熵投影鞅测度的计价单位不变性

中文摘要 AI 辅助

设 \\(P\\) 为固定的物理概率测度，\\(Q\\) 为从与所选计价单位相关的鞅测度集中选取的等价鞅测度。计价单位的改变将 \\(Q\\) 映射为 \\(T_LQ\\)，其中 \\(d(T_LQ)=L\\,dQ\\)，\\(L\\) 为终端似然比。最小化 \\(D_{\mathrm{KL}}(P\Vert Q)\\) 的前向相对熵投影与该变换可交换，因为其目标函数仅改变常数 \\(-E_P\log L\\)。最小熵鞅测度（MEMM）方向 \\(D_{\mathrm{KL}}(Q\Vert P)\\) 不具备此性质，一个三项式反例表明独立重新计算的 MEMM 不必满足似然相容性。我们明确两个经济后果。第一，两种熵方向分别恰为对数效用和指数效用的经典凸对偶目标中的 \\(Q\\) 依赖项。第二，似然相容性等价于在两种计价单位下所得定价泛函相等。因此，前向选择器对所有可积求偿权在跨计价单位下给出一致定价，而反例中的两个 MEMM 对不可复制的数字求偿权给出不同价格。我们还证明了有限状态类级刻画：在初等单期似然比族上的均匀不变性迫使光滑凸 \\(f\\)-散度在缩放和仿射等价意义下为对数形式。最后，在有限状态市场中，在通常的严格正可行点条件下，前向投影存在；其密度 \\(dP/dQ^*\\) 是可达到的对数最优终端财富，且最小前向熵等于最大期望对数增长。

英文摘要

Let \(P\) be a fixed physical law and let \(Q\) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps \(Q\) to \(T_LQ\), where \(d(T_LQ)=L\,dQ\) and \(L\) is the terminal likelihood ratio. The forward relative-entropy projection minimizing \(D_{\mathrm{KL}}(P\Vert Q)\) commutes with this transform because its objective changes only by the constant \(-E_P\log L\). The minimal entropy martingale measure (MEMM) orientation \(D_{\mathrm{KL}}(Q\Vert P)\) does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the \(Q\)-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex \(f\)-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density \(dP/dQ^*\) is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

发表机构

• Charles University（查理大学）

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