# Using SDE uniqueness in law to show that two processes have the same distribution

Let $$B$$ and $$\tilde{B}$$ be independent standard Brownian motions defined on the same probability space with $$B_0=\tilde{B}_0=0$$. Let $$X_t=e^{B_t}\int_0^te^{-B_s}d\tilde{B}_s,\hspace{1cm}Y_t=\sinh(B_t).$$ I am required to show that $$X$$ and $$Y$$ have the same law, with the question giving that I can use that SDEs with Lipschitz coefficients satisfy the uniqueness in law property.

With the implication that the SDE hint gives, I used Ito's fomula (hopefully without making any computational errors) with $$f(x,y)=e^xy$$ to derive that $$X_t$$ satisfies the SDE $$dX_t=\frac{1}{2}X_tdt+X_tdB_t+d\tilde{B}_t.$$ Then the coefficients $$b(x)=x/2$$ and $$\sigma(x)=(x,1)$$ are Lipschitz and so this SDE will have uniqueness in law. I think then that I am supposed to show that $$Y$$ satisfies the SDE also. Using Ito's formula with $$g(x)=\sinh(x)$$ and the fact that $$\cosh(B_t)=\sinh(B_t)+e^{-B_t}$$, I obtained that $$dY_t=\frac{1}{2}Y_tdt+(Y_t+e^{-B_t})dB_t.$$ But, defining $$Z_t$$ by $$dZ_t=e^{-B_t}dB_t$$, we do not have that $$Z$$ is a BM independent of $$B$$, so it does not appear that these SDEs are the same.

I would greatly appreciate if someone could point me to where I've gone wrong with my efforts.

Recall the identity $$\cosh^2x - \sinh^2x =1$$. Then, after applying Itô's lemma, the SDE for $$Y$$ is: \begin{align} dY_t &= \cosh B_t dB_t + \frac{1}{2}\sinh B_t dt \\ &= \sqrt{ 1 + \sinh^2 B_t }dB_t + \frac{1}{2}\sinh B_t dt \\ &= \sqrt{1 + Y_t^2 }dB_t + \frac{1}{2}Y_t dt \end{align}
Now, in the SDE for $$X$$ notice that the term $$d\tilde{B}_t + X_t dB_t$$ defines a martingale whose quadratic variation is $$(1 + X_t^2)dt$$; thus, by the martingale representation theorem, we can construct a Brownian motion $$W$$ such that $$d\tilde{B}_t + X_t dB_t = \sqrt{1 + X_t^2} dW_t$$ Thus, the SDE for $$X$$ becomes $$dX_t = \sqrt{1+X_t^2} dW_t + \frac{1}{2}X_t dt$$ Since $$X$$ and $$Y$$ satisfy the same SDE, if we can prove that the solution to the SDE is unique we may conclude that $$X$$ and $$Y$$ have the same law. Uniqueness for the SDE follows by observing that both coefficients $$x \mapsto \frac{x}{2}$$ and $$x \mapsto \sqrt{1 + x^2}$$ are Lipschitz (for the second one, observe it is $$C^1$$ with bounded derivative), so we conclude the processes agree in law.