Gaussian-damped martingale density construction ($dQ/dP\propto e^{-|X{|}^2-h_{\ast }\cdot X}$)

For a finite-dimensional discounted gain random vector $X$, absence of arbitrage means $h\cdot X\ge 0$ almost surely only when $h\cdot X=0$ almost surely. Let $N=\{h:h\cdot X=0\text{ almost surely}\}$ and minimize $F(h)=\mathbb{E}{e}^{-|X{|}^2-h\cdot X}$ on the orthogonal complement $N^{\perp }$. The Gaussian damping makes $F$ finite and differentiable without physical integrability assumptions on $X$. Every nonzero direction in $N^{\perp }$ has negative gains on an event of positive probability; compactness of directions proves coercivity. At a minimizer, differentiation gives $\mathbb{E}[X{e}^{-|X{|}^2-h_{\ast }\cdot X}]=0$. Normalizing the positive weight produces an equivalent martingale measure with integrable gains. This is a constructive one-period proof of the fundamental theorem of asset pricing even on an infinite probability space.