So we have an Ito Stochastic Differential Equation with $b$ as a constant: $$dX_t = (bX_t +1)dt +2 \sqrt{X_t}dW_t $$
I then am told to let $Y_t = \sqrt{X_t} $ and thus derive the Ito stochastic differential equation $dY_t = A(Y_t) dt + B(Y_t)dW_t$ and to then determine $A(Y_t), B(Y_t)$.
I can see that this is supposed to be an application of Ito's lemma. Which states that if we have the Ito Stochastic Differential Equation $X_t$, we could then define a new Ito stochastic process on $Y)y = g(X_t,t)$ which obeys $dY_t=dg(X_t,t)=d\sqrt{X_t}$.
Anything further from this is where my notes stop and I can't see any examples of how to actually implement this lemma?
I thought of substituting $Y_t$ into the equation to get: \begin{align}dY_t & = A(Y_t)dt + B(Y_t)dW_t \\
& = b \sqrt{X_t}dt+ \sqrt{X_t}dX_t \\
&= b(\frac{1}{b})\sqrt{X_t}dt + \sigma \sqrt{X_t}dW_t \\
& = b(\frac{1}{b}\sqrt{X_t}dt + (\sqrt{X_t})^2dW_t \end{align}
Although I am pretty sure this is incorrect. Can anyone help, thank you.
Best Answer
To see how this follows directly from Itô, write $f(x,t)=\sqrt{x}$. Then, the partial derivatives are: $f_t= 0$, $f_x=\frac{1}{2\sqrt{x}}$ and $f_{xx}=-\frac{1}{4x^{3/2}}$. So since $f : \mathbb{R}^2 \to \mathbb{R}$ is $C^2$, and $X_t$ is semimartingale, so is $Y_t = f(X_t,t)$, with: \begin{equation} Y_t = Y_0 + \int_{0}^{t} \dfrac{\partial f}{\partial t}(X_u,u) du + \int_{0}^{t} \dfrac{\partial f}{\partial x}(X_u,u) dX_u + \dfrac{1}{2} \int_{0}^{t} \dfrac{\partial^2 f}{\partial x^2}(X_u,u) d[X]_u. \end{equation} Using inherent characterisation of stochastic integration and bilinearity of the quadratic variation process, you can calculate that $d[X]_t = 4X_tdt$ (only the Brownian motion term contributes to this quadratic variation). Therefore, in differential form, we have: \begin{equation} dY_t = \dfrac{1}{2\sqrt{X_t}} dX_t- \frac{1}{8X_{t}^{\frac{3}{2}}}d[X]_t = \\ \frac{1}{2}\left(b \sqrt{X_t} + \frac{1}{\sqrt{X_t}}\right )dt + dW_t-\frac{1}{2 \sqrt{X_t}}dt, \end{equation} i.e. \begin{equation} dY_t =\frac{1}{2} bY_t dt + dW_t. \end{equation}