By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.

Let X be a point process and let X denote the filtration generated by X. In this paper we study martingale representation theorems in the filtration G obtained as an initial and progressive enlargement of the filtration X. In particular, the progressive enlargement is done by means of a whole point process H. We work here in full generality, without requiring any further assumption on the process H and we recover the special case in which X is enlarged progressively by a random time τ.

We consider the problem of hedging a contingent claim with a “semistatic” strategy composed of a dynamic position in one asset and static (buy‐and‐hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance optimality and provide tractable formulas using Fourier integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semistatic hedging strategy, i.e., a strategy that only uses a small subset of available hedging assets and discuss parallels to the variable‐selection problem in linear regression. The methods developed are illustrated in an extended numerical example where we compute a sparse semistatic hedge for a variance swap using European options as static hedging assets.

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