We study the dynamics of the exponential utility indifference value process
C(B;\alpha) for a contingent claim B in a semimartingale model with a general
continuous filtration. We prove that C(B;\alpha) is (the first component of)
the unique solution of a backward stochastic differential equation with a
quadratic generator and obtain BMO estimates for the components of this
solution. This allows us to prove several new results about C_t(B;\alpha). We
obtain continuity in B and local Lipschitz-continuity in the risk aversion
\alpha, uniformly in t, and we extend earlier results on the asymptotic
behavior as \alpha\searrow0 or \alpha\nearrow\infty to our general setting.
Moreover, we also prove convergence of the corresponding hedging strategies.Comment: Published at http://dx.doi.org/10.1214/105051605000000395 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
We consider a problem of minimization of a hedging error, measured by a positive convex random function, in an incomplete financial market model, where the dynamics of asset prices is given by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.
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