Solution to stochastic differential eqn

Question

How do you solve this stochastic differential equation?

Not sure how to start on this. Need some guidance.

Answer 1

Hint The solution of the corresponding ordinary differential equation (ODE)

$$dx(t) = - \alpha x(t) \, dt \qquad x(0)=x_0$$

is given by

$$x(t) = x_0 \exp(-\alpha t).$$

In particular,

$$x(t) \cdot \exp(\alpha t) = x_0 = \text{const}$$

Apply Itô's formula to the process

$$Y_t := X_t \cdot e^{\alpha \, t}$$

in order to obtain an expression for $(X_t)_{t \geq 0}$.