Approximation pricing and the variance-optimal martingale measure

Let X be a semimartiangale and let Θ be the space of all predictable X-integrable process ϑ such that ∫ϑ dX is in the space ² of semimartingales. We consider the problem of approximating a given random variable H∈ L²(P) by the sum of a constant c and a stochastic integral ∫₀Tϑₛ dXₛ, with respect to the L²(P)-norm. This problem comes from financial mathematics, where the optimal constant V₀ can be interpreted as an approximation price for the contingent clam H. An elementary computation yields V₀ as the expectation of H under the variance-optimal signed Θ-martingale measure P̃, and this leads us to study P in more detail. In the case of finite discrete time, we explicitly construct P̃ by backward recursion, and we show that P̃ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that P̃ is nonegative is X has continuous paths and satisfies a very mild no-arbitrage condition. As an application, we show how to obtain the optimal integrand ξ∈Θ in feedback form with the help of a backward stochastic differential equation.