Publications
100 results found
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
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 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, 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, Xiong J, 2014, Numerical solution for a class of SPDEs over bounded domains, STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES, Vol: 86, Pages: 450-472, ISSN: 1744-2508
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- Citations: 1
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 πt, the solution of the stochastic filtering problem, to depend continuously on the observed data Y = {Ys, 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 θft, defined on the space of continuous paths C([0, t], ℝd) endowed with the uniform convergence topology such that πt (f) = θft (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 θft, defined on a suitably chosen space of Hölder continuous paths such that πt (f) = θft
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
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, 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, 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 XT, where X = {Xt, 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 = {Yt, t ∈ [0, T]} can be written as functions of time and X, Yt = θ{symbol}t(Xt). 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 Yt 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, 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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- Citations: 21
Crisan D, Obanubi O, 2012, Particle Filters with Random Resampling Times, Stochastic Processes and their Applications, Vol: 122, Pages: 1332-1368
Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the onesthat contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results applyonly to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filterswhere resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge. The conditions are verified when the resampling times are chosen in accordance to effective sample size of thesystem of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum ofthe particles’ weights. We also deduce central-limit theorem type results for the approximating particle system withrandom resampling times.
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.
, 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, Li K, 2011, GENERALISED PARTICLE FILTERS WITH GAUSSIAN MEASURES, 19TH EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO-2011), Pages: 659-663, ISSN: 2076-1465
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- Citations: 1
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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- Citations: 13
Crisan D, Manolarakis K, Touzi N, 2010, On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights, Stochastic Processes and their Applications, Vol: 120, Pages: 1133-1158, ISSN: 0304-4149
We propose a generic framework for the analysis of Monte Carlo simulation schemes of backward SDEs. The general results are used to re-visit the convergence of the algorithm suggested by Bouchard and Touzi (2004) [6]. By keeping the higher order terms in the expansion of the Skorohod integrals resulting from the Malliavin integration by parts in [6], we introduce a variant of the latter algorithm which allows for a significant reduction of the numerical complexity. We prove the convergence of this improved Malliavin-based algorithm, and derive a bound on the induced error. In particular, we show that the price to pay for our simplification is to use a more accurate localizing function. © 2010 Elsevier B.V. All rights reserved.
Crisan D, Xiong J, 2010, Approximate McKean-Vlasov representations for a class of SPDEs, STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES, Vol: 82, Pages: 53-68, ISSN: 1744-2508
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- Citations: 52
Crisan D, Kouritzin M, Xiong J, 2009, Nonlinear filtering with signal dependent observation noise, ELECTRONIC JOURNAL OF PROBABILITY, Vol: 14, Pages: 1863-1883, ISSN: 1083-6489
Míguez J, Maíz CS, Djurić PM, et al., 2009, Sequential Monte Carlo optimization using artificial state-space models, 2009 IEEE 13th Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, DSP/SPE 2009, Proceedings, Pages: 268-273
We introduce a method for sequential minimization of a certain class of (possibly non-convex) cost functions with respect to a high dimensional signal of interest. The proposed approach involves the transformation of the optimization problem into one of estimation in a discrete-time dynamical system. In particular, we describe a methodology for constructing an artificial state-space model which has the signal of interest as its unobserved dynamic state. The model is "adapted" to the cost function in the sense that the maximum a posteriori (MAP) estimate of the system state is also a global minimizer of the cost function. The advantage of the estimation framework is that we can draw from a pool of sequential Monte Carlo methods, for particle approximation of probability measures in dynamic systems, that enable the numerical computation of MAP estimates. We provide examples of how to apply the proposed methodology, including some illustrative simulation results. ©2009 IEEE.
Thomas N, Bradley JT, Knottenbelt WJ, 2009, Proceedings of PASM 2008, 3rd International Workshop on the Practical Application of Stochastic Modelling, Publisher: Elsevier
This issue contains papers that were originally presented at the Third International Workshop on the Practical Application of Stochastic Modelling (PASM), held at the Universitat de les Illes Balears, Palma de Mallorca, in September 2008. The workshop was collocated with the 5th European Performance Engineering Workshop.\r\n\r\n PASM follows in a long tradition of the application of stochastic modelling to real-world problems. Such models have led to significant advances in modelling theory, as well as insights into the specific problem areas concerned. Coming from a computer science background, PASM is particularly concerned with applications of specification and analysis techniques and tools developed for computer science, as well as computing and communications applications. In particular, the aim of PASM is to give a forum for which applies current well-developed formalisms (stochastic Petri nets, stochastic process algebras, layered queueing networks, etc) to real-world case-studies.\r\n
Heine K, Crisan D, 2008, UNIFORM APPROXIMATIONS OF DISCRETE-TIME FILTERS, ADVANCES IN APPLIED PROBABILITY, Vol: 40, Pages: 979-1001, ISSN: 0001-8678
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- Citations: 18
Crisan D, Heine K, 2008, Stability of the discrete time filter in terms of the tails of noise distributions, Journal of the London Mathematical Society, Vol: 78, Pages: 441-458, ISSN: 0024-6107
In recent years, the stability of discrete time filters has been a field of active research. By stability we mean that the effect of the possibly erroneous initial distribution in the filter eventually vanishes as time increases. One of the motivations for our interest in the stability is its close relation to the convergence of various numerical filter approximation schemes, for example, particle filters. In this paper, the main result states easily verifiable conditions that are sufficient for filter stability. Essentially, the conditions state that the filter is stable, if the observation noise is sufficiently light tailed compared with the randomness in the signal process. Compactness of the state space or ergodicity of the signal is not required. © 2008 London Mathematical Society.
Crisan D, Ghazali S, 2007, On the Convergence Rates of a General Class of Weak Approximations of SDEs, Stochastic Differential Equations: Theory and Applications, Editors: Baxendale, Lototsky, Pages: 221-249
Crisan D, 2006, Particle filters in a continuous time framework, IEEE Nonlinear Statistical Signal Processing Workshop, Publisher: IEEE, Pages: 73-78
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- Citations: 2
Crisan D, 2006, Particle approximations for a class of stochastic partial differential equations, Applied Mathematics and Optimization Journal, Vol: 54, Pages: 293-314, ISSN: 0095-4616
Clark JMC, Crisan D, 2005, On a robust version of the integral representation formula of nonlinear filtering, Probability and Related Fields, Vol: 133, Pages: 43-56
Crisan D, 2004, Superprocesses in a Brownian environment, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 460, Pages: 243-270, ISSN: 1364-5021
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- Citations: 5
Crisan D, 2004, Superprocesses in random environments, Stochastic analysis with applications to mathematical finance., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Vol: 460, Pages: 243-270
Crisan D, 2003, Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation, ANNALS OF PROBABILITY, Vol: 31, Pages: 693-718, ISSN: 0091-1798
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- Citations: 25
Crisan D, Doucet A, 2002, A survey of convergence results on particle filtering methods for practitioners, IEEE TRANSACTIONS ON SIGNAL PROCESSING, Vol: 50, Pages: 736-746, ISSN: 1053-587X
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- Citations: 552
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