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Tesi etd-04132021-171113


Tipo di tesi
Tesi di laurea magistrale
Autore
MACCHIAROLI, LUCA
URN
etd-04132021-171113
Titolo
Malliavin Calculus and Pathwise Uniqueness for SDEs
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Flandoli, Franco
Parole chiave
  • Ito-Tanaka trick
  • Ito-Wentzell formula
  • Kolmogorov backward equation
  • Malliavin calculus
  • stochastic calculus
  • stochastic differential equations
Data inizio appello
14/05/2021
Consultabilità
Tesi non consultabile
Riassunto
There are many well-known uniqueness results for SDEs, but usually they require the coefficients to be deterministic; i.e. they only depend on t and X_t itself.
Here, under suitable conditions, we show a uniqueness result for an SDE with stochastic drift; i.e. it might explicitly depend on omega.
In particular, after having introduced Malliavin calculus, we find an adapted solution to a generalised version of the Kolmogorov backward equation. We need to do this because, since we have a stochastic drift coefficient, the solution to the canonical Kolmogorov backward equation is stochastic and non-adapted; however, in order to apply the Ito-Wentzell formula, we need an adapted solution.
Finally, with the help of a generalisation of the Ito-Tanaka trick, we are able to prove a uniqueness result, assuming sufficient conditions on the drift coefficient.
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