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Root number
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11467 |
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Semester
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HS2023 |
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Type of course
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Lecture |
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Allocation to subject
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Statistics |
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Type of exam
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not defined |
| Title |
Stochastic Processes I |
| Description |
Conditional expectation. (Properties, the law of the iterated expectation (the total probability formula. Calculation of the conditional expectation of X given Y if the joint probability density function is given.)
Martingales in discrete time. (Definition of (sub, super) martingale. Main examples (sums, conditional expectations with respect to a filtration, martingale transform. The Doob decomposition, quadratic variation. Stopping time, stopping sigma-algebra. Elementary cases of the optional sampling theorem (bounded stopping time, bounded martingale, uniform integrable martingale). The Doob optional sampling theorem, optional sampling theorem with condition on the increments. Wald's identities. Application to the ruin problem. Uniformly integrable sequences, criterion for the uniform integrability based on the moments. Convergence of supermartingales, and the variants for (sub) martingales. Convergence in L1. The case of L2-bounded martingales. L'evy theorem for convergence of conditional expectations. Applications to the Borel-Cantelli theorem and the zero-one law. Strong law of large numbers and reversed martingales Appearance of patterns in sequences. Doob fundamental inequalities. Square-integrable and Lp-bounded cases.
Markov chains (Definition, transition probabilities, the Chapman-Kolmogorov equation. The strong Markov property. Return times, numbers of visits and the classification of states. Doeblin's theorem on the existence and uniqueness of the stationary distribution. Various interpretations of the stationary distribution. Ergodic theorem. Mean return times. Time spent in transient states. Harmonic and superharmonic functions, relation to martingales. Moving from transient states to recurrent classes. Optimal stopping (recurrent relation for the optimal payofffunction). Time reversal and the stationary distribution. Countable Markov chains, positive and null recurrence. The stationary distribution. Recurrence criterion based on harmonic functions. Random walk, recurrence criterion for random walks on the line. The Chung-Fuchs theorem and its variants. Continuous time finite Markov chains. Infinitesimal matrix, the construction of a Markov chain. Birth and death processes, recurrence and transience. |
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ILIAS-Link (Learning resource for course)
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Registrations are transmitted from CTS to ILIAS (no admission in ILIAS possible).
ILIAS
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Link to another web site
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| Lecturers |
Prof. Dr.
Ilya Molchanov, Institute of Mathematical Statistics and Actuarial Science ✉
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ECTS
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6 |
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Recognition as optional course possible
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Yes |
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Grading
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1 to 6 |
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| Dates |
Monday 08:15-10:00 Weekly
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Thursday 10:15-12:00 Weekly
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Tuesday 23/1/2024 10:00-11:30
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| Rooms |
Hörsaal B005, Exakte Wissenschaften, ExWi
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Hörsaal B007, Exakte Wissenschaften, ExWi
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| Students please consult the detailed view for complete information on dates, rooms and planned podcasts. |
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