= Exponential interest-rate switch
{title2=$R\gamma_1+\lambda(\gamma_1/\gamma_0)^R=D_1$}
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 $\gamma_1w$. 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.
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