We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio optimization under an additional liability and on dynamic utility indifference valuation and partial hedging in incomplete financial markets which are exposed to risk from unpredictable events. In particular, we characterize the limiting behavior of the utility indifference hedging strategy and of the indifference value process for vanishing risk aversion.

Asset prices discounted by a tradable numeraire N should be (local) martingales under some measure Q that is equivalent to the original probability measure P . Instead of studying the set of equivalent martingale measures with respect to a prespecified numeraire, we will look for a tradable numeraire N P such that the discounted asset prices become martingales with respect to the original measure P . N P is called (P -)numeraire portfolio. Since the above martingale condition is too stringent to obtain a general existence result, we define a (generalized) numeraire portfolio by a weaker requirement. This N P is characterized as the solution to several optimization problems.

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Abstract. We consider an investor maximizing his expected utility from terminal wealth with portfolio decisions based on the available information flow. This investor faces the opportunity to acquire some additional initial information G. His subjective fair value of this information is defined as the amount of money that he can pay for G such that this cost is balanced out by the informational advantage in terms of maximal expected utility. We study this value for common utility functions in the setting of a complete market modeled by general semimartingales. The main tools are a martingale preserving change of measure and martingale representation results for initially enlarged filtrations.

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Itô and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuation problems for contingent claims with a recursive payoff structure.

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the inter-temporal resilience of the market in spirit of [PSS11], is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.

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