Workshop on Stochastic Differential Equations and Statistical Inference

for Markov Processes (Day 2)

Day 1: January 19th , Day 2: January 28th

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

• Continuous Time Continuous Space Processes

• Stochastic Integrals

• Ito Stochastic Differential Equations

• Analysis of Ito SDE

CONTINUOUS TIME CONTINUOUS SPACE PROCESSES

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

Transition pdf

• The transition pdf is given by

• Process is homogenous if

• In this case

Chapman Kolmogorov Equations

• For a continuous time continuous space process the Chapman Kolmogorov Equations are

• If

• The C-K equation in this case become

From Random Walk to Brownian Motion

• Let X(t) be a DTMC (governing a random walk)

• Note that if

• Then satisfies

Provided

Symmetric Random Walk: ‘Brownian Motion’

• In the symmetric case satisfies

• If the initial data satisfies

• The pdf of evolves in time as

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 =

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)

Equivalent Conditions

Equivalently

Kolmogorov Equations

• Using the C-K equations and the finiteness conditions we can derive the Backward Kolmogorov Equation

• For a homogenous process

The Forward Equation

• THE FKE (Fokker Planck equation) is given by

• If the BKE is written as

• The FKE is given by

Brownian Motion Revisited

• The FKE and BKE are the same in this case

• If X(0)=0, then the pdf is given by

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)

Weiner Process is a Diffusion Process

• Let

• Then

• These are the conditions for a diffusion process

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

Properties of Ito Stochastic Integral

• Linearity

• Zero Mean

• Ito Isometry

Evaluation of some Ito Integrals

Not equal to Riemann Integrals!!!!

Ito Stochastic Differential Equations

• A Stochastic Process is said to satisfy an Ito SDE

if it is a solution of

Riemann Ito

Existence & Uniqueness Results

• Stochastic Process X(t) which is a solution of

if the following conditions hold

Similarity to Lipchitz Conditions!!

Evolution of the pdf

• The solution of an Ito SDE is a diffusion process

• It’s pdf then satisfies the FKE

Some Ito Stochastic Differential Equations

• Arithmetic Brownian Motion

• Geometric Brownian Motion

• Simple Birth and Death Process

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!!

Solving SDE using Ito’s Lemma

• Geometric Brownian Motion

• Let

• Then the solution is

• Note that