Notebooks

## Inference for Stochastic Differential Equations

28 Jun 2021 00:18

Spun off from stochastic differential equations, and/or inference for Markov models.

To be clear, I'm considering situations in which we observe a trajectory $x(t)$ of a stochastic process $X$ that obeys an SDE $dx = a(x;\theta) dt + b(x;\theta) dW$ and want to do inference on the parameter $\theta$. (The "parameter" here might be a whole function.)

The "easy" case is discrete-time, equally-spaced data, without loss of generality $x(0), x(h), x(2h), \ldots x(nh)$. Because $X$ is (by hypothesis) a Markov process, there is a conditional probability kernel $P_h(y|x;\theta)$, which one could find by integrating the generator of the SDE, and the log-likelihood is just $L(\theta) = \sum_{t=1}^{n}{\log{P_h(x(th)|x((t-1)h); \theta)}}$ Of course, "just" integrating the generator is not necessarily an easy issue...

Recommended, big picture:
• Stefano M. Iacus, Simulation and Inference for Stochastic Differential Equations: With R Examples
Recommended, close-ups:
• Luca Capriotti
• "A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion", physics/0703180
• "The Exponent Expansion: An Effective Approximation of Transition Probabilities of Diffusion Processes and Pricing Kernels of Financial Derivatives", International Journal of Theoretical and Applied Finance 9 (2006): 1179--1199, physics/0602107
• Jose Bento, Morteza Ibrahimi and Andrea Montanari
• Xi Chen, Ilya Timofeyev, "Non-parametric estimation of Stochastic Differential Equations from stationary time-series", arxiv:2007.08054
• Daan Crommelin, "Estimation of Space-Dependent Diffusions and Potential Landscapes from Non-equilibrium Data", Journal of Statistical Physics 149 (2012): 220--233
• Serguei Dachian, Yury A. Kutoyants, "On the Goodness-of-Fit Tests for Some Continuous Time Processes", arxiv:0903.4642 ["We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes"]
• Arnak Dalalyan and Markus Reiss, "Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case", math.ST/0505053
• A. De Gregorio and S. M. Iacus, "Adaptive Lasso-type estimation for ergodic diffusion processes", arxiv:1002.1312
• D. Dehay and Yu. A. Kutoyants, "On confidence intervals for distribution function and density of ergodic diffusion process", Journal of Statistical Planning and Inference 124 (2004): 63--73
• D. Florens and H. Pham, "Large Deviations in Estimation of an Ornstein-Uhlenbeck Model," Journal of Applied Probability 36 (1999): 60--77
• Shota Gugushvili, Peter Spreij, "Parametric inference for stochastic differential equations: a smooth and match approach", arxiv:1111.1120
• Stefano M. Iacus
• "Statistical analysis of stochastic resonance with ergodic diffusion noise," math.PR/0111153
• "On Lasso-type estimation for dynamical systems with small noise", arxiv:0912.5078
• D. Kleinhans, R. Friedrich, A. Nawroth and J. Peinke, "An iterative procedure for the estimation of drift and diffusion coefficients of Langevin processes", Physics Letters A 346 (2005): 42--46, physics/0502152
• Yury A. Kutoyants
• Statistical Inference for Ergodic Diffusion Processes
• "On the Goodness-of-Fit Testing for Ergodic Diffusion Processes", arxiv:0903.4550
• "Goodness-of-Fit Tests for Perturbed Dynamical Systems", arxiv:0903.4612
• Chenxu Li, "Maximum-likelihood estimation for diffusion processes via closed-form density expansions", Annals of Statistics 41 (2013): 1350--1380
• Martin Lysy, Natesh S. Pillai, "Statistical Inference for Stochastic Differential Equations with Memory", arxiv:1307.1164
• Javier R. Movellan, Paul Mineiro, and R. J. Williams, "A Monte Carlo EM Approach for Partially Observable Diffusion Processes: Theory and Applications to Neural Networks," Neural Computation 14 (20020: 1507--1544
• Ilia Negri, "Efficiency of a class of unbiased estimators for the invariant distribution function of a diffusion process", math.ST/0609590
• Ilia Negri and Yoichi Nishiyama, "Goodness of fit test for ergodic diffusions by tick time sample scheme", Statistical Inference for stochastic Processes 13 (2010): 81--95
• Jun Ohkubo, "Nonparametric model reconstruction for stochastic differential equations from discretely observed time-series data", Physical Review E 84 (2011): 066702
• B. L. S. Prakasa Rao, Statistical Inference for Diffusion-Type Proccesses
• E. Racca and A. Porporato, "Langevin equations from time series", Physical Review E 71 (2005): 027101
• Aad van der Vaart and Harry van Zanten, "Donsker theorems for diffusions: Necessary and sufficient conditions", Annals of Probability 33 (2005): 1422--1451, math.PR/0507412
• Harry van Zanten, "On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators", Statistical Inference for Stochastic Processes 6 (2003): 199--213 ["In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the 'size' of the class [of functions] in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive."]
• J. H. van Zanten, "On the Uniform Convergence of the Empirical Density of an Ergodic Diffusion", Statistical Inference for Stochastic Processes 3 (2000): 251--262