Exponential interest-rate switch ($R{\gamma }_1+\lambda ({\gamma }_1/{\gamma }_0{)}^R=D_1$)

Consider an investment-consumption problem with an independent one-way rate change of intensity $\lambda$ from $r_1$ to $r_0$. For a stock of constant drift $\mu$ and positive volatility $\sigma$, set $B_j=r_j+(\mu -r_j{)}^2/(2R{\sigma }^2)$, ${\gamma }_0=[\rho -(1-R)B_0]/R$, and $D_1=\rho +\lambda -(1-R)B_1$. In the finite-value regime ${\gamma }_0,D_1>0$, the prechange consumption coefficient is the unique positive root displayed above. The left side increases continuously from zero to infinity. Before the change, the value is ${\gamma }_1^{-R}{w}^{1-R}/(1-R)$, the stock fraction is $(\mu -r_1)/(R{\sigma }^2)$, and consumption is ${\gamma }_{1}w$. At the change these become the ordinary Merton consumption-investment problem controls for $r_0$. The jump term in the Hamilton-Jacobi-Bellman equation is $\lambda (V_0-V_1)$; knowing only the marginal law of the change time does not justify that term without a conditional intensity assumption.