Transforms of RVs

I have a nonlinear transformation of a random process. What is its distribution?

Stochastic Itō-Taylor expansion

See stochastic taylor expansion. tl;dr: More trouble than it is worth.


As seen in the Ensemble Kalman Filter.

Unscented transform

The great invention of Uhlmann and Julier is unscented transform, which uses a ‘\(\sigma\)-point approximation.’

In the context of Kalman filtering,

What the Unscented Transform does is to replace the mean vector and its associated error covariance matrix with a special set of points with the same mean and covariance. In the case of the mean and covariance representing the current position estimate for a target, the UT is applied to obtain a set of points, referred to as sigma points, to which the full nonlinear equations of motion can be applied directly. In other words, instead of having to derive a linearized approximation, the equations could simply be applied to each of the points as if it were the true state of the target. The result is a transformed set of points, and the mean and covariance of that set represents the estimate of the predicted state of the target.

See, e.g., Roth, Hendeby, and Gustafsson (2016).


Aït-Sahalia, Yacine, Lars Peter Hansen, and José A. Scheinkman. 2010. “Operator Methods for Continuous-Time Markov Processes.” In Handbook of Financial Econometrics: Tools and Techniques, 1–66. Elsevier.
Ariffin, Noor Amalina Nisa, and Norhayati Rosli. 2017. “Stochastic Taylor Expansion of Derivative-Free Method for Stochastic Differential Equations.” Malaysian Journal of Fundamental and Applied Sciences 13 (3).
Collard, Fabrice, and Michel Juillard. 2001. “Accuracy of Stochastic Perturbation Methods: The Case of Asset Pricing Models.” Journal of Economic Dynamics and Control 25 (6-7): 979–99.
Gustafsson, Fredrik, and Gustaf Hendeby. 2008. “On Nonlinear Transformations of Stochastic Variables and Its Application to Nonlinear Filtering.” In 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 3617–20.
———. 2012. “Some Relations Between Extended and Unscented Kalman Filters.” IEEE Transactions on Signal Processing 60 (2): 545–55.
Hendeby, Gustaf, and Fredrik Gustafsson. 2007. “On Nonlinear Transformations of Gaussian Distributions,” 3.
Jacob, Niels, and René L. Schilling. 2001. “Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser.
Jin, He-hui, Kenneth L Judd, and Hoover Insitution. n.d. “Perturbation Methods for General Dynamic Stochastic Models,” 44.
Kloeden, P. E., and E. Platen. 1991. “Stratonovich and Ito Stochastic Taylor Expansions.” Mathematische Nachrichten 151 (1): 33–50.
Kloeden, P. E., E. Platen, and I. W. Wright. 1992. “The Approximation of Multiple Stochastic Integrals.” Stochastic Analysis and Applications 10 (4): 431–41.
Kloeden, Peter E., and Eckhard Platen. 1992. “Stochastic Taylor Expansions.” In Numerical Solution of Stochastic Differential Equations, edited by Peter E. Kloeden and Eckhard Platen, 161–226. Applications of Mathematics. Berlin, Heidelberg: Springer.
———. 2010. Numerical Solution of Stochastic Differential Equations. Berlin, Heidelberg: Springer Berlin Heidelberg.
Papapantoleon, Antonis, and Maria Siopacha. 2010. “Strong Taylor Approximation of Stochastic Differential Equations and Application to the Lévy LIBOR Model.” October 4, 2010.
Roth, Michael, Gustaf Hendeby, and Fredrik Gustafsson. 2016. “Nonlinear Kalman Filters Explained: A Tutorial on Moment Computations and Sigma Point Methods 11 (1): 24.
Rößler, Andreas. 2004. “Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes.” Stochastic Analysis and Applications 22 (6): 1553–76.
Schmitt-Grohe, Stephanie, and Martın Uribe. n.d. “Perturbation Methods for the Numerical Analysis of DSGE Models: Lecture Notes,” 38.
Schmitt-Grohé, Stephanie, and Martı́n Uribe. 2004. “Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function.” Journal of Economic Dynamics and Control 28 (4): 755–75.
Schoutens, Wim, K U Leuven, and Michael Studer. 2001. “Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management,” February, 24.

Warning! Experimental comments system! If is does not work for you, let me know via the contact form.

No comments yet!

GitHub-flavored Markdown & a sane subset of HTML is supported.