## 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...

See also: Equations of Motion from a Time series

- 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

- To read:
- Jose Bento, Morteza Ibrahimi and Andrea Montanari
- "Learning Networks of Stochastic Differential Equations", NIPS 23 (2010), arxiv:1011.0415
- "Information Theoretic Limits on Learning Stochastic Differential Equations", arxiv:1103.1689

- 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