Study-unit STOCHASTIC PROCESS AND STOCHASTIC DIFFERENTIAL EQUATIONS
| Course name | Mathematics |
|---|---|
| Study-unit Code | A002324 |
| Curriculum | Matematica per l'economia e la finanza |
| Lecturer | Irene Benedetti |
| Lecturers |
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| Hours |
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| CFU | 6 |
| Course Regulation | Coorte 2023 |
| Supplied | 2023/24 |
| Supplied other course regulation | |
| Learning activities | Caratterizzante |
| Area | Formazione teorica avanzata |
| Sector | MAT/05 |
| Type of study-unit | Obbligatorio (Required) |
| Type of learning activities | Attività formativa monodisciplinare |
| Language of instruction | Italian on request the course can be held in English |
| Contents | Notions and techniques of Probability. Random walks, Markov chains. Stationary Processes, Martingales, Gaussian Processes. Brownian Motion and elements of Stochastic Calculus. |
| Reference texts | Grimmett-Stirzaker: Probability and Random Processes; Clarendon Press, Oxford (1982). |
| Educational objectives | Generally, after passing the exam, the student has a deep knowledge of the general properties of the main stochastic processes, and skillness in the methods of studying and connecting them, together with some ability in stochastic calculus. The students particularly motivated could be invited to face also some first-level research problems. |
| Prerequisites | Some basic notions of Elementary Probability and Measure Theory should be already known to the students. |
| Teaching methods | Lectures in classroom |
| Other information | Student office: https://www.unipg.it/personale/irene.benedetti/didattica visit the webpage: www.unistudium.unipg.it |
| Learning verification modality | Oral exam: the test usually lasts about 40 minutes. The student should give definitions, theorems and proofs contained in the program as well as solve some very simple exercises. The aim of the colloquium is to evaluate if and to what extent the student is acquainted with the main topics studied, and check his/her capability in handling them, establishing connections and consequences. |
| Extended program | A partial survey of Calculus of Probability. Generating functions and their utility. Random walks: distributions, first return time, reflecting properties and applications. Markov chains: transition matrix, recurrent and transient states, classification of states. Stationary distributions and their links with mean recurrence times. Applications to random walks. Stationary processes, ergodic theorems and application. Generation of random sequences. Martingales: general properties, convergence theorems, characterization in L_2. Optional theorem and Wald Formula. Gaussian processes: general theory, examples, Wiener process and its properties. Brownian Motion: existence and approximation, properties if its trajectories, scale invariance, Iterated Logarithm Theorem and the Arcsin Law. Stochastic Integration: Stieltjes and Ito integrals. Ito formulas and stochastic differentials. Stochastic differential equations: existence and uniqueness theorem, methods of solution in the linear case. |