ETD

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

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.