@jlewk: While the argument is formally correct, I do not think it is useful here. A Lipschitz martingale is of bounded total variation, hence of zero quadratic variation. So the only martingales satisfying this conditions are constants anyway.

A maybe useful reformulation of the SDE can be found in Lemma 3.1 of the following paper: Sven Rady. Option Pricing in the Presence of Natural Boundaries and a Quadratic Diffusion Term. Finance & Stochastics 1, 331-344 (1997)

You might also try to look up “suicide strategy” in the financial math literature. It is intimately tied to this problem as well as the classic (D. Bernoulli in the 18th century) St. Petersburg paradox.

Consider a time change $u = Log(T-t)$. Then this is equivalent to that Brownian motion hits 1 in finite time.

@香结丁 No issue. Thank you!

I tried to address your question by expanding my answer.

And I suspect that there might be problems. At least the natural bound $2yz \leq y^2 + z^2$ will not work as the heat equation with quadratic nonliearity blows up (see, e.g., Evan's PDE book, section 9.4). Thus if there is an existence for the solution, the argument might be quite subtle.

If $a(u)$, $b(u)$ would be random processes, the according Feynman-Kac formula would be in a three dimensional state space $(x,a,b)$. But I think one has first to understand what is happening in easy (but not trivial) model cases. If the solution doesn't exist there, we cannot expect much in general.

(just to be clear, assuming $a=0, \, b=1$).

Just a quick note: Looking at the "simple" case where $\xi = W_T$ this can be seen as FBSDE which (assuming Feynman-Kac) should correspond to the semilinear PDE $-u_t + \frac{1}{2}u_{xx} = uu_x$. A quick search in Polyanin-Zaitsev didn't produce any results.

@MattF. Agrred.

Indeed, doesn't any right-continuous nondecreasing function $g$ with domain $[0,1]$ define a quantile function? (Just check that $g(U)$, for $U$ a standard uniform random variable defines a random variable with cdf $g^{-1}$, the (right-) generalized inverse function of $g$)?