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An introduction to stochastic differential equations. Lecture notes (available electronically at math.berkeley.edu/evans
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Diffusionbased motion planning for a nonholonomic flexible needle model
 Proc. IEEE Int. Conf. Robot. Autom. (ICRA); 2005
"... Abstract — Fine needles facilitate diagnosis and therapy because they enable minimally invasive surgical interventions. This paper formulates the problem of steering a very flexible needle through firm tissue as a nonholonomic kinematics problem, and demonstrates how planning can be accomplished us ..."
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Cited by 41 (12 self)
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Abstract — Fine needles facilitate diagnosis and therapy because they enable minimally invasive surgical interventions. This paper formulates the problem of steering a very flexible needle through firm tissue as a nonholonomic kinematics problem, and demonstrates how planning can be accomplished using diffusionbased motion planning on the Euclidean group, SE(3). In the present formulation, the tissue is treated as isotropic and no obstacles are present. The bevel tip of the needle is treated as a nonholonomic constraint that can be viewed as a 3D extension of the standard kinematic cart or unicycle. A deterministic model is used as the starting point, and reachability criteria are established. A stochastic differential equation and its corresponding FokkerPlanck equation are derived. The EulerMaruyama method is used to generate the ensemble of reachable states of the needle tip. Inverse kinematics methods developed previously for hyperredundant and binary manipulators that use this probability density information are applied to generate needle tip paths that reach the desired targets. Index Terms — needle steering, nonholonomic path planning, probability density function, EulerMaruyama method, medical robotics I.
Stochastic modelling of reactiondiffusion processes: algorithms for bimolecular reactions
 Phys. Biol
"... Abstract. Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reactiondiffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an onlattice model described by the reactiondiffusion master equat ..."
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Cited by 37 (11 self)
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Abstract. Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reactiondiffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an onlattice model described by the reactiondiffusion master equation. The second SSA is an offlattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site).
Modeling and simulating chemical reactions
 SIAM Review
, 2007
"... Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain largescale limit of a sequence of finerscale probabilistic models. In studying this hierarchy of models, st ..."
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Cited by 34 (1 self)
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Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain largescale limit of a sequence of finerscale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner, and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described. 1
Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation
 Analysis and Computation, SIAM Books
, 2007
"... Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions usi ..."
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Cited by 33 (7 self)
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Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jumpdiffusions. Next, markedjumpdiffusions are treated using compound Poisson processes to include random marked jumpamplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measuretheoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding nonsmooth and jump discontinuities to the model. Finally, the stochastic optimal control of markedjumpdiffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jumpdiffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There
A practical guide to stochastic simulations of reactiondiffusion processes, 35 pages, available as http://arxiv.org/abs/0704.1908
, 2007
"... Abstract. A practical introduction to stochastic modelling of reactiondiffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modell ..."
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Cited by 29 (13 self)
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Abstract. A practical introduction to stochastic modelling of reactiondiffusion processes is presented. No prior knowledge of stochastic simulations is assumed. The methods are explained using illustrative examples. The article starts with the classical Gillespie algorithm for the stochastic modelling of chemical reactions. Then stochastic algorithms for modelling molecular diffusion are given. Finally, basic stochastic reactiondiffusion methods are presented. The connections between stochastic simulations and deterministic models are explained and basic mathematical tools (e.g. chemical master equation) are presented. The article concludes with an overview of more advanced methods and problems. Key words. stochastic simulations, reactiondiffusion processes AMS subject classifications. 60G05, 92C40, 60J60, 92C15
A contraction theory approach to stochastic incremental stability
 IEEE Transactions on Automatic Control
, 2009
"... We investigate the incremental stability properties of Itô stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can ..."
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Cited by 19 (8 self)
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We investigate the incremental stability properties of Itô stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noisefree system. We illustrate these results in the contexts of stochastic nonlinear observers design and stochastic synchronization. 1
Intercarrier interference due to Phase Noise
 in OFDM  estimation and suppression,” in Proc. IEEE VTC Fall 2004
, 2004
"... Abstract — In this paper we provide an analysis of the intercarrier interference (ICI) due to phase noise in OFDM systems and present an algorithm for its suppression. We examine the general case where phase noise can take any values, thus the ”small ” phase noise model is dropped. The statistical p ..."
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Cited by 15 (5 self)
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Abstract — In this paper we provide an analysis of the intercarrier interference (ICI) due to phase noise in OFDM systems and present an algorithm for its suppression. We examine the general case where phase noise can take any values, thus the ”small ” phase noise model is dropped. The statistical properties of the intercarrier interference are analyzed, showing that the ICI is generally a nongaussian random process which has a large impact on the system performance. Closed form expressions which describe the correlation properties of the constituents of ICI are calculated. An MMSE approach for suppressing ICI in the frequency domain is presented. This approach avoids error propagation to which our previously proposed algorithm was prone. The performance of the suppression algorithm is shown, pointing out the limits for the ICI suppression algorithms in general. I.
A.: Efficient strong integrators for linear stochastic systems
 SIAM J. Numer. Anal
"... Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by two Wiener processes and with noncommutative vector fields. These schemes are based on the Neumann and Magnus expansions. We prove that for a sufficiently small stepsize, the half o ..."
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Cited by 13 (6 self)
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Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by two Wiener processes and with noncommutative vector fields. These schemes are based on the Neumann and Magnus expansions. We prove that for a sufficiently small stepsize, the half order Magnus and a new modified order one Magnus integrator are globally more accurate than classical stochastic numerical schemes or Neumann integrators of the corresponding order. These Magnus methods will therefore always be preferable provided the cost of computing the matrix exponential is not significant. Further, for small stepsizes the accurate representation of the Lévy area between the two driving processes dominates the computational cost for all methods of order one and higher. As a consequence, we show that the accuracy of all stochastic integrators asymptotically scales like the squareroot of the computational cost. This has profound implications on the effectiveness of higher order integrators. In particular in terms of efficiency, there are generic scenarios where order one Magnus methods compete with and even outperform higher order methods. We consider the consequences in applications such as linearquadratic optimal control, filtering problems and the pricing of pathdependent financial derivatives.