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

2. Schedule Day 1 ( Saturday 21 st Jan ): Review of Probability and Markov Chains Day 2 ( Saturday 28 th Jan ): Theory of Stochastic Differential Equations Day 3 ( Saturday 4 th Feb ): Numerical Methods for Stochastic Differential Equations Day 4 ( Saturday 11 th 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) b) are independent c)

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 Riemann Ito

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