Mimicking martingales
Given the univariate marginals of a real valued, continuous time martingale, (respectively, 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 (respectively, 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 family of martingales which are consistent with a given set of measures) has the smallest expected total variation. We also give a path wise inequality, which in the mathematical finance context yields a modelindependent sub hedge for an exotic security with payoff equal to the total variation along a realisation of the price process. 1 The problem In this article we are concerned with the following problem: given a martingale M (Mt)0<=t<=T on a stochastic basis (O ,FM ,F ,P construct a fake version of M (or equivalently a process which mimics M) i.e. construct (potentially on a new filtered probability space (O,F ,F,P) a stochastic process X (Xt)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 (mt)0<=t<=T which are increasing in convex order. Then the aim is to construct a filtered probability space (O,F ,F,P) and a F martingale X (Xt)0<=t<=T on that space such that P(Xt x) mt( x] for all t and all x. Then we say that X is consistent with the measures (mt)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 limk| C(t, k) 0; C(t, 0) is a positive constant, independent of t; and C(t, k) is non decreasing in t, for all k. Then the aim is to find a model which is consistent with the given