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Interpret the exponential distribution change time as an exogenous time independent of the stock's Brownian motion, or, equivalently for this calculation, as having constant conditional intensity $\lambda$ while the rate remains high. Its marginal distribution alone would not imply that conditional intensity if the change time carried information about stock returns. Write $r_1=r+\epsilon$, $p=1-R$, $B_1=B(r_1)$, and let $V_1$ be the prechange value function.

Over a short interval $dt$, a rate change occurs with probability $\lambda dt+o(dt)$ and changes continuation value from $V_1(w)$ to $V_0(w)$. The dynamic programming equation is consequentlyThis jump term distinguishes the exponential interest-rate switch from a permanent higher bank rate.

By the homogeneity of CRRA utility, put $V_1(w)=K_1{w}^p/p$, with $K_1>0$. Differentiating the strictly concave expressions in $c$ and $\pi$ givesThe optimized consumption terms contribute $R{K}_1^{1-1/R}{w}^p/p$, and the optimized investment terms contribute $K_1{B}_1{w}^p$. With $K_0={\gamma }_0^{-R}$, the Hamilton-Jacobi-Bellman equation reduces toEquivalently,The left side is continuous and strictly increasing for ${\gamma }_1>0$, with limits zero and infinity. Thus $D_1>0$ gives one and only one positive root. This scalar equation is an explicit characterization for arbitrary real $R$; elementary closed forms are not generally available. For example, when $R=2$,Use $({\pi }_1^{\ast },{\gamma }_1)$ until the rate changes, and $({\pi }_0^{\ast },{\gamma }_0)$ afterwards. If the two rates coincide, the root is ${\gamma }_1={\gamma }_0$; as $\lambda$ tends to infinity, it also tends to ${\gamma }_0$.

For a direct attainability check, use the proposed prechange constant fractions. The geometric Brownian motion moment is $E[w_t^p]=w^p\exp [p(B_1-{\gamma }_1)t]$ on the prechange evolution. Conditioning on the independent exponential distribution time therefore givesThe denominator is positive because the scalar equation gives $D_1+p{\gamma }_1={\gamma }_1+\lambda K_0/K_1>0$. Hence the candidate policy actually achieves the stated value.

One can also verify the upper bound for both signs of $p$ without relying on a sign-sensitive terminal-value argument. Let $j_t$ be the current regime, $w^{\ast }$ the candidate portfolio wealth, andIf $N_t=1_{\{\tau \le t\}}$ and ${\tilde{N}}_t=N_t-\lambda (t\wedge \tau )$, the Itô formula and the scalar equation givewhere the jump term stops after the switch and continuous coefficients are understood predictably. Thus $M$ is a positive martingale deflator: for any admissible portfolio wealth $w$, the process $M_t{w}_t+{\int }_0^t{M}_s{c}_{s}ds$ is a nonnegative local martingale. It follows thatFor the candidate policy, $E[M_t{w}_t^{\ast }]\to 0$. Indeed its surviving prechange term decays at rate $D_1+p{\gamma }_1>0$, and its postchange terms decay at rate ${\gamma }_0>0$; integrating over the switch time preserves convergence to zero. The finite-time moment computations also make its budget equality exact. Consequently $E{\int }_0^{\infty }{M}_t{c}_t^{\ast }dt=M_0{w}_0$. Apply the concave supporting-tangent inequalityand integrate in time and expectation. The budget inequality proves that no admissible policy exceeds the candidate. This verifies the value function, including when $R>1$.

Both ${\gamma }_0>0$ and $D_1>0$ are needed for a finite prechange answer. If the postchange value is already infinite, that problem persists after the almost surely finite switch. If ${\gamma }_0>0$ but $D_1\le 0$, then for $p>0$ a small constant prechange consumption fraction makes the expected utility function before the switch diverge, or makes its supremum diverge when $D_1=0$. For $p<0$, the optimal no-consumption terminal moment before the switch is $w^p{e}^{p{B}_{1}t}$; withdrawals only increase this negative-power moment. The expected magnitude of the negative postchange continuation utility function therefore contains the divergent integral $\lambda K_0{w}^p{\int }_0^{\infty }{e}^{-D_{1}t}dt/|p|$. Thus these cases yield respectively $+\infty$ or $-\infty$, rather than a positive-root solution.