## Thesis etd-04132021-171113 |

Link copiato negli appunti

Thesis type

Tesi di laurea magistrale

Author

MACCHIAROLI, LUCA

URN

etd-04132021-171113

Thesis title

Malliavin Calculus and Pathwise Uniqueness for SDEs

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Flandoli, Franco

Keywords

- Ito-Tanaka trick
- Ito-Wentzell formula
- Kolmogorov backward equation
- Malliavin calculus
- stochastic calculus
- stochastic differential equations

Graduation session start date

14/05/2021

Availability

None

Summary

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.

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.

File

Nome file | Dimensione |
---|---|

Thesis not available for consultation. |