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
¼
0.468
1/8
0.152
1/16
0.082
0.277
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
Process if
–
– For
– For
is a Weiner
the increment
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