165 research outputs found
Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance
We are concerned with a new type of supermartingale decomposition in the
Max-Plus algebra, which essentially consists in expressing any supermartingale
of class as a conditional expectation of some running supremum
process. As an application, we show how the Max-Plus supermartingale
decomposition allows, in particular, to solve the American optimal stopping
problem without having to compute the option price. Some illustrative examples
based on one-dimensional diffusion processes are then provided. Another
interesting application concerns the portfolio insurance. Hence, based on the
``Max-Plus martingale,'' we solve in the paper an optimization problem whose
aim is to find the best martingale dominating a given floor process (on every
intermediate date), w.r.t. the convex order on terminal values.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
An Exact Connection between two Solvable SDEs and a Nonlinear Utility Stochastic PDE
Motivated by the work of Musiela and Zariphopoulou \cite{zar-03}, we study
the It\^o random fields which are utility functions for any
. The main tool is the marginal utility and its inverse
expressed as the opposite of the derivative of the Fenchel conjuguate
\tU(t,y). Under regularity assumptions, we associate a and
its adjoint SPDE in divergence form whose and its
inverse -\tU_y(t,y) are monotonic solutions. More generally, special
attention is paid to rigorous justification of the dynamics of inverse flow of
SDE. So that, we are able to extend to the solution of similar SPDEs the
decomposition based on the solutions of two SDEs and their inverses. The second
part is concerned with forward utilities, consistent with a given incomplete
financial market, that can be observed but given exogenously to the investor.
As in \cite{zar-03}, market dynamics are considered in an equilibrium state, so
that the investor becomes indifferent to any action she can take in such a
market. After having made explicit the constraints induced on the local
characteristics of consistent utility and its conjugate, we focus on the
marginal utility SPDE by showing that it belongs to the previous family of
SPDEs. The associated two SDE's are related to the optimal wealth and the
optimal state price density, given a pathwise explicit representation of the
marginal utility. This new approach addresses several issues with a new
perspective: dynamic programming principle, risk tolerance properties, inverse
problems. Some examples and applications are given in the last section
Ramsey Rule with Progressive utility and Long Term Affine Yields Curves
The purpose of this paper relies on the study of long term affine yield
curves modeling. It is inspired by the Ramsey rule of the economic literature,
that links discount rate and marginal utility of aggregate optimal consumption.
For such a long maturity modelization, the possibility of adjusting preferences
to new economic information is crucial, justifying the use of progressive
utility. This paper studies, in a framework with affine factors, the yield
curve given from the Ramsey rule. It first characterizes consistent progressive
utility of investment and consumption, given the optimal wealth and consumption
processes. A special attention is paid to utilities associated with linear
optimal processes with respect to their initial conditions, which is for
example the case of power progressive utilities. Those utilities are the basis
point to construct other progressive utilities generating non linear optimal
processes but leading yet to still tractable computations. This is of
particular interest to study the impact of initial wealth on yield curves.Comment: arXiv admin note: substantial text overlap with arXiv:1404.189
On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation
We study the class of Az\'ema-Yor processes defined from a general
semimartingale with a continuous running maximum process. We show that they
arise as unique strong solutions of the Bachelier stochastic differential
equation which we prove is equivalent to the drawdown equation. Solutions of
the latter have the drawdown property: they always stay above a given function
of their past maximum. We then show that any process which satisfies the
drawdown property is in fact an Az\'ema-Yor process. The proofs exploit group
structure of the set of Az\'ema-Yor processes, indexed by functions, which we
introduce. We investigate in detail Az\'ema-Yor martingales defined from a
nonnegative local martingale converging to zero at infinity. We establish
relations between average value at risk, drawdown function, Hardy-Littlewood
transform and its inverse. In particular, we construct Az\'ema-Yor martingales
with a given terminal law and this allows us to rediscover the Az\'ema-Yor
solution to the Skorokhod embedding problem. Finally, we characterize
Az\'ema-Yor martingales showing they are optimal relative to the concave
ordering of terminal variables among martingales whose maximum dominates
stochastically a given benchmark.Comment: Published in at http://dx.doi.org/10.1214/10-AOP614 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dynamics of multivariate default system in random environment
We consider a multivariate default system where random environmental
information is available. We study the dynamics of the system in a general
setting and adopt the point of view of change of probability measures. We also
make a link with the density approach in the credit risk modelling. In the
particular case where no environmental information is concerned, we pay a
special attention to the phenomenon of system weakened by failures as in the
classical reliability system
Maturity randomization for stochastic control problems
We study a maturity randomization technique for approximating optimal control
problems. The algorithm is based on a sequence of control problems with random
terminal horizon which converges to the original one. This is a generalization
of the so-called Canadization procedure suggested by Carr [Review of Financial
Studies II (1998) 597--626] for the fast computation of American put option
prices. In addition to the original application of this technique to optimal
stopping problems, we provide an application to another problem in finance,
namely the super-replication problem under stochastic volatility, and we show
that the approximating value functions can be computed explicitly.Comment: Published at http://dx.doi.org/10.1214/105051605000000593 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Ramsey Rule with Progressive Utility in Long Term Yield Curves Modeling
The purpose of this paper relies on the study of long term yield curves
modeling. Inspired by the economic litterature, it provides a financial
interpretation of the Ramsey rule that links discount rate and marginal utility
of aggregate optimal consumption. For such a long maturity modelization, the
possibility of adjusting preferences to new economic information is crucial.
Thus, after recalling some important properties on progressive utility, this
paper first provides an extension of the notion of a consistent progressive
utility to a consistent pair of progressive utilities of investment and
consumption. An optimality condition is that the utility from the wealth
satisfies a second order SPDE of HJB type involving the Fenchel-Legendre
transform of the utility from consumption. This SPDE is solved in order to give
a full characterization of this class of consistent progressive pair of
utilities. An application of this results is to revisit the classical backward
optimization problem in the light of progressive utility theory, emphasizing
intertemporal-consistency issue. Then we study the dynamics of the marginal
utility yield curve, and give example with backward and progressive power
utilities
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