= Gaussian-damped martingale density construction
{c}
{title2=$dQ/dP\propto e^{-|X|^2-h_*\cdot X}$}
For a finite-dimensional discounted gain <random vector> $X$, absence of <arbitrage> means $h\cdot X\geq0$ 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 $\mathbb E[Xe^{-|X|^2-h_*\cdot X}]=0$. 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>.
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