 Deflator-based claim replication

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.