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

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

Today Numerical Schemes for ODE Numerical Evaluation of Stochastic Integrals Euler Maruyama Method for SDE Milstein and Higher Order Methods for SDE

Numerical Methods for Ordinary Differential Equations

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

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

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

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

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

How Error Varies with ∆t Claim : We saw theoretically Euler’s Method is O(∆t) accurate Error 1 0.718 ½ 0.468 ¼ 0.277 1/8 0.152 1/16 0.082

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

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

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

The Weiner Process

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

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

Sample Paths for Weiner Process

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

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

Euler Maruyama Scheme for SDE

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

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)”

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

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

E-M Scheme for Exponential Growth

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

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

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

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

Mean for the Stochastic Oscillator

Variance for the Stochastic Oscillator

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

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

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. 525-546 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 www.mathstat.strath.ac.uk/d.j.higham