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Last update: T_KPMS (01.06.2016)
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Last update: T_KPMS (17.05.2013)
The main objective is to introduce the fundamentals of probability theory that are used in finance and insurance mathematics. |
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Last update: RNDr. Jitka Zichová, Dr. (25.04.2018)
Oral exam. |
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Last update: T_KPMS (11.05.2015)
P. Lachout: Diskrétní martingaly, skripta MFF UK B. Oksendal: Stochastic Differential Equations, Springer-Verlag, 2010 (sedmé vydání) I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus, Springer-Verlag, 1988 (první vydání) J. M. Steele, Stochastic Calculus and Financial Applications, Springer-Verlag, 2001 |
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Last update: T_KPMS (17.05.2013)
Lecture. |
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Last update: prof. RNDr. Bohdan Maslowski, DrSc. (26.02.2018)
The examination will be oral.
Requirements (may be slightly modified according to the stuff talked over):
Basic Notions and Theorems: 1. Basics from the theory of random processes : Conditional expectations w.r.t. sigma-algebra, its basic properties and interpretation., Daniell-Kolmogorov, the principle of construction of a random process by its finite-dimensional distributions. Kolmogorov-Chentsov continuity test and application to the existence proof of the Wiener process (the basic idea).
2. Martingales: Stochastic basis, filtrations, usual conditions. Adapted and progressively measurable processes and relations between them. The concept of (sub-, super-) martingales, L^2- martingales, local martingales, examples. Basic properties of martingales. Stopping times and their basic properties, examples. Measurability of the stopped process. Optional Sampling Thm. Maximal inequalities. Concept of increasing natural (predictable) process, the class (DL). Doob-Meyer decomposition of a contuinuous submartingale and applications in the definition of quadratic variation. Examples of D-M decompositions of martingales. Quadratic variation as a limit of partial sums.
3. Stochastic Analysis: Wiener process - motivation, definition and basic properties, what is the relation the the concept of white noise? Stochastic integral - definition and a basic properties, connection with the Lebesgue - Stieltjes integral. Stochastic differential and Ito formula - formulation and applications in the examples. Stochastic bilinear equation (dX= f(t)X dt + g(t)X dW) and geometric Brownian motion, the formula for solution . Linear equation (dX=f(t)X dt + g(t)dW), the formula for solution (variation of constants) .
Proofs: Show that Wiener and the "compensated" Poisson process are martingales and find their Doob-Meyer decomposition, proof of Ito isometry for step functions. Intuitive derivation of Ito formula. Proof of the formulas for solutions to bilinear and linear equations (the two examples following the Ito lemma). Obviously, the knowledge of all definitions and basically all statements presented during the lecture is necessary for the exam to be successful. The level of understanding of the topic is important as well. |
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Last update: T_KPMS (11.05.2015)
1. Conditional expectation w.r.t. sigma-algebra, random process, finite-dimensional distributions, Daniell-Kolmogorov and Kolmogorov-Chentsov theorems. 2. Martingales, definition of super- and submartingales, filtration, basic examples. Stopping times and hitting times of a subset of the state space by a random process. Maximal inequalities, Doob-Meyer decomposition. 3. Quadratic variation of martingales, Wiener process and its basic properties. 4. Stochastic integration w.r.t. Wiener process, definition and basic properties. Stochastic differential and Ito formula, examples. 5. Stochastic integration w.r.t. martingales - an introduction. |
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Last update: prof. RNDr. Bohdan Maslowski, DrSc. (24.05.2018)
In order to enroll, basic knowledge of standard calculus and probability fundmantals (based on the concept of probability as a measure and mathematical expectation as the Lebesgue integral) is required. |