Stochastic analysis I, spring 2017


Teacher: Dario Gasbarra 

Scope: 5 cr

Type: Advanced studies



Kolmogorov extension theorem and stochastic processes, Kolmogorov continuity criterium, Paul Levy Construction of Brownian motion. Processes with  jumps: Poisson process and counting processes, random measures.  General theory in  continuous time, filtration, predictable σ-algebra, stopping times and predictable times. Stochastic integrals with respect to processes with locally finite variation. Continuous time martingales, existence of  right continuous modification,  predictable and dual predictable projections,  compensator and Doob-Meyer decomposition, locally square integrable martingales, quadratic and predictable variation. Stochastic integration with respect to continuous martingales,  Kunita-Watanabe inequality, Ito isometry and Ito integral, Ito formula and generalizations.  Burkholder Davis Gundy inequalities. Ito-Tanaka formula and local times. Change of measure and Girsanov theorem. Stochastic differential equations, weak and strong solutions. Partial differential equations and Feynman-Kac formula. Applications: stochastic filtering, stochastic control, option pricing in mathematical finance.

Prerequisites:  Probability theory.


Teaching schedule (old).

Weeks 3-9, Tuesday 12-14 and Thursday 10-12 in room B120. Exercise class on Wednesdays 10-12 in B120. The first lecture is on tuesday 17.1,

The course continues in the IV teaching period as Stochastic analysis II,  with the new teaching schedule:

Weeks 11-18, lectures on Wednesday 12-14 in B120 and Thursday 12-14 in room D123, and exercises on Tuesday 10-12 in room D123

Easter holiday 13.-19.4.



Exam lasts 2,5 hours.

You can use (lecturer will fill in) in the exam.

Course materials:

lecture notes (last update on 10.5.2017)

levybm.m  octave function showing  Paul Levy construction of Brownian motion.

Main references:

 Richard Bass, Stochastic processes, Cambridge University Press 2011.

Fabrice Baudoin, Diffusion Processes and Stochastic Calculus. European Mathematical Society Ems Textbooks in Mathematics 2014.

Alexander Gushchin, Stochastic calculus for quantitative finance. ISTE Press, Optimization in insurance and finance 2015.

René L Schilling Lothar Partzsch, Brownian motion, an introduction to stochastic processes, De Gruyter 2012.

Also recommended:

Sheng-wu He, Jia-gang Wang, Jia-an Yan, Semimartingale Theory and Stochastic Calculus, CRC 1992.

Jean Jacod and Albert Shiryaev, Limit theorems for stochastic processes, 2nd edition Springer 2003.

Hui-Hsiung Kuo, Introduction to stochastic analysis, Springer 2006.

Course diary:    

  week 1:    We have discussed Kolmogorov extension theorem for a family of consistent finite dimensional probability distributions (without proof), proved Paul Levy theorem about the construction of Brownian motion, introducing also the Cameron Martin space.

 week 2,3: We have proved Kolmogorov continuity theorem, almost sure non-differentiability and Hölder-continuity of Brownian paths. We have also proved  Doob backward martingale convergence theorem (in discrete time), and shown that Brownian motion has quadratic variation  [ B ]t   = t , where the convergence is in Land  also almost surely when the sequence of discretizing partitions is refining (see Prop 7,8 in the lecture notes).


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Exercise classes

1.Wednesday 10-12 B120 Dario Gasbarra 

Course feedback

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