Mathematical Homepage of Jochen Voss

By Jochen Voss, last updated November 15, 2012.

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.

• Stuart, Andrew M., Voss, Jochen and Wiberg, Petter: 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…
• Hairer, Martin, Stuart, Andrew M., Voss, Jochen and Wiberg, Petter: 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, preprint:pdf, more…
• Hairer, Martin, Stuart, Andrew M. and Voss, Jochen: 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, preprint:pdf, more…
• Apte, Amit, Hairer, Martin, Stuart, Andrew M. and Voss, Jochen: 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…
• Apte, Amit, Jones, C. K. R. T., Stuart, Andrew M. and Voss, Jochen: 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…
• Beskos, Alexandros, Roberts, Gareth O., Stuart, Andrew M. and Voss, Jochen: MCMC Methods for Diffusion Bridges. Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350, 2008.
  DOI:10.1142/S0219493708002378, preprint:pdf, more…
• Hairer, Martin, Stuart, Andrew M. and Voss, Jochen: Sampling Conditioned Diffusions. Pages 159–186 in Trends in Stochastic Analysis, Cambridge University Press, vol. 353 of London Mathematical Society Lecture Note Series, 2009.
  link, preprint:pdf, more…
• Hairer, Martin, Stuart, Andrew M. and Voss, Jochen: Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods. Pages 833–873 in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, 2011.
  link, preprint:pdf, more…
• Hairer, Martin, Stuart, Andrew M. and Voss, Jochen: Sampling Conditioned Hypoelliptic Diffusions. Annals of Applied Probability, vol. 21, no. 2, pp. 669–698, 2011.
  DOI:10.1214/10-AAP708, arXiv:0908.0162, preprint:pdf, 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.

• Voss, Andreas, Rothermund, Klaus and Voss, Jochen: Interpreting the Parameters of the Diffusion Model: An Empirical Validation. Memory & Cognition, vol. 32, no. 7, pp. 1206–1220, 2004.
  link, preprint:pdf, more…
• Voss, Andreas and Voss, Jochen: Fast-Dm: a Free Programm for Efficient Diffusion Model Analysis. Behavior Research Methods, vol. 39, no. 4, pp. 767–775, 2007.
  link, preprint:pdf, more…
• Voss, Andreas and Voss, Jochen: 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…
• Voss, Andreas, Voss, Jochen and Klauer, Karl Christoph: 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.

• Voß, Jochen: Über die Asymptotik des Bayesrisikos bei Diffusionsprozessen. 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.

• Voss, Jochen: Some Large Deviation Results for Diffusion Processes. PhD thesis, University of Kaiserslautern, Germany, 2004.
  link, preprint:pdf, more…
• Voss, Jochen: 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…
• Voss, Jochen: Upper and Lower Bounds in Exponential Tauberian Theorems. Tbilisi Mathematical Journal, vol. 2, pp. 41–50, 2009.
  link, arXiv:0908.0642, preprint:pdf, 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):
  • Jochen Voß: Übungsaufgaben zur Analysis. Available for download from the author's homepage, 2003.
    preprint:pdf, more…