Definition of a filtration and a stochastic process in discrete time.
Definition of a trajectory/sample path, an adapted stochastic process and a predictable stochastic process in discrete time.
Definition of a martingale in discrete time.
Definition of a supermartingale and a submartingale in discrete time.
Doob decomposition theorem.
Definition of a stopping time in descrete time.
Definition and porperties of a sigma-field in discrete time.
Price bounds for a Call Option, Put-Call-Parity and Merton’s Theorem.
Description a financial market model.
Definition of an admissible trading strategy, its value and a selffinancing trading strategy.
Characterization of selffinancing trading strategies.
Definition of an arbitrage opportunity.
Definition of an equivalent martingale measure.
Lemma about martingale measures and selffinancing strategies.
First fundamental theorem of asset pricing.
Definition of an attainable contingent claim, its fair price and market completion.
Theorem of risk neutral pricing.
Second fundamental theorem of asset pricing.
Formal description of the CRR model and probability of an up-movement.
Definition of the Snell envelope.
Characterization of optimal stopping times.
Theorem about the fair price of an american option in an arbitrage-free and complete security market.
Definition of a stochastic process, an adapted stochastic processes, a marginal distribution of stochastic processes and the trajectory of a stochastic process in continuous time.
Definition of submartingales, supermartingales and martingales in continuous time.
Definition of semimartingales in continuous time.
Definition of a markov process in continuous time.
Definition of stopping times in continuous time.
Definition of a sigma-field in continuous time.
Optional sampling theorem.
Definition of a Brownian motion.
Elementary properties of a Brownian motion.
Stochastic properties of a Brownian motion.
Definition of a partition and the mesh of a partition.
Definition of the first variation.
Definition of the quadratic variation.
Proposition about continuous function with either finite variation or or which are strictly increasing.
Quadratic variation of a semimartingale.
Theorem about the quadratic variation of Brownian motion.
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