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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.

Specify the trading conventions first. Let the bank account have value $R_r>0$, with $R_0=1$ and its next gross return known at time $r$. Write $B_r=1/R_r$ for the discount factor. Let $S_r$ be the ex-dividend asset-price vector, and $d_r$ the vector of cash dividends paid per unit at time $r$; set $d_r=0$ for non-dividend-paying assets. Under an equivalent martingale measure $Q$, each discounted cumulative asset gainis a martingale. Equivalently, ${\mathbb{E}}_Q[B_{r+1}(S_{r+1}+d_{r+1})\mid {\mathcal{F}}_r]=B_r{S}_r$. This is the gain-process form of risk-neutral pricing, rather than a martingale assumption on raw ex-dividend prices.

Let $X_r$ be the ${\mathcal{F}}_r$-measurable holdings after trading and distributions at time $r$. The previous holdings $X_{r-1}$ earn the next price change and dividends. Define the strategy's net cash payment and its residual portfolio wealth byA negative payment is an external infusion; a self-financing portfolio with no consumption has $D_r^X=0$. The discounted budget equation isAssume these gains and payments are integrable under $Q$, for example by bounded predictable holdings and integrable discounted asset gains. Thenis a true martingale, by conditioning the displayed one-step gains. Merely assuming a local martingale without appropriate integrability would not justify the conditional equality.

Conditioning $M_n$ on ${\mathcal{F}}_r$ proves the general multi-period dividend pricing identityFor a strategy liquidated at time $n$, $X_n=0$ and all liquidation proceeds are included in $D_n^X$. The terminal value disappears, giving exactly the requested dividend-only relation. That liquidation convention is necessary: holding one unit of the bank account with no payments would otherwise have a positive left side and zero right side.

An attainable claim $H$ paid at $n$ can be replicated by a strategy with no earlier payments and final liquidation payment $D_n^X=H$. Its time-$r$ value is thereforeTwo admissible replicating strategies have the same value by this identity, and the value is the same under every equivalent martingale measure for which the replication gains are true martingales. For unattainable claims this expectation is a possible model price, but the replication argument does not establish a unique price. With density process $Z_r={\mathbb{E}}_P[dQ/dP\mid {\mathcal{F}}_r]$, the equivalent state-price density form is $V_r=(B_r{Z}_r{)}^{-1}{\mathbb{E}}_P[B_n{Z}_{n}H\mid {\mathcal{F}}_r]$.