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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