Abstract
Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution μ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of μ.
Highlights
We contribute to the literature on Skorokhod embedding problem (SEP)
If we let τB,X = t on {σt ≤ T m < σt+1} it follows that τB,X is a stopping time relative to the natural filtration of X enlarged with a suitable family of independent random variables, it has the precise structure of τ (r) in (2), embeds μ and is uniformly integrable (UI)
While (11) may be deduced from known bounds, as explained before, we provide a quick self –contained proof
Summary
The SEP, in general, refers to the problem of finding a stopping time τ , such that a given stochastic process X , when. The coins are suitably biased, with state dependent probabilities, which can be readily computed using an explicit algorithm we provide Such a strategy is easy to compute and easy to implement, which is important for applications, e.g. to justify its use by economic agents, see [12]. Our second construction, presented, is a discretetime analogue of the Azema and Yor [2] embedding It is explicit and its first appeal lies in the fact that it only stops when the loss, relative to the last maximum, gets too large.
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