# Statistical inference for difference and differential equations

ST3201 Master Seminar in Statistics on Tuesdays 14:15-15 in room 1138 in Sentralbygg 2 with Professor Gunnar Taraldsen.

This is given as a guided self-study course leading to the Kalman filter and more general prediction and inference methods for difference and differential equations. The contents will be selected topics from the reading list given below.

**Reading list:**

Brockwell and Davis (1991): Time Series: Theory and Methods, Springer

Brockwell and Davis (2016): Introduction to Time Series and Forecasting, Springer

Øksendal (2013): Stochastic Differential Equations, Springer

Schervish (1995): Theory of statistics, Springer

**Reading history and plan:**

27.11.2018: **EXAM** in room 1138 Sentralbygg 2. Filip 0915-10, Marcus 1015-11.

20.11.2018: Schervish: Chap 5, Estimation

13.11.2018: Øksendal: Chap 6, The filtering problem, Theorem 6.2.8 (The 1-dimensional Kalman-Bucy filter)

06.11.2018: Brockwell and Davis, 2016, Chap 9: State-Space Models, the Kalman filter and estimation

30.10.2018: Brockwell and Davis, 2016, Chap 11.5 Continuous-Time ARMA Processes

23.10.2018: Brockwell and Davis, 2016, Chap 7.5-.6 Time Series Models for Financial Data

16.10.2018: No meeting. Self-study of Øksendal: Chap 3, Itô Integrals

09.10.2018: Brockwell and Davis, 2016, Appendix D, Lévy Processes, Brownian Motion and Itô Calculus

02.10.18: Øksendal: Chap 6, The filtering problem, Outline, Example 6.2.1 and Lemma 6.2.2

25.09.18: B&D: Ch 3 and Ch 5.7. The Wold decomposition. Existence and uniqueness for ARMA.

18.09.18: Brockwell and Davis, Chap 3, ARMA processes. Homogeneous Linear Difference Equations with Constant
Coefficients. The Autocovariance Generating Function.

11.09.18: Brockwell and Davis, Chap 2, Hilbert space theory for time series. Linear Regression and the General Linear Model.
Mean Square Convergence, Conditional Expectation and Best Linear Prediction

04.09.18: Brockwell and Davis, Chap 2, Elementary Hilbert space theory.

**Curriculum: **

Beware that it is assumed that basic concepts needed for an understanding of the curriculum is known a priori, and if not you must prepare for this in addition. For Chap 5 in Schervish this means in particular that you must be familiar with basic probability and statistics including for instance the concept of a random variable and a sufficient statistic as defined by Schervish.

1) Brockwell and Davis (2016): Chap 7.5-7.6, 9.1-9.5

2) Brockwell and Davis (1991): Chap 2, 3, 5.1, 5.5, 5.7, 5.8

3) Øksendal (2013): Chap 3, 6

4) Schervish (1995): Chap 5 minus 5.1.2, 5.1.5, 5.2.3, 5.2.5