### Mimicking martingales

David Hobson
2016 The Annals of Applied Probability
Given the univariate marginals of a real-valued, continuous-time martingale, (resp., a family of measures parameterised by t ∈ [0, T ] which is increasing in convex order, or a double continuum of call prices), we construct a family of pure-jump martingales which mimic that martingale (resp., are consistent with the family of measures, or call prices). As an example, we construct a fake Brownian motion. Then, under a further "dispersion" assumption, we construct the martingale which (within the
more » ... e which (within the family of martingales which are consistent with a given set of measures) has the smallest expected total variation. We also give a pathwise inequality, which in the mathematical finance context yields a model-independent sub-hedge for an exotic security with payoff equal to the total variation of the price process. 1. The problem. In this article, we are concerned with the following problem: construct a fake version of M (or equivalently a process which mimics M), that is, construct [potentially on a new filtered probability space ( , F, F, P)] a stochastic process X = (X t ) 0≤t≤T such that X is a F-martingale and the univariate marginals of X are the same as those of M, but such that the joint marginals are different. The problem can be reformulated in two further ways. First, instead of beginning with a martingale M, we can begin with a family of laws (μ t ) 0≤t≤T which are increasing in convex order. Then the aim is to construct a filtered probability space ( , F, F, P) and a F-martingale X = (X t ) 0≤t≤T on that space such that P(X t ≤ x) = μ t ((−∞, x]) for all t and all x. Then we say that X is consistent with the measures (μ t ) 0≤t≤T . Second, but closely related, instead of beginning with a process or a set of measures we can work in the setting of mathematical finance and start with a double continuum of European call prices {C(t, k); 0 ≤ t ≤ T , 0 ≤ k < ∞} which satisfy no-arbitrage conditions. [We assume that we are working with discounted prices, and then the no-arbitrage conditions are that: for each t, C(t, k) is a decreasing convex function with C (t, 0+) ≥ 1 and lim k↑∞ C(t, k) = 0; C(t, 0) is a positive constant, independent of t; and C(t, k) is nondecreasing in t, for all k.] Then the aim is to find a model which is consistent with the given call prices, that is, a filtered probability space and a martingale X