Multi-period dividend pricing identity ($B_r{V}_r={\mathbb{E}}_Q[B_n{V}_n+{\sum }_{j=r+1}^n{B}_j{D}_j\mid {\mathcal{F}}_r]$)

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.