Exponential Vaek (1977) model (1). Consider a Vasicek process (left(r_{t}ight)_{t in mathbb{R}_{+}}) solving of the stochastic differential

Exponential Vašíček (1977) model (1). Consider a Vasicek process \(\left(r_{t}ight)_{t \in \mathbb{R}_{+}}\) solving of the stochastic differential equation

\[d r_{t}=\left(a-b r_{t}ight) d t+\sigma d B_{t}, \quad t \geqslant 0\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion and \(\sigma,

a, b>0\) are positive constants. Show that the exponential \(X_{t}:=\mathrm{e}^{r_{t}}\) satisfies a stochastic differential equation of the form

\[

d X_{t}=X_{t}\left(\widetilde{a}-\widetilde{b} f\left(X_{t}ight)ight) d t+\sigma g\left(X_{t}ight) d B_{t}

\]

where the coefficients \(\widetilde{a}\) and \(\widetilde{b}\) and the functions \(f(x)\) and \(g(x)\) are to be determined.