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Skorokhod embeddings via stochastic flows on the space of Gaussian measures. (English. French summary) Zbl 1350.60039
Summary: We present a new construction of a Skorokhod embedding, namely, given a probability measure \(\mu\) with zero expectation and finite variance, we construct an integrable stopping time \(T\) adapted to a filtration \(\mathcal{F}_{t}\), such that \(W_{T}\) has the law \(\mu\), where \(W_{t}\) is a standard Wiener process adapted to the same filtration. We find several sufficient conditions for the stopping time \(T\) to be bounded or to have a sub-exponential tail. In particular, our embedding seems rather natural for the case that \(\mu\) is a log-concave measure and \(T\) satisfies several tight bounds in that case. Our embedding admits the property that the stochastic measure-valued process \(\{\mu_{t}\}_{t\geq0}\), where \(\mu_{t}\) is as the law of \(W_{T}\) conditioned on \(\mathcal{F}_{t}\), is a Markov process. In view of this property, we will consider a more general family of Skorokhod embeddings which can be constructed via a kernel generating a stochastic flow on the space of measures. This family includes existing constructions such as the ones by J. Azema and M. Yor [Lect. Notes Math. 721, 90–115 (1979; Zbl 0414.60055)] and by R. F. Bass [ibid. 986, 221–224 (1983; Zbl 0509.60080)], and thus suggests a new point of view on these constructions.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
60H20 Stochastic integral equations
60J65 Brownian motion
60J25 Continuous-time Markov processes on general state spaces
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