Deflator-based claim replication ($X_t=\mathbb{E}[Y_T{\xi }_T\mid {\mathcal{F}}_t]/Y_t$)

In a one-factor market with the usual natural Brownian filtration, assume nonzero traded diffusion exposure and a strictly positive local martingale deflator $Y$, normalized by $Y_0=1$. Let $H\ge 0$ be a terminal contingent claim with $\mathbb{E}[Y_{T}H]<\infty$. DefineThe Brownian martingale representation theorem, with localization when needed, represents the true conditional-expectation martingale $M$ as a stochastic integral. Solving the exposure equation against the nonzero traded diffusion constructs a self-financing portfolio with wealth $X\ge 0$ and terminal value $H$. The dollar portfolio from a deflated wealth martingale gives this calculation explicitly.

For any competing nonnegative-wealth self-financing portfolio, the deflated wealth is a nonnegative local martingale, hence a supermartingale. Its initial capital is therefore at least $\mathbb{E}[Y_{T}H]$, and the constructed strategy attains that cost. Without normalization, the cost is $\mathbb{E}[Y_{T}H]/Y_0$. Boundedness of $H$ alone does not replace the required deflated-payoff integrability in an arbitrary model.

In the completed natural Brownian filtration, suppose $d{S}_t/S_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t$ with ${\sigma }_t\ne 0$, and let $d{\zeta }_t=-{\zeta }_t(r_{t}dt+{\kappa }_{t}d{W}_t)$, ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. For an affordable nonnegative terminal claim $X$, represent the conditional-expectation martingale $N_t=\mathbb{E}[{\zeta }_{T}X\mid {\mathcal{F}}_t]$ as $d{N}_t=h_{t}d{W}_t$. Then $w_t=N_t/{\zeta }_t$ and the displayed dollar amount in the stock replicate $X$. The Itô product rule gives $d(\zeta w)=\zeta (\sigma \theta -\kappa w)dW$, proving the formula; share holdings are ${\theta }_t/S_t$.