How to check if $E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty$

$$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$$ where $$W_t$$ is $$P-$$Brownian motion (Wiener process).

I have defined a new measure $$Q$$ where the Kernel density (In Girsanov theorem) is $$\phi_t = \frac{Y_t}{\sqrt{1+Y_t^2}}$$ Now I need to assure that the Novikov condition is satisfied. Hence I need to make sure: $$E^P [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty.$$ Is it? Is it possible to show that and how can I show that?

• I don’t understand the first equation. Please correct
– Ezy
Jan 13 '19 at 12:59
• $dY_t$ instead of $Y_t$. Thanks for the reminder! I have now corrected. Jan 13 '19 at 14:12
• Note that $0 \le \frac{Y_u^2}{1+Y_u^2} < 1$. Then $\exp\int_0^t \frac{Y_u^2}{1+Y_u^2} du < \exp(t)$, and $E\left( \exp\int_0^t \frac{Y_u^2}{1+Y_u^2} du\right) \le \exp(t)$. Jan 13 '19 at 20:23

If you make the change of variable $$Y_t = \sinh U_t$$ and apply Ito then you immediately get

$$dU_t = 2dW_t$$

so the solution of your SDE is $$Y_t = \sinh\left(2W_t + C\right)$$

with $$C$$ a constant.

$$\frac{Y_u}{\sqrt{1+Y_u^2}}=\tanh(U_t)$$