Stochastic Processes Course Notes

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By: Anton Wakolbinger

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1 Discrete Markov chains

1.1 Random paths; stochastic and Markovian dynamics

1.2 Excursions from a state; recurrence and transience

1.3 Renewal chains

1.4 Equilibrium distributions

1.5 The ergodic theorem for Markov chains

1.6 Convergence to equilibrium

1.7 Optimal Stopping

1.8 Renewal chains revisited

2 Renewal processes

2.1 The renewal points and the residual lifetime process

2.2 Stationary renewal processes

2.3 Convergence to equilibrium

2.4 Homogeneous Poisson processes on the line

3 Poisson processes

3.1 Heuristics

3.2 Characterization

3.3 Construction

3.4 Independent labelling and thinning

3.5 Poisson integrals, subordinators

and Levy processes

4 Markov chains in continuous time

4.1 Jump rates

4.2 The minimal process and its transition semigroup

4.3 Backward and forward equations

4.4 Revival after explosion

4.5 Standard transition semigroups and their Q-matrices

5 Conditional Expectation

6 Martingales

6.1 Basic concepts

6.2 The supermartingale convergence theorem

6.3 Doob's submartingale inequalities

6.4 Stopping times

6.5 Stopped supermartingales

7 The Wiener Process

7.1 Heuristics and basics

7.2 Levy's construction of W

7.3 Quadratic variation of Wiener paths

7.4 Intermezzo: Filtrations and stopping in continuous time

7.5 The Ito-integral for simple integrands

7.6 Integrators of locally finite variation

7.7 Continuous local martingales as integrators

7.8 Stochastic calculus for continuous local semimartingales

7.9 Levy's characterisation of W

7.10 Reweighting the probability = changing the drift

7.11 A strategy for (almost) all cases

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