Mathematical Homepage of Jochen Voss

By Jochen Voss, last updated 2012-11-15

This page gives a short summary of my mathematical interests.

Conditional Path Sampling of SDEs

With Andrew Stuart and Martin Hairer I am exploring how infinite dimensional Langevin sampling can be used to sample from conditioned distributions of solutions of stochastic differential equations. Our method is based on constructing a partial stochastic differential equation which has the conditioned target density as its stationary distribution.

Example: the following picture shows a path from the solution of a stochastic differential equation on the time interval [0,100], conditioned on having the value -1 at time 0 and having the value +1 at time 100. The drift has -1 and +1 as stable equilibrium points and 0 as an unstable equilibrium point.

This method yields a new algorithm for the (non-linear) Kalman filter/smoother. One can view the pair of signal and observation as the solution of a two-dimensional stochastic differential equation. Our method can then be used to study the distribution of the signal conditioned on a given observation.

A.M. Stuart, J. Voss and P. Wiberg: Conditional Path Sampling of SDEs and the Langevin MCMC Method. Communications in Mathematical Sciences, vol. 2, no. 4, pp. 685–697, 2004.
link, preprint:pdf, more…
M. Hairer, A.M. Stuart, J. Voss and P. Wiberg: Analysis of SPDEs arising in Path Sampling, Part I: The Gaussian Case. Communications in Mathematical Sciences, vol. 3, no. 4, pp. 587–603, 2005.
link, arXiv:math/0601095, more…
M. Hairer, A.M. Stuart and J. Voss: Analysis of SPDEs Arising in Path Sampling, Part II: The Nonlinear Case. Annals of Applied Probability, vol. 17, no. 5, pp. 1657–1706, 2007.
DOI:10.1214/07-AAP441, arXiv:math/0601092, more…
A. Apte, M. Hairer, A.M. Stuart and J. Voss: Sampling The Posterior: An Approach to Non-Gaussian Data Assimilation. Physica D: Nonlinear Phenomena, vol. 230, no. 1–2, pp. 50–64, 2007.
DOI:10.1016/j.physd.2006.06.009, preprint:pdf, more…
A. Apte, C.K.R.T. Jones, A.M. Stuart and J. Voss: Data Assimilation: Mathematical and Statistical Perspectives. International Journal for Numerical Methods in Fluids, vol. 56, no. 8, pp. 1033–1046, 2008.
DOI:10.1002/fld.1698, preprint:pdf, more…
A. Beskos, G.O. Roberts, A.M. Stuart and J. Voss: MCMC Methods for Diffusion Bridges. Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350, 2008.
DOI:10.1142/S0219493708002378, preprint:pdf, more…
M. Hairer, A.M. Stuart and J. Voss: Sampling Conditioned Diffusions. Pages 159–186 in Trends in Stochastic Analysis, Cambridge University Press, vol. 353 of London Mathematical Society Lecture Note Series, 2009.
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M. Hairer, A.M. Stuart and J. Voss: Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods. Pages 833–873 in The Oxford Handbook of Nonlinear Filtering, Dan Crisan and Boris Rozovsky (editors), Oxford University Press, 2011.
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M. Hairer, A.M. Stuart and J. Voss: Sampling Conditioned Hypoelliptic Diffusions. Annals of Applied Probability, vol. 21, no. 2, pp. 669–698, 2011.
DOI:10.1214/10-AAP708, arXiv:0908.0162, more…

Diffusion Processes

Together with my brother Andreas Voss I work on a parameter estimation problem for diffusion processes which arises in psychology when modelling speeded binary decision processes.

A. Voss, K. Rothermund and J. Voss: Interpreting the Parameters of the Diffusion Model: An Empirical Validation. Memory & Cognition, vol. 32, no. 7, pp. 1206–1220, 2004.
link, preprint:pdf, more…
A. Voss and J. Voss: Fast-Dm: a Free Programm for Efficient Diffusion Model Analysis. Behavior Research Methods, vol. 39, no. 4, pp. 767–775, 2007.
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A. Voss and J. Voss: A Fast Numerical Algorithm for the Estimation of Diffusion-Model Parameters. Journal of Mathematical Psychology, vol. 52, pp. 1–9, 2008.
DOI:10.1016/j.jmp.2007.09.005, preprint:pdf, more…
A. Voss, J. Voss and K.C. Klauer: Separating Response-Execution Bias from Decision Bias: Arguments for an Additional Parameter in Ratcliff's Diffusion Model. British Journal of Mathematical and Statistical Psychology, vol. 63, no. 3, pp. 539–555, 2010.
DOI:10.1348/000711009X477581, preprint:pdf, more…
We published our fast-dm programm for parameter estimation in the diffusion model.

My Diplomarbeit (approximately equivalent to an MSc thesis) deals with a topic related to diffusion processes: There I consider the question how fast one can distinguish between two different given diffusions, when observing a single path over long intervals of time.

J. Voß: Über die Asymptotik des Bayesrisikos bei Diffusionsprozessen. Diplomarbeit, Universität Kaiserslautern, 1997.
link, preprint:ps, more…

Large Deviations

In my PhD thesis I prove a large deviation result about the behaviour of diffusions under a strong drift.

J. Voss: Some Large Deviation Results for Diffusion Processes. PhD thesis, University of Kaiserslautern, Germany, 2004.
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J. Voss: Large Deviations for One Dimensional Diffusions with a Strong Drift. Electronic Journal of Probability, vol. 13, no. 53, pp. 1479–1526, 2008.
link, preprint:pdf, more…
J. Voss: Upper and Lower Bounds in Exponential Tauberian Theorems. Tbilisi Mathematical Journal, vol. 2, pp. 41–50, 2009.
link, arXiv:0908.0642, more…

Simulations and Numerical Analysis

The Ziggurat Method for Generating Gaussian Random Numbers.
I typed the lecture notes for the numerical linear algebra course I gave in 2004 at the University of Warwick.
I wrote a tutorial about how to compute the square root of a matrix in a C program.
My program XIsing simulates states of the Ising model.

Miscellaneous

My mathematical gallery displays some pictures of mathematical objects.
An overview over common probability distributions (German) lists expectations, variances and characteristic functions of many distributions.
I wrote part of our vector analysis lecture notes (German).
I have designed numerous exercises about calculus (in German):
- J. Voß: Übungsaufgaben zur Analysis. Available for download from the author's homepage, 2003.
  preprint:pdf, more…