Linear stochastic differential equation

Consider linear SDE process.

$dX_t=( (-\mu +\frac{\sigma^2}{2})X_t+\kappa)dt+\sigma (X_t+\eta)dW_t$

Here, $\kappa, \eta$ are constant.

The ODE for this process,

$\frac{\sigma^2}{2}\frac{d^2}{dx^2}(x^2 g(x) )+\frac{d}{dx}( ( (\mu-\frac{1}{2}\sigma^2)x+\tau)g(x) )+ikg(t)=-\delta(x-\xi)$ eq(2)

Here, $x$ represents $x+\eta$ , and $\xi$ represents $\xi+\eta$ .

$\tau=-\kappa-(\mu-\frac{\sigma^2}{2})\eta$

We write the independent solutions for this ODE after geometric Brownian motion.

$y_1(x)=x^{\alpha-1}w(x)$

$y_2(x)=x^{\beta-1}z(x)$

$\alpha, \beta$ are the solution of quadratic equation.

$\frac{\sigma^2}{2}\lambda^2+\mu\lambda+ik=0$

We define the result of differentiating $y_1, y_2$ as

$y_1'(x)=(\alpha-1)x^{\alpha-2}w_d(x)$

$y_2'(x)=(\beta-1) x^{\beta-2}z_d(x)$

Eq(2) has fixed singular point at $x=\infty$ , so we can apply Frobenius method.

Therefore $w, w_d, z, z_d$ are polynomial of $x^{-1}$ , and constant part is 1.

The coefficients of $x^{-l}$ only depend to "k", the differentiating result by k is O(1/k).

We transform the equation like Sturm-Liouville equation in the same way as geometric Brownian motion,

$r(x)\frac{d}{dx}(p(x)g(x))+q(x)=-\delta(x-\xi)$

Here,

$r(x)=\frac{\sigma^2}{2}x^{-1-\frac{2\mu}{\sigma^2}}\exp(-\frac{2\tau}{\sigma^2 x})$

$p(x)=x^{3+\frac{2\mu}{\sigma^2}}\exp(\frac{2\tau}{\sigma^2 x})$

The independent solutions which satisfy boundary condition are

In x< $\xi$ ,

$y_1(x)+cy_2(x)$

$w(b+\eta)(b+\eta)^{\alpha}+cz(b+\eta)(b+\eta)^{\beta}=0$

In x> $\xi$ ,

$y_1(x)$

Solve the Green function in the same way as geometric Brownian motion

In x< $\xi$

$g(x)=-\frac{y_1(\xi)(y_1(x)+cy_2(x))}{A(\xi)}$

In x> $\xi$

$g(x)=-\frac{y_1(x)(y_1(\xi)+cy_2(\xi))}{A(\xi)}$

$A(\xi)=\frac{c\sigma^2}{2}\xi^{-1-\frac{2\mu}{\sigma^2}}\Psi_k(\xi)$

Here,

$\Psi_k(x)=(\alpha-1)w_d(x)z(x)-(\beta-1)w(x)z_d(x)$

in x< $\xi$ g(x) is

$g(x)=-\frac{2w(\xi+\eta){(\xi+\eta)}^{-\beta}({(x+\eta)}^{\beta-1}z(x+\eta)w(b+\eta)-{(x+\eta)}^{\alpha-1}w(x+\eta)z(b+\eta){(b+\eta)}^{\beta-\alpha})}{\sigma^2\Psi_k(\xi+\eta)w(b+\eta)}$

in x> $\xi$ g(x) is

$g(x)=-{w(x+\eta)(x+\eta)}^{\alpha-1}\frac{2({(\xi+\eta)}^{-\alpha}z(\xi+\eta)w(b+\eta)-{(\xi+\eta)}^{-\beta}w(\xi+\eta)z(b+\eta){(b+\eta)}^{\beta-\alpha}}{\sigma^2\Psi_k(\xi+\eta)w(b+\eta)})$

The characteristi function of first passage time is

$\int_{0}^{\infty}\exp(ikt)f(t)dt=-w(\xi+\eta)(\xi+\eta)^{-\beta}\frac{(\alpha-1)(b+\eta)^{\alpha}w_d(b+\eta)+c(\beta-1)(1+\eta)^{\beta}z_d(b+\eta)}{c( (\alpha-1) w_d(\xi+\eta)z(\xi+\eta)-(\beta-1)w(\xi+\eta)z_d(\xi+\eta) )}$