Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
X is said to mimic ξ on the time interval [0,T], if ξ and X have the same
marginal distributions:
d
∀t ∈ [0,T], ξt=Xt.
(1.1)
The construction of a Markov process X with the above property is called
the “mimicking” problem.
First suggested by Brémaud[21] inthe context ofqueues (under thename
of ’first-order equivalence), this problem has been the focus of a considerable
literature, using a variety of probabilistic and analytic methods [2, 7, 23, 36,
52, 55, 74, 64, 69, 51]. These results have many applications, in particular
related to option pricing problems [34, 36] and the related inverse problem of
calibration of pricing models given observed option prices [34, 22, 28, 27, 79].
1.1.1 Stochastic processes with given marginal distributions
The mimicking problem is related to the construction of martingales with a
given flow of marginals, which dates back to Kellerer [64]. It is known that
the flow of marginal distributions of a martingale is increasing in the convex
order i.e. for any convex function φ : Rd ↦→ R,
∀t ≥ s, φ(y)pt(y)dy ≥ φ(y)ps(y)dy, (1.2)
Kellerer [64] shows that, conversely, any family of probability distributions
with this property can be realized as the flow of marginal densities of a
(sub)martingale:
Theorem 1.1 ([64], p.120). Let pt(y), y ∈ Rd , t ≥ 0, be a family of marginal
densities, with finite first moment, which is increasing in the convex order:
for any convex function φ : Rd ↦→ R, and any s < t,
∀t ≥ s, φ(y)pt(y)dy ≥ φ(y)ps(y)dy. (1.3)
Then there exists a Markov process (Xt)t≥0 such that the marginal density of
Xt is pt(y). Furthermore, Xt is a martingale.
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