302 research outputs found
Infinite horizon optimal control of forward-backward stochastic differential equations with delay
We consider a problem of optimal control of an infinite horizon system
governed by forward-backward stochastic differential equations with delay.
Sufficient and necessary maximum principles for optimal control under partial
information in infinite horizon are derived. We illustrate our results by an
application to a problem of optimal consumption with respect to recursive
utility from a cash flow with delay
A Donsker delta functional approach to optimal insider control and applications to finance
We study \emph{optimal insider control problems}, i.e. optimal control
problems of stochastic systems where the controller at any time in addition
to knowledge about the history of the system up to this time, also has
additional information related to a \emph{future} value of the system. Since
this puts the associated controlled systems outside the context of
semimartingales, we apply anticipative white noise analysis, including forward
integration and Hida-Malliavin calculus to study the problem. Combining this
with Donsker delta functionals we transform the insider control problem into a
classical (but parametrised) adapted control system, albeit with a
non-classical performance functional. We establish a sufficient and a necessary
maximum principle for such systems. Then we apply the results to obtain
explicit solutions for some optimal insider portfolio problems in financial
markets described by It\^ o-L\' evy processes. Finally, in the Appendix we give
a brief survey of the concepts and results we need from the theory of white
noise, forward integrals and Hida-Malliavin calculus
Malliavin calculus and optimal control of stochastic Volterra equations
Solutions of stochastic Volterra (integral) equations are not Markov
processes, and therefore classical methods, like dynamic programming, cannot be
used to study optimal control problems for such equations. However, we show
that by using {\em Malliavin calculus} it is possible to formulate a modified
functional type of {\em maximum principle} suitable for such systems. This
principle also applies to situations where the controller has only partial
information available to base her decisions upon. We present both a sufficient
and a necessary maximum principle of this type, and then we use the results to
study some specific examples. In particular, we solve an optimal portfolio
problem in a financial market model with memory.Comment: 18 page
Optimal insider control of stochastic partial differential equations
We study the problem of optimal inside control of an SPDE (a stochastic
evolution equation) driven by a Brownian motion and a Poisson random measure.
Our optimal control problem is new in two ways: (i) The controller has access
to inside information, i.e. access to information about a future state of the
system, (ii) The integro-differential operator of the SPDE might depend on the
control.
In the first part of the paper, we formulate a sufficient and a necessary
maximum principle for this type of control problem, in two cases: (1) When the
control is allowed to depend both on time t and on the space variable x. (2)
When the control is not allowed to depend on x.
In the second part of the paper, we apply the results above to the problem of
optimal control of an SDE system when the inside controller has only noisy
observations of the state of the system. Using results from nonlinear
filtering, we transform this noisy observation SDE inside control problem into
a full observation SPDE insider control problem.
The results are illustrated by explicit examples
Optimal insider control and semimartingale decompositions under enlargement of filtration
We combine stochastic control methods, white noise analysis and
Hida-Malliavin calculus applied to the Donsker delta functional to obtain new
representations of semimartingale decompositions under enlargement of
filtrations. The results are illustrated by explicit examples
Dynamic robust duality in utility maximization
A celebrated financial application of convex duality theory gives an explicit
relation between the following two quantities:
(i) The optimal terminal wealth of the problem
to maximize the expected -utility of the terminal wealth
generated by admissible portfolios in a market
with the risky asset price process modeled as a semimartingale;
(ii) The optimal scenario of the dual problem to minimize
the expected -value of over a family of equivalent local
martingale measures , where is the convex conjugate function of the
concave function .
In this paper we consider markets modeled by It\^o-L\'evy processes. In the
first part we use the maximum principle in stochastic control theory to extend
the above relation to a \emph{dynamic} relation, valid for all .
We prove in particular that the optimal adjoint process for the primal problem
coincides with the optimal density process, and that the optimal adjoint
process for the dual problem coincides with the optimal wealth process, . In the terminal time case we recover the classical duality
connection above. We get moreover an explicit relation between the optimal
portfolio and the optimal measure . We also obtain that the
existence of an optimal scenario is equivalent to the replicability of a
related -claim.
In the second part we present robust (model uncertainty) versions of the
optimization problems in (i) and (ii), and we prove a similar dynamic relation
between them. In particular, we show how to get from the solution of one of the
problems to the other. We illustrate the results with explicit examples
A white noise approach to insider trading
We present a new approach to the optimal portfolio problem for an insider
with logarithmic utility. Our method is based on white noise theory, stochastic
forward integrals, Hida-Malliavin calculus and the Donsker delta function.Comment: arXiv admin note: text overlap with arXiv:1504.0258
Stochastic differential games with inside information
We study stochastic differential games of jump diffusions, where the players
have access to inside information. Our approach is based on anticipative
stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals
and the Donsker delta functional. We obtain a characterization of Nash
equilibria of such games in terms of the corresponding Hamiltonians. This is
used to study applications to insider games in finance, specifically optimal
insider consumption and optimal insider portfolio under model uncertainty.Comment: arXiv admin note: text overlap with arXiv:1504.0258
- …