214 research outputs found
Optimal transportation under controlled stochastic dynamics
We consider an extension of the Monge-Kantorovitch optimal transportation
problem. The mass is transported along a continuous semimartingale, and the
cost of transportation depends on the drift and the diffusion coefficients of
the continuous semimartingale. The optimal transportation problem minimizes the
cost among all continuous semimartingales with given initial and terminal
distributions. Our first main result is an extension of the Kantorovitch
duality to this context. We also suggest a finite-difference scheme combined
with the gradient projection algorithm to approximate the dual value. We prove
the convergence of the scheme, and we derive a rate of convergence. We finally
provide an application in the context of financial mathematics, which
originally motivated our extension of the Monge-Kantorovitch problem. Namely,
we implement our scheme to approximate no-arbitrage bounds on the prices of
exotic options given the implied volatility curve of some maturity.Comment: Published in at http://dx.doi.org/10.1214/12-AOP797 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Irreducible convex paving for decomposition of multi-dimensional martingale transport plans
Martingale transport plans on the line are known from Beiglbock & Juillet to
have an irreducible decomposition on a (at most) countable union of intervals.
We provide an extension of this decomposition for martingale transport plans in
R^d, d larger than one. Our decomposition is a partition of R^d consisting of a
possibly uncountable family of relatively open convex components, with the
required measurability so that the disintegration is well-defined. We justify
the relevance of our decomposition by proving the existence of a martingale
transport plan filling these components. We also deduce from this decomposition
a characterization of the structure of polar sets with respect to all
martingale transport plans.Comment: 52 pages, 2 figure
Optimal Stopping under Nonlinear Expectation
Let be a bounded c\`adl\`ag process with positive jumps defined on the
canonical space of continuous paths. We consider the problem of optimal
stopping the process under a nonlinear expectation operator \cE defined
as the supremum of expectations over a weakly compact family of nondominated
measures. We introduce the corresponding nonlinear Snell envelope. Our main
objective is to extend the Snell envelope characterization to the present
context. Namely, we prove that the nonlinear Snell envelope is an
\cE-supermartingale, and an \cE-martingale up to its first hitting time of
the obstacle . This result is obtained under an additional uniform
continuity property of . We also extend the result in the context of a
random horizon optimal stopping problem.
This result is crucial for the newly developed theory of viscosity solutions
of path-dependent PDEs as introduced in Ekren et al., in the semilinear case,
and extended to the fully nonlinear case in the accompanying papers (Ekren,
Touzi, and Zhang, parts I and II).Comment: 36 page
Moral Hazard in Dynamic Risk Management
We consider a contracting problem in which a principal hires an agent to
manage a risky project. When the agent chooses volatility components of the
output process and the principal observes the output continuously, the
principal can compute the quadratic variation of the output, but not the
individual components. This leads to moral hazard with respect to the risk
choices of the agent. We identify a family of admissible contracts for which
the optimal agent's action is explicitly characterized, and, using the recent
theory of singular changes of measures for It\^o processes, we study how
restrictive this family is. In particular, in the special case of the standard
Homlstr\"om-Milgrom model with fixed volatility, the family includes all
possible contracts. We solve the principal-agent problem in the case of CARA
preferences, and show that the optimal contract is linear in these factors: the
contractible sources of risk, including the output, the quadratic variation of
the output and the cross-variations between the output and the contractible
risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent
contracts naturally arise when there is moral hazard with respect to risk
management. In a numerical example, we show that the loss of efficiency can be
significant if the principal does not use the quadratic variation component of
the optimal contract.Comment: 36 pages, 3 figure
Tightness and duality of martingale transport on the Skorokhod space
The martingale optimal transport aims to optimally transfer a probability
measure to another along the class of martingales. This problem is mainly
motivated by the robust superhedging of exotic derivatives in financial
mathematics, which turns out to be the corresponding Kantorovich dual. In this
paper we consider the continuous-time martingale transport on the Skorokhod
space of cadlag paths. Similar to the classical setting of optimal transport,
we introduce different dual problems and establish the corresponding dualities
by a crucial use of the S-topology and the dynamic programming principle
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