Workshop on Stochastic Differential Equations and...
Transcript of Workshop on Stochastic Differential Equations and...
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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
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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
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Today
• Continuous Time Continuous Space Processes
• Stochastic Integrals
• Ito Stochastic Differential Equations
• Analysis of Ito SDE
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CONTINUOUS TIME CONTINUOUS SPACE PROCESSES
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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
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Transition pdf
• The transition pdf is given by
• Process is homogenous if
• In this case
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Chapman Kolmogorov Equations
• For a continuous time continuous space process the Chapman Kolmogorov Equations are
• If
• The C-K equation in this case become
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From Random Walk to Brownian Motion
• Let X(t) be a DTMC (governing a random walk)
• Note that if
• Then satisfies
Provided
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Symmetric Random Walk: ‘Brownian Motion’
• In the symmetric case satisfies
• If the initial data satisfies
• The pdf of evolves in time as
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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 =
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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)
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Equivalent Conditions
Equivalently
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Kolmogorov Equations
• Using the C-K equations and the finiteness conditions we can derive the Backward Kolmogorov Equation
• For a homogenous process
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The Forward Equation
• THE FKE (Fokker Planck equation) is given by
• If the BKE is written as
• The FKE is given by
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Brownian Motion Revisited
• The FKE and BKE are the same in this case
• If X(0)=0, then the pdf is given by
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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)
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Weiner Process is a Diffusion Process
• Let
• Then
• These are the conditions for a diffusion process
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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
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Properties of Ito Stochastic Integral
• Linearity
• Zero Mean
• Ito Isometry
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Evaluation of some Ito Integrals
Not equal to Riemann Integrals!!!!
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Ito Stochastic Differential Equations
• A Stochastic Process is said to satisfy an Ito SDE
if it is a solution of
Riemann Ito
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Existence & Uniqueness Results
• Stochastic Process X(t) which is a solution of
if the following conditions hold
Similarity to Lipchitz Conditions!!
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Evolution of the pdf
• The solution of an Ito SDE is a diffusion process
• It’s pdf then satisfies the FKE
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Some Ito Stochastic Differential Equations
• Arithmetic Brownian Motion
• Geometric Brownian Motion
• Simple Birth and Death Process
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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!!
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Solving SDE using Ito’s Lemma
• Geometric Brownian Motion
• Let
• Then the solution is
• Note that