Query regarding existence and uniqueness of SDE solutions

[yip]

Hi,

Just wanted to ask a question regarding existence and uniqueness of solutions to SDEs. Say you have shown existence and uniqueness of a solution to an SDE that the process [tex ] X_{t} [/tex ]a particular process follows (by showing drift and diffusion coefficients are Lipschitz). If you take a one to one transformation of the process, let's say, exponentiating it, do we have existence and uniqueness of a solution to the new SDE that the transformed process follows? The reason I am asking is because I am trying to show existence and uniqueness of a solution to the Black Karasinski model for interest rates, where the log of the short rate follows an Ornstein Uhlenbeck process. Existence and uniqueness is simple to show if you look at the SDE the log of the short rate follows, but if you use Ito's Lemma to derive the SDE that the short rate follows, then showing the coefficients are Lipschitz seems to be substantially more difficult.

Thanks

 

[mmwave]

for any insight!



Thank you for your question about existence and uniqueness of solutions to SDEs. The short answer to your question is yes, if you have shown existence and uniqueness of a solution to an SDE for a particular process, then a one-to-one transformation of that process will also have existence and uniqueness of a solution to the new SDE. This is due to the fact that Lipschitz continuity is preserved under one-to-one transformations.

In your specific case, if you are trying to show existence and uniqueness of a solution to the Black Karasinski model for interest rates, where the log of the short rate follows an Ornstein Uhlenbeck process, you can use the fact that the log function is one-to-one and Lipschitz continuous to show existence and uniqueness of a solution to the transformed SDE. This approach may be easier than trying to directly show the coefficients are Lipschitz for the transformed process.

I hope this helps and best of luck with your research. Happy studying!