Download Workshop on Stochastic Differential Equations and

Workshop on Stochastic
Differential Equations and
Statistical Inference for Markov
Processes
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
• The index set is
or
X(t) has pdf that satisfies
X(t) satisfies the Markov Property if
or
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
Provided
satisfies
Symmetric Random Walk: ‘Brownian
Motion’
• In the symmetric case
• If the initial data satisfies
• The pdf of
evolves in time as
satisfies
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)
c)
are independent
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