1. Workshop on Stochastic Differential Equations and Statistical Inference for Markov Processes Day 3: Numerical Methods for Stochastic Differential Equations
Day 1: January 19th , Day 2: January 28th
Day 3: February 9th
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 Numerical Schemes for ODE
Numerical Evaluation of Stochastic Integrals
Euler Maruyama Method for SDE
Milstein and Higher Order Methods for SDE

4. Numerical Methods for Ordinary Differential Equations

5. Euler’s Scheme Consider the following IVP
Using a forward difference approximation we get
This is called the Forward Euler Scheme

6. A Simple Example
Consider the IVP
The solution to the IVP is

7. Solving the IVP by Euler’s Method
For the IVP
The Euler Scheme is

9. Error How to characterize the error ? Factors which introduce an error
Discretization
Round off
Maximum of error over the interval
How does the error depend on

10. Discretization Error in Forward Euler
Consider the IVP
Satisfying the conditions
Also consider the Euler Scheme
Then the error satisfies

11. How Error Varies with ∆t
Claim : We saw theoretically Euler’s Method is O(∆t) accurate
Error 1
0.718
0.468
0.277
1/8
0.152
1/16
0.082

12. Stability Consider The Euler Scheme is
For the solution to die out need
For

13. Stability of Euler Scheme
For
Discretize using Euler’s Scheme
At some stage of the solution assume a small error is introduced
The error evolves according to
Thus need for stability

14. Challenge
Write a code to verify the order of accuracy of the Euler Scheme
Experiment with different values of to explore the stability of the Euler Scheme
Note: You may use the IVP discussed here

15. The Weiner Process

16. Weiner Process Recall a random variable is a Weiner Process if
For the increment
For the increments
are independent

17. Simulating Weiner Processes
Consider the discretization
where and
Also each increment is given by

18. Sample Paths for Weiner Process

19. Numerical Expectation and Variance
Theoretically on the interval [0,t]

20. Stochastic Exponential Growth
The Exponential Growth Model is
Let
Then the solution is
Note that

22. Euler Maruyama Scheme for SDE

23. Sources of Error in Numerical Schemes
Errors in Numerical Schemes for SDE
Discretization
Monte Carlo
Round off
Discretization determines the order of the scheme as in the ODE case
Also want a handle on the Monte Carlo errors

24. Some Numerical Schemes for SDE
Euler Maruyama
Half order accurate
Milstein
Order one accurate
Reference: “Numerical Solution of Stochastic Differential Equations by Kloeden and Platen (Springer)”

25. Euler Maruyama Scheme Consider an autonomous SDE
A Simple (Euler-Maruyama) discretization is

26. E-M Applied of Exponential Growth
Consider
This has the solution
The Euler Murayama Scheme takes the form

27. E-M Scheme for Exponential Growth

28. Strong Accuracy of E-M
A method converges with strong order if there exists C such that
For the Euler Maruyama Scheme the following holds
i.e. E-M is order accurate

30. Weak Accuracy of E-M
A method converges with weak order if there exits C such that
For the Euler Maruyama Scheme the following holds true

32. Stochastic Oscillator
Consider the stochastically forced oscillator
The mean and variance are given by

33. Numerical Scheme
We simulate the oscillator using the following scheme (Higham & Melbo)
Note the semi implicit nature of the method

34. Mean for the Stochastic Oscillator

35. Variance for the Stochastic Oscillator

36. Challenge I
Derive the exact mean and variance for the stochastic oscillator
Use Euler Maruyama to simulate trajectories and calculate the mean and variance
Show numerically that the variance blow up with decreasing for the E-M method

37. Challenge II Exploring the Stochastic SIR Model
Use the references provided on the webpage to simulate sample paths for the infected class for different parameters
Calculate the numeric mean and variance

38. References and Credits
Kloeden. P.E & Platen.E, Numerical Solution of Stochastic Differential Equations, Springer (1992)
Desmond J. Higham. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , SIAM Rev. 43, pp
Atkinson. K, Han W. & Stewart D.E, Numerical Solution of Ordinary Differential Equations, Wiley
Many of the codes are available at Desmond Higham's webpage