Utility duality with martingale deflators

The concave conjugate $\hat{u}(y)={\sup }_{x>0}\{u(x)-xy\}$ gives the weak-duality bound $\mathbb{E}u(X)\le \mathbb{E}\hat{u}(Y)+X_0{Y}_0$. Equality holds when $u^{\prime }(X)=Y$ and the payoff exhausts the deflated budget.

In a complete market with a unit cash asset and normalized strictly positive state-price density $Z$, the optimal positive terminal portfolio wealth is $x/Z$ whenever it is replicable in the admissible class. The state-price budget constraint is an equality: $\mathbb{E}[Z(x/Z)]=x$. The utility conjugate inequality gives $\mathbb{E}\log H\le \log x-\mathbb{E}\log Z$ for every affordable positive claim. This value can be infinite, because $\mathbb{E}(\log Z{)}^{+}<\infty$ while $\mathbb{E}(\log Z{)}^{-}$ need not be finite. Even completeness for bounded claims suffices to approximate the value: set $b_n=\mathbb{E}[Z/(Z\vee n^{-1})]$ and $H_n=x/[b_n(Z\vee n^{-1})]$. Each claim is bounded and costs $x$, while its expected logarithmic utility tends to the displayed value by monotone convergence after an integrable lower bound is subtracted.

For a negative, strictly increasing, concave differentiable utility with finite expected utility at every holding, a gain having both signs makes the objective coercive at both ends. At a finite optimum its derivative is zero. Secant domination for expected utility derivatives gives integrability of $|X|U^{\prime }({\theta }_{\ast }X)$; boundedness of $U^{\prime }$ on the remaining compact set gives integrability of the positive marginal utility itself. Its normalization has expectation one and prices the gain at zero.

In a one-period utility problem with gross riskless return $R$, the marginal utility price is the infinitesimal claim price leaving optimized expected utility unchanged. If optimal terminal wealth is $W^{\ast }$, the pricing density is $Z=U^{\prime }(W^{\ast })/(R\mathbb{E}{U}^{\prime }(W^{\ast }))$, and $p_H=\mathbb{E}[ZH]$. The denominator is the marginal utility of initial wealth. Under the standard first-order optimality conditions this density prices the traded assets and gives the state-price density associated with the investor.

At a nonzero optimum, the derivative of expected utility of a proportional transaction cost payoff is zero. Normalized marginal utility therefore prices $X$ at $\epsilon$ for a positive optimum or $-\epsilon$ for a negative optimum. At a zero optimum the one-sided derivative inequalities put $\mathbb{E}X$ inside that interval, so density one works. The competing one-sided-payoff alternative prevents the coercive-maximizer argument from being assumed when a net gain has only one sign.

For a positive state-price density $Z$ pricing every attainable payoff, a feasible positive terminal payoff $W^{\ast }$ is optimal at its budget if its marginal utility is $yZ$ for some scalar $y>0$. The pointwise conjugate inequality gives $\mathbb{E}U(W)\le \mathbb{E}\hat{U}(yZ)+yx$ for every competitor; the marginal-utility identity makes equality hold for $W^{\ast }$. Finite states make integrability straightforward. In a complete market an interior optimum also necessarily has this proportionality, by variations preserving the state-price budget.

For concave increasing utility, the concave supporting-tangent inequality gives $e^{-bt}[U({\hat{c}}_t)-U(c_t)]\le Y_t({\hat{c}}_t-c_t)$. If the proposed consumption exhausts the state-price budget and every competitor has no larger budget, integration proves optimality. The objective must be well defined; a positive discount rate and finite utility bounds are a convenient sufficient condition.

For increasing strictly concave wealth value $v$, the dual $J(z)={\sup }_x[v(x)-xz]$ is convex. At an interior optimum $z=v^{\prime }(x)$, $J^{\prime }=-x$, $J^{\prime \prime }=-1/v^{\prime \prime }>0$, and $v=J-z{J}^{\prime }$. This sign convention is the negative of the concave Legendre dual. It can turn the optimized portfolio term of a Hamilton-Jacobi-Bellman equation into a linear second derivative.