Explosion time of $dX_t=X_t(adW_1+bdW_2)$

Question

I found in Karatzas & Shreve (1991), $dX=\sigma(X_t)dW_t$ cannot explode. But what about $dX_t=X_t(adW_1+bdW_2)$? Here $W_1$ and $W_2$ are independent.

Feller's test for explosion seems to work when SDE is driven by one BM (Karatzas & Shreve (1991)). I am not sure whether my question can be transformed into the setting of one BM, such as $dX_t=cX_tdW_3$.

Thanks for any suggestions.

Answer 1

As long as the coefficient $\sigma$ is globally Lipschitz (regardless of dimension), there will not be explosion see for example http://en.wikipedia.org/wiki/Stochastic_differential_equation#Existence_and_uniqueness_of_solutions.

In your example the coefficient is $\sigma:\mathbb{R}\to\mathbb{R}^2$, $\sigma(x):=(ax, bx)$ which is clearly Lipschitz.