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In a zero-interest one-period market with a continuous terminal stock law under a pricing measure, $C(K)={\mathbb{E}}_Q(S-K{)}^{+}$ implies $-C^{\prime }(K)=Q(S>K)$ and $C^{\prime \prime }(K)=f_Q(K)$. Thus a full differentiable call curve determines its pricing density. A finite collection of strikes generally does not. With a deterministic nonunit discount factor, divide the call curve by that factor before recovering the probability density.

The value of an account that continuously reinvests at the short rate $r$, with $B_0=1$. Its reciprocal is the discount factor $D_t=B_t^{-1}$.

In the one-factor Heath-Jarrow-Morton model, put $b_t^T={\int }_t^T\sigma (t,u)du$. The Itô formula gives $dP/P=r_{t}dt-b_t^{T}d{W}_t$. Thus multiplying by the discount factor yields the displayed stochastic exponential. Bounded forward volatility on a finite maturity horizon implies the Novikov condition, so the discounted bond is a true martingale.

Write the discount factor as $D_t=B_t^{-1}=\exp (-{\int }_0^t{r}_{s}ds)$. Splitting the time integral at $t$ givesThe random variable $D_T$ lies in $(0,1]$ because the short rate is nonnegative and continuous on the finite maturity interval. A process of conditional expectations of an integrable terminal variable is a martingale, by the tower property of conditional expectation. Therefore

Concavity gives the supporting-tangent inequalityMultiply by the discount factor $e^{-bt}$ and use the first-order condition $e^{-bt}{U}^{\prime }(c_t)=Y_t$. This gives the pointwise marginal-utility verification of optimal consumption inequalityApply the budget inequality from part (d) to the competing admissible strategy, and use equality for the proposed one:The two weighted consumption integrals are finite, so their difference is integrable. Under the usual positive discount-rate assumption $b>0$, the utility integrals are also integrable: $U(0)\le U(x)\le L$ for a finite upper bound $L$, and ${\int }_0^{\infty }{e}^{-bt}dt=1/b$. Integrating the tangent inequality and taking expectations therefore givesEconomically, both consumers face the same state-price budget, and the candidate spends it exactly where its discounted marginal utility equals the state price. This is utility duality with martingale deflators in its consumption form.

A positive $b$, or another hypothesis making the infinite-horizon objectives well defined and permitting this integration, is needed. The printed question does not specify the sign of $b$. Bounded utility alone does not ensure that an undiscounted infinite time integral exists: a bounded integrand can have both infinite positive and negative parts. Thus the conclusion is established under the standard discount convention $b>0$, and also whenever the displayed objectives satisfy the stated integrability conditions; without either convention the literal infinite-horizon comparison need not be a defined mathematical expression.

This can occur within an admissible financial model, not just for an abstract bounded integrand. Take $U(x)=1-2e^{-x}$ and $b=0$. Choose a deterministic smooth nonnegative $c$ with successive unit-length plateaus alternating between zero and the integer $n$, and connect them over intervals whose lengths have finite sum. Then ${\int }_0^{\infty }2c_t{e}^{-c_t}dt<\infty$: the high-plateau contributions sum to ${\sum }_{n\ge 1}2n{e}^{-n}$ and the transition integrand is bounded by $2/e$. SetThe bank account is a positive deterministic Itô process and $Y_t{B}_t=Y_0$, so $Y$ is a state-price density. Holding $X_t/B_t$ bank units finances consumption with nonnegative wealth and exact budget equality. Also $U^{\prime }(c_t)=Y_t$. Nevertheless every zero plateau contributes $-1$ to the utility integral, while each sufficiently high plateau contributes a fixed positive amount. Both its negative and positive parts are infinite, so the undiscounted objective is undefined. This establishes why the missing discount or objective-integrability hypothesis is substantive.