Workshop on Stochastic Differential Equations and Statistical Inference for Markov Processes

1. Workshop on Stochastic Differential Equations and Statistical Inference for Markov Processes Day 1: January 19th , Day 2: January 28th Lahore University of Management Sciences

2. Schedule • Day 1 (Saturday 21st Jan): Review of Probability and Markov Chains • Day 2 (Saturday 28th Jan): Theory of Stochastic Differential Equations • Day 3 (Saturday 4th Feb): Numerical Methods for Stochastic Differential Equations • Day 4 (Saturday 11th Feb): Statistical Inference for Markovian Processes

3. Today • Continuous Time Continuous Space Processes • Stochastic Integrals • Ito Stochastic Differential Equations • Analysis of Ito SDE

4. Continuous Time Continuous space processes

5. Mathematical Foundations X(t) is a continuous time continuous space process if • The State Space is or or • The index set is X(t) has pdf that satisfies X(t) satisfies the Markov Property if

6. Transition pdf • The transition pdf is given by • Process is homogenous if • In this case

7. Chapman Kolmogorov Equations • For a continuous time continuous space process the Chapman Kolmogorov Equations are • If • The C-K equation in this case become

8. From Random Walk to Brownian Motion • Let X(t) be a DTMC (governing a random walk) • Note that if • Then satisfies Provided

9. Symmetric Random Walk: ‘Brownian Motion’ • In the symmetric case satisfies • If the initial data satisfies • The pdf of evolves in time as

10. Standard Brownian Motion • If and the process is called standard Brownian Motion or ‘Weiner Process’ • Note over time period • Mean = • Variance = • Over the interval [0,T] we have • Mean = • Variance =

11. Diffusion Processes • A continuous time continuous space Markovian process X(t), having transition probability is a diffusion process if the pdf satisfies • i) • ii) • Iii)

12. Equivalent Conditions Equivalently

13. Kolmogorov Equations • Using the C-K equations and the finiteness conditions we can derive the Backward Kolmogorov Equation • For a homogenous process

14. The Forward Equation • THE FKE (Fokker Planck equation) is given by • If the BKE is written as • The FKE is given by

15. Brownian Motion Revisited • The FKE and BKE are the same in this case • If X(0)=0, then the pdf is given by

16. Weiner Process • W(t) CT-CS process is a Weiner Process if W(t) depends continuously on t and the following hold a) • are independent

17. Weiner Process is a Diffusion Process • Let • Then • These are the conditions for a diffusion process

18. Ito Stochastic Integral • Let f(x(t),t) be a function of the Stochastic Process X(t) • The Ito Stochastic Integral is defined if • The integral is defined as • where the limit is in the sense that given means

19. Properties of Ito Stochastic Integral • Linearity • Zero Mean • Ito Isometry

20. Evaluation of some Ito Integrals Not equal to Riemann Integrals!!!!

21. Ito Stochastic Differential Equations • A Stochastic Process is said to satisfy an Ito SDE if it is a solution of RiemannIto

22. Existence & Uniqueness Results • Stochastic Process X(t) which is a solution of if the following conditions hold Similarity to Lipchitz Conditions!!

23. Evolution of the pdf • The solution of an Ito SDE is a diffusion process • It’s pdf then satisfies the FKE

24. Some Ito Stochastic Differential Equations • Arithmetic Brownian Motion • Geometric Brownian Motion • Simple Birth and Death Process

25. Ito’s Lemma • If X(t) is a solution of and F is a real valued function with continuous partials, then Chain Rule of Ito Calculus!!

26. Solving SDE using Ito’s Lemma • Geometric Brownian Motion • Let • Then the solution is • Note that