Compute expected cost and variance under log-normal probabilistic strategy selection

Compute the expected total cost of trading and its variance in the probabilistic strategy selection framework where the competitor’s position size λ is distributed log-normally with parameters μ and σ, the market impact coefficient κ is fixed, and the trading strategies are the two-trader λ-scaled equilibrium strategy bλ(t) and its best-response aλ(t). Specifically, evaluate E[C] and Var[C], where C(aλ, bλ; κ) = ∫₀¹ [(ẋaλ(t) + λ ẋbλ(t)) ẋaλ(t) + κ (aλ(t) + λ bλ(t)) ẋaλ(t)] dt, with the expectation taken over λ under the log-normal distribution and aλ(t), bλ(t) determined by the two-trader equilibrium for the given λ and κ.

Background

The paper introduces a probabilistic strategy selection framework to handle uncertainty about an adversary’s trading size by equipping the set of possible λ-scaled strategies with a probability measure. In the log-normal example, the distribution of the adversary’s size λ is specified and the expected total cost of trading is defined as an integral of the two-trader equilibrium best-response cost over λ.

While the integrals defining the expected cost and variance are formulated, the authors explicitly defer their computation, indicating an unresolved quantitative step necessary to operationalize mean–variance strategy selection under uncertainty.