Solving the SDE $\mathrm dX_t=\frac1{X_t}\,\mathrm dt+kX_t\,\mathrm d{X_t}$

Question

I'm looking to solve the stochastic differential equation $$\mathrm dX_t=\frac1{X_t}\,\mathrm dt+kX_t\,\mathrm d{W_t},$$ where $k\in\mathbb R$ is a constant. I am quite certain that I will need to substitute something along the lines $X_t\exp\,(\cdots)$ and possibly prove $X_t>0$ a.s. (perhaps via $X_t\exp\,(\cdots)>0$ a.s.) but I haven't been able to come up with a successful substitution to do so. Could someone provide some guidance please? Thanks!

Answer 1

Let us denote $Y_t = X_t^2$, using Ito' lemma, we deduce easily that $$dY_t = Y_t (k^2dt+2kdW_t) + 2dt \tag{1}$$

The equation $(1)$ is a special case of the equation $$dY_t =Y_t(b(t)dt +\sigma(t)dW_t)+c(t)+\delta(t)dW_t \tag{2}$$ with

• $b(t)=k^2$
• $\sigma(t) = 2k$
• $c(t) = 2$
• $\delta(t)=0$

The solution of $(2)$ is $$Y_t = Z_t\left(Y_0+\int_0^t \frac{c(s)-\sigma(s)\delta(s)}{Z_s}ds +\int_0^t\frac{\delta(s)}{Z_s}ds \right)$$ with $$Z_t :=\exp{\left(\int_0^t b(s)ds +\int_0^t \sigma(s)dW_s -\frac{1}{2}\int_0^t \sigma^2(s)ds \right)}$$

The solution of $(1)$ seems to be $$Y_t = e^{-k^2 t + 2kW_t} \left(Y_0 + 2\int_0^t e^{k^2 s - 2kW_s} ds \right)$$

Q.E.D