= Solution
For the first equation use an integrating factor. The <Itô product rule> gives $d(e^{\lambda t}X_t)=\sigma e^{\lambda t}dB_t$, so
$$
X_t=e^{-\lambda t}\left(1+\sigma\int_0^t e^{\lambda s}\,dB_s\right).
$$
The <Gaussianity of deterministic Brownian stochastic integrals> follows by approximating its deterministic integrand by step functions and using the <Itô isometry>. Hence
$$
\boxed{X_t\sim N\left(e^{-\lambda t},\begin{cases}\displaystyle\frac{\sigma^2(1-e^{-2\lambda t})}{2\lambda},&\lambda\ne0,\\\sigma^2t,&\lambda=0.\end{cases}\right).}
$$
The variance is positive also when $\lambda<0$, since numerator and denominator then both have negative sign. This is the <explicit Ornstein-Uhlenbeck solution> with initial value one.
For the second equation, take independent standard <Brownian motions> $U,V$ and define $R_t=\sqrt{(1+U_t)^2+V_t^2}$. The associated planar Brownian motion does not hit the origin. To check this rather than silently differentiate at zero, stop its radius upon hitting $\varepsilon$ or $L$, with $0<\varepsilon<1<L$. The function $\log r$ is harmonic on that annulus, so the <optional stopping theorem> gives
$$
\mathbb P_1(T_\varepsilon<T_L)=\frac{\log L}{\log L-\log\varepsilon}.
$$
Let $\varepsilon\downarrow0$: hitting zero before $T_L$ has probability zero. Taking integer $L\to\infty$ covers every possible finite hitting time, since continuous paths are bounded on compact intervals. Thus <planar Brownian motion avoids a fixed point> and $R$ stays positive.
The radial <Itô formula> is now legitimate and gives
$$
dR_t=\frac{1+U_t}{R_t}\,dU_t+\frac{V_t}{R_t}\,dV_t+\frac{1}{2R_t}\,dt.
$$
The sum of the two martingale terms has bracket $t$ and starts at zero, so it is a standard Brownian motion $\beta$ by <Lévy characterization of Brownian motion>. Thus $dR=d\beta+(2R)^{-1}dt$ with $R_0=1$. The law-determination assumption permitted in the question identifies the law of $Y$ with that of this dimension-two <Bessel process>. For independent standard <normal random variables> $G_1,G_2$,
$$
\boxed{Y_t\overset d=\sqrt{(1+\sqrt t\,G_1)^2+tG_2^2}.}
$$
This is the <two-dimensional Bessel transition law> and answers the requested distribution without needing a special-function density.
Back to article page
