ont></strong><br>
  <strong><font size="3">Professor Dr. Malkhaz Shashiashvili</font></strong></div>
<br>
<p><strong>Topic 1. Martingale, Stopping times and Filtrations</strong><br>
  1.1 Stochastic Processes and - Fields<br>
  1.2 Stopping times<br>
  1.3 Continuous-times Martingales<br>
  1.4 The Doob-Mayer Decomposition<br>
  1.5 Continuous, Square-Integrable Martingales</p>
<p><strong>Topic 2. Brownian Motion</strong><br>
  2.1 The construction of Brownian Motion<br>
  2.2 The Space , Weak Convergence and Wiener Measure<br>
  2.3 The Markov Property<br>
  2.4 The Strong Markov Property and the reflection Principal<br>
  2.5 Brownian Filtration<br>
  2.6 The Brownian Sample Paths</p>
<p><strong>Topic 3. Stochastic Integration</strong><br>
  3.1 Construction of the Stochastic Integral<br>
  3.2 The Change-of-Variable Formula (The Ito rule)<br>
  3.3 Representations of Continuous Martingales in term of Brownian Motion<br>
  3.4 The Girsanov Theorem</p>
<p><strong>Topic 4. Brownian Motion and Partial Differentoial Equations</strong><br>
  4.1 Harmonic Functions and the Dirichlet Problem<br>
  4.2 The one Dimensional Heat equation<br>
  4.3 The Formulas of Feynman and Kac</p>
<p><strong>Topic 5. Stochastic Differential Equations</strong><br>
  5.1 Strong Solutions<br>
  5.2 Weak Solutions<br>
  5.3 Gauss-Markov Processes<br>
  5.4 The General, one-dimensional linear equation<br>
  5.5 Connections with Partial Differential Equations: The Dirichlet problem. 
  The Cauchy problem and a Feynman-Kac representation.</p>
<p><strong>References </strong><br>
  1. I. Karatzas, St. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 
  1998.<br>
  2. Ph. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, 
  1992.<br>
  3. A. N. Shiryaev, Probability, Springer, 1995.</p>
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