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Fix a finite horizon $N$, a filtration $({\mathcal{F}}_n)$, a positive bank account $B_n$ with predictable one-step growth, ex-dividend risky prices $S_n$, and asset dividends $D_n$ paid at date $n$. Put $s_n=S_n/B_n$ and $d_n=D_n/B_n$. An equivalent martingale measure $Q$ makes the discounted cum-dividend gainsmartingales; equivalently ${\mathbb{E}}_Q[s_n-s_{n-1}+d_n\mid {\mathcal{F}}_{n-1}]=0$. It is these gains, not necessarily the discounted ex-dividend prices alone, which enter the definition.

Let ${\theta }_n$ be risky holdings chosen at date $n-1$ and carried to date $n$. Let $C_n$ be the portfolio's net cash distribution to its owner at date $n$, and $V_n$ its remaining value after that distribution and rebalancing. A self-financing portfolio with these distributions has the budget identityTo see this, value the previous stock and bank holdings at the new prices, add the asset dividends, subtract the owner distribution, and observe that the discounted bank holding is unchanged. Rebalancing then alters holdings but not value. An asset dividend reinvested in the portfolio is already in the gain term; it is not also an owner distribution.

For a predictable strategy whose gains are integrable under $Q$ (bounded holdings in a finite-state market suffice), conditioning the gain increment gives zero. Therefore discounted remaining wealth plus cumulative discounted owner payments is a martingale. Conditioning its terminal value proves the multi-period dividend pricing identityThe relation holds under every equivalent martingale measure for which these integrability conditions hold. Each payment is discounted from its own payment date, not from one common terminal date. The expression inside the expectation is in bank account units; multiplication by $B_n$ returns current currency units.

When the portfolio is liquidated and all its final value is paid out, $V_N=0$ after liquidation and that value is included in $C_N$. The displayed identity then involves only future discounted distributions. Without liquidation, the terminal value must remain. A bank account holding with no coupons before redemption has positive value despite having no intervening dividends; omitting its redemption or terminal holding value would plainly give the wrong result. In an infinite horizon, deleting the terminal term instead requires a dividend-price transversality condition, namely convergence of its conditional discounted expectation to zero.

For a traded asset the same proof gives its discounted expected dividends plus terminal sale value. For a replicated claim paid only at $N$, it gives $V_n=B_n{\mathbb{E}}_Q[H/B_N\mid {\mathcal{F}}_n]$. In a complete market this prices every admissible claim uniquely. In an incomplete market only the portfolio's attainable payout stream has this common value under all measures; an arbitrary unattainable payoff can have different expectations under different measures. For a merely dominated measure the conditional identity holds $Q$-almost surely; it makes no assertion on physical states assigned zero mass by $Q$. External capital injections must also be counted, as negative owner distributions. Thus the valuation principle follows from the gain martingale and the trading budget, together with the terminal convention and integrability, rather than from unconditional averaging of arbitrary future cash flows.

Take $S_t$ to be the ex-dividend stock price: the date-$t$ output has already been paid when trading occurs. Receiving ${\theta }_t{d}_t$ units of fruit and selling ${\theta }_t-{\theta }_{t+1}$ shares gives the discrete dividend budget equationThe entering holding ${\theta }_t$ is chosen with date-$(t-1)$ information, while $C_t$ and the next holding ${\theta }_{t+1}$ may use date-$t$ information. The output is the consumption good and price numéraire; its stochastic law is exogenous to the individual's price-taking choice.

For the interior differentiable case, attach an adapted Lagrange multiplier ${\Lambda }_t$ to each budget constraint. On a finite horizon, with the terminal holding fixed or its continuation value included, the optimization Lagrangian isAn adapted variation in $C_t$ gives ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. An arbitrary date-$t$ measurable variation in ${\theta }_{t+1}$ affects two adjacent terms. Its coefficient must have zero conditional mean, soThese are the Euler equations of marginal utility pricing in a dividend economy. If the utility function is merely concave, a supporting slope replaces $U^{\prime }$; binding consumption or trading restrictions give the corresponding inequalities rather than unconstrained equalities.

With known initial information and ${\Lambda }_0>0$, normalize ${\zeta }_t={\Lambda }_t/{\Lambda }_0$. This is a state-price density for the traded asset. In particular,The ratio is the stochastic discount factor: payment in a state with greater marginal utility has a higher marginal value. For a priced contingent payoff $H$ at $t+1$, its price is ${\mathbb{E}}_t[({\zeta }_{t+1}/{\zeta }_t)H]$; pricing arbitrary claims uniquely also requires market completeness. The deflated cumulative gains ${\zeta }_t{S}_t+{\sum }_{s=1}^t{\zeta }_s{d}_s$ form a martingale whenever the displayed terms are integrable.

Iterating the Euler equation givesUnder the dividend-price transversality condition and the needed integrability, this becomes the fundamental priceThus an infinite-horizon price is not automatically just a dividend sum: its terminal term must vanish. Telescoping the individual budget constraints similarly givesA no-Ponzi lower restriction on this final term gives the consumption budget inequality; its vanishing gives equality. Together with the supporting-line inequality for a concave utility function, the Euler equations and these terminal conditions supply the usual sufficiency argument for an admissible candidate. In an incomplete market, each optimal agent's marginal utility can supply a different state-price density pricing the same traded tree.

Iteration of the pricing Euler equation givesThe displayed fundamental price follows when the future dividend series is integrable and ${\mathbb{E}}_t[{\zeta }_N{S}_N]\to 0$. Without this dividend-price transversality condition, a deflated price martingale component can survive. For individual holdings, the analogous terminal term is $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$. Telescoping the discrete dividend budget equation gives the initial wealth cost of consumption minus this terminal term; its vanishing is needed for budget equality.