732 research outputs found
Path-dependent equations and viscosity solutions in infinite dimension
Path-dependent PDEs (PPDEs) are natural objects to study when one deals with
non Markovian models. Recently, after the introduction of the so-called
pathwise (or functional or Dupire) calculus (see [15]), in the case of
finite-dimensional underlying space various papers have been devoted to
studying the well-posedness of such kind of equations, both from the point of
view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g.
[16]). In this paper, motivated by the study of models driven by path-dependent
stochastic PDEs, we give a first well-posedness result for viscosity solutions
of PPDEs when the underlying space is a separable Hilbert space. We also
observe that, in contrast with the finite-dimensional case, our well-posedness
result, even in the Markovian case, applies to equations which cannot be
treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit
The maximum maximum of a martingale with given marginals
We obtain bounds on the distribution of the maximum of a martingale with
fixed marginals at finitely many intermediate times. The bounds are sharp and
attained by a solution to -marginal Skorokhod embedding problem in
Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many
marginals (2013) Preprint]. It follows that their embedding maximizes the
maximum among all other embeddings. Our motivating problem is superhedging
lookback options under volatility uncertainty for an investor allowed to
dynamically trade the underlying asset and statically trade European call
options for all possible strikes and finitely-many maturities. We derive a
pathwise inequality which induces the cheapest superhedging value, which
extends the two-marginals pathwise inequality of Brown, Hobson and Rogers
[Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by
elementary arguments, is derived by following the stochastic control approach
of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014)
312-336].Comment: Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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