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If a discrete-time martingale deflator obeys $Y_t/Y_{t-1}=e^{X_t}$ for an affine process $X$, unit zero-coupon bonds have exponential-affine prices. For one-step coefficients $A,B$, the recursion is $\alpha (0)=\beta (0)=0$, $\alpha (n+1)=A(1+\alpha (n))$, $\beta (n+1)=\beta (n)+B(1+\alpha (n))$.

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.

Use the conditional second-moment matrix $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ from the PDF. By part (b), the filtration is finite-state at every finite time. On a time-$(t-1)$ atom let $p_1,\dots ,p_m$ be the successor price vectors and $q_1,\dots ,q_m>0$ their conditional probabilities. Then $m\le n$ andPositive definiteness gives rank $n$, so $m\ge n$. Hence $m=n$, and the $n$ successor vectors have linear independence.

We use the finite-horizon fundamental theorem of asset pricing in deflator form: an arbitrage-free finite market admits a strictly positive adapted process $D$, with $D_0=1$, such that $DP$ is a martingale. Equivalently, on every step,Fix a finite horizon containing the step in question. On the parent atom, write $d_i=D_t/D_{t-1}>0$ on successor $i$ and $p_0=P_{t-1}$. ThenThe proposed values $z_i=p_i^{\top }{V}^{-1}{p}_0$ satisfy precisely the same equation:Because the $p_i$ have linear independence and the $q_i$ are positive, this linear system has a unique solution. Thus $z_i=d_i>0$, provingThis is the positive regression deflator in a complete finite market. It also proves the suggested conclusion: with $Y_0=1$ and $Y_t={\prod }_{u=1}^t{Z}_u$,Hence $Y$ is a strictly positive martingale deflator. Finite-state structure makes these expectations integrable on each finite horizon. Positive definiteness alone would not ensure positivity of the regression factor; market completeness and the positive deflator are essential.

A unit-face-value zero-coupon bond pays $1$ at maturity. The martingale deflator pricing identity isUse the affine process Markov property with respect to the market filtration; if that filtration contains extra predictive information, the natural Markov property alone would not suffice. Part (b), with every future coefficient equal to $1$, gives the exponential-affine form. More explicitly, the exponential-affine bond pricing recursion isIndeed, conditioning the first future step in an $(n+1)$-step horizon transforms $e^{(1+\alpha (n))X_{t+1}+\beta (n)}$ into $e^{A(1+\alpha (n))X_t+B(1+\alpha (n))+\beta (n)}$. ThereforeThis uses the true martingale deflator pricing identity; a merely local martingale deflator would not by itself justify replacing prices by conditional terminal expectations.

The discrete-time fundamental theorem of asset pricing supplies a strictly positive martingale deflator $Y$. Since the maturity-$T$ bond pays one unit at $T$, its deflated price is a martingale:Division by $Y_t>0$ gives the formula.

A predictable strategy $H$ is self-financing when its wealth $X=H\cdot P$ satisfiesIt is admissible when its wealth obeys the stipulated lower bound, here taken to be nonnegative. A strictly positive Itô process $Y$ is a martingale deflator when every deflated asset price $Y{P}^i$ is a local martingale.

A martingale deflator is a strictly positive adapted process $Y$ such that every deflated cum-dividend asset gain has zero conditional drift:Using the definitions of ${\pi }^H$ and ${\xi }^H$,The holdings $H_t$ are ${\mathcal{F}}_{t-1}$-measurable, so the right side is a martingale transform of the deflated asset-gain local martingale. Hence $Z$ is a local martingale.

The fundamental theorem of asset pricing says, in this discrete-time formulation, that the market has no arbitrage if and only if it admits a strictly positive martingale deflator. Under a chosen positive numeraire this is equivalent to the existence of an equivalent martingale measure for numeraire-discounted gains.

Let $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ be positive definite. Market completeness restricts each parent atom to at most $n$ successors, while positive definiteness forces exactly $n$ independent successor price vectors. The positive martingale deflator supplied by the fundamental theorem of asset pricing then has the unique one-step ratio $Z_t=P_t^{\top }{V}_t^{-1}{P}_{t-1}>0$. Its product deflates the prices into martingales.