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Stochastic Finance
Introduction Assets, Portfolios and Arbitrage Definitions and Formalism Portfolio Allocation and Short-Selling Arbitrage Risk-Neutral Measures Hedging of Contingent Claims Market Completeness Example…
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Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales
The Discrete Time Case.- Continuous Time Normal Martingales.- Gradient and Divergence Operators.- Annihilation and Creation Operators.- Analysis on the Wiener Space.- Analysis on the Poisson Space.-…
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Stochastic analysis of Bernoulli processes
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered…
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White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance
TLDR
A white noise approach to Malliavin calculus is used to prove the following white noise generalization of the Clark-Haussmann-Ocone formula: the replicating portfolio for a European call option in a Poissonian Black & Scholes type market is computed.
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Understanding Markov Chains: Examples and Applications
Introduction 1 Probability Background 1.1 Probability Spaces and Events 1.2 Probability Measures 1.3 Conditional Probabilities and Independence 1.4 Random Variables 1.5 Probability Distributions 1.6…
Determinantal Point Processes
In this survey we review two topics concerning determinantal (or fermion) point processes. First, we provide the construction of diffusion processes on the space of configurations whose invariant…
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Computations of Greeks in a market with jumps via the Malliavin calculus
TLDR
Using the Malliavin calculus on Poisson space, Greeks in a market driven by a discontinuous process with Poisson jump times and random jump sizes is computed, and it is shown that Asian options can be considered due to a smoothing effect of the integral over time.
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Concentration and deviation inequalities in infinite dimensions via covariance representations
Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on…
Convex concentration inequalities and forward-backward stochastic calculus
Given $(M_t)_{t\in \mathbb{R}_+}$ and $(M^*_t)_{t\in \mathbb{R}_+}$ respectively a forward and a backward martingale with jumps and continuous parts, we prove that $E[\phi (M_t+M^*_t)]$ is…
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An extension of stochastic calculus to certain non-Markovian processes
By time changes of Lévy processes we construct two operators on Fock space whose sum is a second quantized operator, and that complement the annihilation and creation operators whose probabilistic…
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