook deduced sharp small time asymptotic bounds for the derivatives of the semigroup in the directions where smoothing occurs. In this paper, we study the large time asymptotics for the gradients of the diffusion semigroup in the same set of directions and under the same UFG condition. In particular, we identify conditions under which the derivatives of the diffusion semigroup in the smoothing directions decay exponentially in time. This paper constitutes, therefore, a stepping stone in the analysis of the long-Time behaviour of diffusions which do not satisfy the Hörmander condition.

Crisan D, Li K, 2015, Generalised particle filters with Gaussian mixtures, *STOCHASTIC PROCESSES AND THEIR APPLICATIONS*, Vol: 125, Pages: 2643-2673, ISSN: 0304-4149

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Crisan D, Litterer C, Lyons T, 2015, Kusuoka-Stroock gradient bounds for the solution of the filtering equation, *Journal of Functional Analysis*, Vol: 268, Pages: 1928-1971, ISSN: 0022-1236

© 2014 Elsevier Inc. We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [13,14,16,17], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup following Ocone [22] . The estimates we derive have sharp small time asymptotics.

Crisan D, Otobe Y, Peszat S, 2015, Inverse problems for stochastic transport equations, *Inverse Problems*, Vol: 31, ISSN: 0266-5611

© 2015 IOP Publishing Ltd. Inverse problems for stochastic linear transport equations driven by a temporal or spatial white noise are discussed. We analyse stochastic linear transport equations which depend on an unknown potential and have either additive noise or multiplicative noise. We show that one can approximate the potential with arbitrary small error when the solution of the stochastic linear transport equation is observed over time at some fixed point in the state space.

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Miguez J, Crisan D, Marino IP, 2015, Particle filtering for Bayesian parameter estimation in a high dimensional state space model, Pages: 1241-1245

© 2015 EURASIP. Researchers in some of the most active fields of science, including, e.g., geophysics or systems biology, have to deal with very-large-scale stochastic dynamic models of real world phenomena for which conventional prediction and estimation methods are not well suited. In this paper, we investigate the application of a novel nested particle filtering scheme for joint Bayesian parameter estimation and tracking of the dynamic variables in a high dimensional state space model-namely a stochastic version of the two-scale Lorenz 96 chaotic system, commonly used as a benchmark model in meteorology and climate science. We provide theoretical guarantees on the algorithm performance, including uniform convergence rates for the approximation of posterior probability density functions of the fixed model parameters.

Beskos A, Crisan D, Jasra A, 2014, On the stability of sequential Monte Carlo methods in high dimensions, *Annals of Applied Probability*, Vol: 24, Pages: 1396-1445, ISSN: 1050-5164

We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on Rd for large d. It is well known [Bengtsson, Bickel and Li, In Probability and Statistics: Essays in Honor of David A. Freedman, D. Nolan and T. Speed, eds. (2008) 316-334 IMS; see also Pushing the Limits of Contemporary Statistics (2008) 318-329 IMS, Mon. Weather Rev. (2009) 136 (2009) 4629-4640] that using a single importance sampling step, one produces an approximation for the target that deteriorates as the dimension d increases, unless the number of Monte Carlo samples N increases at an exponential rate in d. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a "simple" density and moving to the one of interest, using an SMC method to sample from the sequence; see, for example, Chopin [Biometrika 89 (2002) 539-551] ; see also [J. R. Stat. Soc. Ser. B Stat. Methodol. 68 (2006) 411-436, Phys. Rev. Lett. 78 (1997) 2690-2693, Stat. Comput. 11 (2001) 125-139]. Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable ε N as d → ∞ with 1 < ε N < N. The convergence is achieved with a computational cost proportional to Nd 2 . If ε N ≥ N, we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). In this case, the ESS converges to a random variable ε N,m as d → ∞ and lim m→ε ε N,m = N. Also, we show that the Monte Carlo error for estimating a fixed-dimensional marginal expectation is of order 1/N uniformly in d. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed-dimensional marginals at a cost which is less than exponential

Beskos A, Crisan DO, Jasra A,
et al., 2014, Error bounds and normalising constants for sequential monte carlo samplers in high dimensions, *Advances in Applied Probability*, Vol: 46, Pages: 279-306, ISSN: 0001-8678

In this paper we develop a collection of results associated to the analysis of the sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional independent and identically distributed target probabilities. TheSMCsamplers algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density in d dimensions our results are concerned with d while the number of Monte Carlo samples, N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using theSMCsampler and the exact asymptotic relative L2-error of the estimate of the normalising constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm is O(Nd2). © Applied Probability Trust 2014.

Cass T, Clark M, Crisan D, 2014, The filtering equations revisited, *Springer Proceedings in Mathematics and Statistics*, Vol: 100, Pages: 129-162, ISSN: 2194-1009

© Springer International Publishing Switzerland 2014. The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the changeof– probability-measure method introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by amartingale-problem formulation.Two specific applications are treated.

Chassagneux J-F, Crisan D, Delarue F, 2014, A Probabilistic approach to classical solutions of the master equation for large population equilibria

We analyze a class of nonlinear partial differential equations (PDEs) definedon $\mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d),$ where$\mathcal{P}_2(\mathbb{R}^d)$ is the Wasserstein space of probability measureson $\mathbb{R}^d$ with a finite second-order moment. We show that suchequations admit a classical solutions for sufficiently small time intervals.Under additional constraints, we prove that their solution can be extended toarbitrary large intervals. These nonlinear PDEs arise in the recentdevelopments in the theory of large population stochastic control. Moreprecisely they are the so-called master equations corresponding to asymptoticequilibria for a large population of controlled players with mean-fieldinteraction and subject to minimization constraints. The results in the paperare deduced by exploiting this connection. In particular, we study thedifferentiability with respect to the initial condition of the flow generatedby a forward-backward stochastic system of McKean-Vlasov type. As a byproduct,we prove that the decoupling field generated by the forward-backward system isa classical solution of the corresponding master equation. Finally, we giveseveral applications to mean-field games and to the control of McKean-Vlasovdiffusion processes.

Chassagneux JF, Crisan D, 2014, Runge-kutta schemes for backward stochastic differential equations, *Annals of Applied Probability*, Vol: 24, Pages: 679-720, ISSN: 1050-5164

We study the convergence of a class of Runge.Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability property. As a consequence, the overall rate of the convergence of these schemes is controlled by their local truncation error. The schemes are categorized by the number of intermediate stages implemented between consecutive partition time instances. We show that the order of the schemes matches the number p of intermediate stages for p ≤ 3. Moreover, we show that the so-called order barrier occurs at p = 3, that is, that it is not possible to construct schemes of order p with p stages, when p > 3. The analysis is done under sufficient regularity on the final condition and on the coefficients of the BSDE. © Institute of Mathematical Statistics, 2014.

Crisan D, 2014, The stochastic filtering problem: A brief historical account, Pages: 13-22, ISSN: 0021-9002

© Applied Probability Trust 2014 Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Crisan D, Hambly B, Zariphopoulou T, 2014, Stochastic Analysis and Applications 2014: In Honour of Terry Lyons, ISSN: 2194-1009

Crisan D, Kurtz TG, Lee Y, 2014, Conditional distributions, exchangeable particle systems, and stochastic partial differential equations, *ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES*, Vol: 50, Pages: 946-974, ISSN: 0246-0203

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Crisan D, Manolarakis K, 2014, Second order discretization of backward sdes and simulation with the cubature method, *Annals of Applied Probability*, Vol: 24, Pages: 652-678, ISSN: 1050-5164

We propose a second order discretization for backward stochastic differential equations (BSDEs) with possibly nonsmooth boundary data. When implemented, the discretization method requires essentially the same computational effort with the Euler scheme for BSDEs of Bouchard and Touzi [Stochastic Process. Appl. 111 (2004) 175-206] and Zhang [Ann. Appl. Probab. 14 (2004) 459-488]. However, it enjoys a second order asymptotic rate of convergence, provided that the coefficients of the equation are sufficiently smooth. In the second part of the paper, we combine this discretization with higher order cubature formulas on Wiener space to produce a fully implementable second order scheme. © Institute of Mathematical Statistics, 2014.

Crisan D, Míguez J, 2014, Particle-kernel estimation of the filter density in state-space models, *Bernoulli*, Vol: 20, Pages: 1879-1929, ISSN: 1350-7265

© 2014 ISI/BS. Sequential Monte Carlo (SMC) methods, also known as particle filters, are simulation-based recursive algorithms for the approximation of the a posteriori probability measures generated by state-space dynamical models. At any given time t, a SMC method produces a set of samples over the state space of the system of interest (often termed "particles") that is used to build a discrete and random approximation of the posterior probability distribution of the state variables, conditional on a sequence of available observations. One potential application of the methodology is the estimation of the densities associated to the sequence of a posteriori distributions. While practitioners have rather freely applied such density approximations in the past, the issue has received less attention from a theoretical perspective. In this paper, we address the problem of constructing kernel-based estimates of the posterior probability density function and its derivatives, and obtain asymptotic convergence results for the estimation errors. In particular, we find convergence rates for the approximation errors that hold uniformly on the state space and guarantee that the error vanishes almost surely as the number of particles in the filter grows. Based on this uniform convergence result, we first show how to build continuous measures that converge almost surely (with known rate) toward the posterior measure and then address a few applications. The latter include maximum a posteriori estimation of the system state using the approximate derivatives of the posterior density and the approximation of functionals of it, for example, Shannon's entropy.

Crisan D, Xiong J, 2014, Numerical solution for a class of SPDEs over bounded domains, *STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC REPORTS*, Vol: 86, Pages: 450-472, ISSN: 1744-2508

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Crisan D, Diehl J, Friz PK,
et al., 2013, Robust filtering: Correlated noise and multidimensional observation, *Annals of Applied Probability*, Vol: 23, Pages: 2139-2160, ISSN: 1050-5164

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for π t , the solution of the stochastic filtering problem, to depend continuously on the observed data Y = {Y s , s ∈ [0, t]}. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function f, there exists a continuous map θ f t , defined on the space of continuous paths C([0, t], ℝ d ) endowed with the uniform convergence topology such that π t (f) = θ f t (Y), almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56] , Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167] , Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260- 278] , Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press] , this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process Y is "lifted" to the process Y that consists of Y and its corresponding Lévy area process, and we show that there exists a continuous map θ f t , defined on a suitably chosen space of Hölder continuous paths such that

Crisan D, Manolarakis K, Nee C, 2013, Cubature methods and applications, *Lecture Notes in Mathematics*, Vol: 2081, Pages: 203-316, ISSN: 0075-8434

We present an introduction to a new class of numerical methods for approximating distributions of solutions of stochastic differential equations. The convergence results for these methods are based on certain sharp gradient bounds established by Kusuoka and Stroock under non-Hörmader constraints on diffusion semigroups. These bounds and some other subsequent refinements are covered in these lectures. In addition to the description of the new class of methods and the corresponding convergence results, we include an application of these methods to the numerical solution of backward stochastic differential equations. As it is well-known, backward stochastic differential equations play a central role in pricing financial derivatives. © Springer International Publishing Switzerland 2013.

Crisan D, Ortiz-Latorre S, 2013, A Kusuoka-Lyons-Victoir particle filter, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 469, ISSN: 1364-5021

The aim of this paper is to introduce a new numerical algorithm for solving the continuous time nonlinear filtering problem. In particular, we present a particle filter that combines the Kusuoka-Lyons-Victoir (KLV) cubature method on Wiener space to approximate the law of the signal with a minimal variance 'thinning' method, called the tree-based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle filter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Beneš filter). © 2013 The Author(s) Published by the Royal Society. All rights reserved.

Míguez J, Crisan D, Djurić PM, 2013, On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization, *Statistics and Computing*, Vol: 23, Pages: 91-107, ISSN: 0960-3174

This paper addresses the problem of maximum a posteriori (MAP) sequence estimation in general state-space models. We consider two algorithms based on the sequential Monte Carlo (SMC) methodology (also known as particle filtering). We prove that they produce approximations of the MAP estimator and that they converge almost surely. We also derive a lower bound for the number of particles that are needed to achieve a given approximation accuracy. In the last part of the paper, we investigate the application of particle filtering and MAP estimation to the global optimization of a class of (possibly non-convex and possibly non-differentiable) cost functions. In particular, we show how to convert the cost-minimization problem into one of MAP sequence estimation for a state-space model that is "matched" to the cost of interest. We provide examples that illustrate the application of the methodology as well as numerical results. © 2011 Springer Science+Business Media, LLC.

Crisan D, Delarue F, 2012, Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations, *JOURNAL OF FUNCTIONAL ANALYSIS*, Vol: 263, Pages: 3024-3101, ISSN: 0022-1236

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Crisan D, Manolarakis K, 2012, Solving backward stochastic differential equations using the cubature method: Application to nonlinear pricing, *SIAM Journal on Financial Mathematics*, Vol: 3, Pages: 534-571

We are concerned with the numerical solution of a class of backward stochastic differential equations (BSDEs), where the terminal condition is a function of X T , where X = {X t , t ∈ [0, T]} is the solution to a standard stochastic differential equation (SDE). A characteristic of these type of BSDEs is that their solutions Y = {Y t , t ∈ [0, T]} can be written as functions of time and X, Y t = θ{symbol} t (X t ). Moreover, the function θ{symbol} t can be represented as the expected value of a functional of X. Therefore, since the forward component Xt is "known" at time t, the problem of estimating Y t amounts to obtaining an approximation of the expected value of the corresponding functional. The approximation of the solution of a BSDE requires an approximation of the law of the solution of the SDE satisfied by the forward component. We introduce a new algorithm, combining the Euler style discretization for BSDEs and the cubature method of Lyons and Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), pp. 169-198]. The latter is an approximation method for the law of the solution of an SDE that generates a tree on which expectations and conditional expectations are evaluated. To treat the exponential growth in the number of leaves on the tree, we appeal to the tree based branching algorithm introduced in [D. Crisan and T. Lyons, Monte Carlo Methods Appl., 8 (2002), pp. 343-355] . The convergence results are proved under very general assumptions. In particular, the vector fields defining the forward equation do not necessarily satisfy the Hörmander condition. Numerical examples are also provided. Copyright © 2012 by SIAM.

Crisan D, Obanubi O, 2012, Particle filters with random resampling times, *STOCHASTIC PROCESSES AND THEIR APPLICATIONS*, Vol: 122, Pages: 1332-1368, ISSN: 0304-4149

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Crisan D, 2011, Stochastic analysis 2010, *Stochastic Analysis 2010*, Pages: 1-299

Stochastic Analysis aims to provide mathematical tools to describe and model high dimensional random systems. Such tools arise in the study of Stochastic Differential Equations and Stochastic Partial Differential Equations, Infinite Dimensional Stochastic Geometry, Random Media and Interacting Particle Systems, Super-processes, Stochastic Filtering, Mathematical Finance, etc. Stochastic Analysis has emerged as a core area of late 20th century Mathematics and is currently undergoing a rapid scientific development. The special volume ?Stochastic Analysis 2010? provides a sample of the current research in the different branches of the subject. It includes the collected works of the participants at the Stochastic Analysis section of the 7th ISAAC Congress organized at Imperial College London in July 2009. © Springer-Verlag Berlin Heidelberg 2011.

Crisan D, Li K, 2011, GENERALISED PARTICLE FILTERS WITH GAUSSIAN MEASURES, *19TH EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO-2011)*, Pages: 659-663, ISSN: 2076-1465

, 2011, The Oxford Handbook of Nonlinear Filtering, ISBN: 9780199532902

In many areas of human endeavour, the systems involved are not available for direct measurement. Instead, by combining mathematical models for a system's evolution with partial observations of its evolving state, we can make reasonable inferences about it. The increasing complexity of the modern world makes this analysis and synthesis of high-volume data an essential feature in many real-world problems.The celebrated Kalman-Bucy filter, designed for linear dynamical systems with linearly structured measurements, is the most famous Bayesian filter. Its generalizations to nonlinear systems and/or observations are collectively referred to as nonlinear filtering (NLF), an extension of the Bayesian framework to the estimation, prediction, and interpolation of nonlinear stochastic dynamics. NLF uses a stochastic model to make inferences about an evolving system and is a theoretically optimal algorithm.The breadth of its applications, firmly established and still emerging, is simply astounding. Early uses such as cryptography, tracking, and guidance were mostly of a military nature. Since then, the scope has exploded. It includes the study of global climate, estimating the state of the economy, identifying tumours using non-invasive methods, and much more.The Oxford Handbook of Nonlinear Filtering is the first comprehensive written resource for the subject. It contains classical and recent results and applications, with contributions from 58 authors. Collated into 10 parts, it covers the foundations of nonlinear filtering, connections to stochastic partial differential equations, stability and asymptotic analysis, estimation and control, approximation theory and numerical methods for solving the nonlinear filtering problem (including particle methods). It also contains a part dedicated to the application of nonlinear filtering to several problems in mathematical finance.Contributors to this volume - R. Atar - Department of Electrical Engineering, Technion, Haifa, IsraelA.

Crisan D, Manolarakis K, 2010, PROBABILISTIC METHODS FOR SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS. APPLICATIONS TO FINANCE, *ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE*, Vol: 44, Pages: 1107-1133, ISSN: 0764-583X

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