There is a relatively old, well-known, result that for a GBM process, [$]dX_t = m X_t \, dt + \sigma X_t \, dB_t[$],we have [$]E[e^{\alpha \int_0^T X_t \, dt}] = +\infty[$] for every [$]T > 0[$] with any [$]\alpha > 0[$].(The parameter [$]m[$] is an arbitrary real, and any [$]\sigma > 0[$] will do).For example, the result is more or less the same as the proven explosion of the expected value of the money market account inlognormal one-factor interest rate models: see Hogan and Weintraub (1998)The result also implies the instantaneous explosion of all moments greater than one in the uncorrelated lognomal SABR model: [$]E[S_T^p] = \infty[$] for every [$]T > 0[$] with any [$]p > 1[$]. ([$]dS_t = \sigma_t S_t dW_t, \quad d \sigma_t = \nu \, \sigma_t \, dB_t[$], assuming any [$]\nu > 0[$]).A look at the Hogan and Weintraub proof (see the link) shows it is rather involved and relies on an integral representation of a Bessel function.Does anyone know or see a more direct (but rigorous) proof?Thanks! ==============================================================p.s. Let me add that I am aware of a heuristic hand-wavy proof:Suppose [$]T > 0[$] but very small. Then[$]E[e^{\alpha \int_0^T X_t \, dt} \approx E[e^{\alpha \frac{T}{2}(X_0 + X_T)}] = +\infty[$] by an elementary application of the normal density for [$]\log X_T[$]; i.e. [$]E[e^{\alpha \, e^z}]=\infty[$], [$]z \sim[$] Normal.While this captures the essence of what is going on, it is not rigorous. Can the argument be cleaned up?

I am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?

See attached for some old notes that I once took while studying for a term structure modelling class. They deal with the bank account case you mentioned.(Please note that this is not my own work but reproduced from some other source that unfortunately I do not have anymore.)

QuoteOriginally posted by: LocalVolatilitySee attached for some old notes that I once took while studying for a term structure modelling class. They deal with the bank account case you mentioned.(Please note that this is not my own work but reproduced from some other source that unfortunately I do not have anymore.)Perfect -- thank you!

QuoteOriginally posted by: frolloosI am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?Interesting idea -- can you carry this through?

QuoteOriginally posted by: AlanQuoteOriginally posted by: frolloosI am feeling overconfident today, so dont take this seriously, but starting with binomial model calculate the expectation, and then take the continuous limit?Interesting idea -- can you carry this through?Just saw your reply, thanks. Yes, I could try this out once I have a bit more time. Still need to spend time on the stoch ir-equity vol mixing from the other thread plus some other things.

Hello Alan,
If you just write out the integrals, you have something of the form
$$
\int_{\mathbb{R}}\exp\left\{\alpha\int_{0}^{T}e^{z\sqrt{t}}dt\right\}\exp\left\{-\frac{z^2}{2}\right\}dz.
$$
As z tends to infinity, the integrand is positive and does not tend zero, for any $\alpha>0$.

Thanks, although I have to say I don't see the 'writing out the integrals' step.
(There was a nice attachment that Local Volatility provided, which had a more involved but rigorous argument).