Volterra processes in finance
Processus de Volterra en Finance
Résumé
Empirical studies indicate the presence of memory and strong inter-temporal dependence across various phenomena in the fields of finance and economics. The Brownian motion and Poisson processes, characterized by independent increments, are not suitable for modeling such phenomena.
We will consider Stochastic Volterra processes: a class of processes which extends the standard Brownian motion and Poisson processes to include memory; the fractional Brownian motion and Hawkes processes constitute a special case.
First, we develop the mathematical tools needed to deal with these stochastic Volterra integral equations that go beyond the standard stochastic calculus theory of Markovian processes and semimartingales.
Second, we explore the modeling flexibility of such equations in introducing memory in a broad range of problem in finance and economy including: volatilit modeling, portfolio allocation, optimal execution, principal agency, mean-field games...
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