961 research outputs found
A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps
We consider a stochastic functional delay differential equation, namely an
equation whose evolution depends on its past history as well as on its present
state, driven by a pure diffusive component plus a pure jump Poisson
compensated measure. We lift the problem in the infinite dimensional space of
square integrable Lebesgue functions in order to show that its solution is an
valued Markov process whose uniqueness can be shown under standard
assumptions of locally Lipschitzianity and linear growth for the coefficients.
Coupling the aforementioned equation with a standard backward differential
equation, and deriving some ad hoc results concerning the Malliavin derivative
for systems with memory, we are able to derive a non--linear Feynman--Kac
representation theorem under mild assumptions of differentiability
Operational risk management and new computational needs in banks
Basel II banking regulation introduces new needs for computational schemes. They involve both optimal stochastic control, and large scale simulations of decision processes of preventing low-frequency high loss-impact events. This paper will first state the problem and present its parameters. It then spells out the equations that represent a rational risk management behavior and link together the variables: Levy processes are used to model operational risk losses, where calibration by historical loss databases is possible ; where it is not the case, qualitative variables such as quality of business environment and internal controls can provide both costs-side and profits-side impacts. Among other control variables are business growth rate, and efficiency of risk mitigation. The economic value of a policy is maximized by resolving the resulting Hamilton-Jacobi-Bellman type equation. Computational complexity arises from embedded interactions between 3 levels: * Programming global optimal dynamic expenditures budget in Basel II context, * Arbitraging between the cost of risk-reduction policies (as measured by organizational qualitative scorecards and insurance buying) and the impact of incurred losses themselves. This implies modeling the efficiency of the process through which forward-looking measures of threats minimization, can actually reduce stochastic losses, * And optimal allocation according to profitability across subsidiaries and business lines. The paper next reviews the different types of approaches that can be envisaged in deriving a sound budgetary policy solution for operational risk management, based on this HJB equation. It is argued that while this complex, high dimensional problem can be resolved by taking some usual simplifications (Galerkin approach, imposing Merton form solutions, viscosity approach, ad hoc utility functions that provide closed form solutions, etc.) , the main interest of this model lies in exploring the scenarios in an adaptive learning framework ( MDP, partially observed MDP, Q-learning, neuro-dynamic programming, greedy algorithm, etc.). This makes more sense from a management point of view, and solutions are more easily communicated to, and accepted by, the operational level staff in banks through the explicit scenarios that can be derived. This kind of approach combines different computational techniques such as POMDP, stochastic control theory and learning algorithms under uncertainty and incomplete information. The paper concludes by presenting the benefits of such a consistent computational approach to managing budgets, as opposed to a policy of operational risk management made up from disconnected expenditures. Such consistency satisfies the qualifying criteria for banks to apply for the AMA (Advanced Measurement Approach) that will allow large economies of regulatory capital charge under Basel II Accord.REGULAR - Operational risk management, HJB equation, Levy processes, budget optimization, capital allocation
Asymptotic Power Utility-Based Pricing and Hedging
Kramkov and Sirbu (2006, 2007) have shown that first-order approximations of
power utility-based prices and hedging strategies can be computed by solving a
mean-variance hedging problem under a specific equivalent martingale measure
and relative to a suitable numeraire. In order to avoid the introduction of an
additional state variable necessitated by the change of numeraire, we propose
an alternative representation in terms of the original numeraire. More
specifically, we characterize the relevant quantities using semimartingale
characteristics similarly as in Cerny and Kallsen (2007) for mean-variance
hedging. These results are illustrated by applying them to exponential L\'evy
processes and stochastic volatility models of Barndorff-Nielsen and Shephard
type.Comment: 32 pages, 4 figures, to appear in "Mathematics and Financial
Economics
Existence and uniqueness results for BSDEs with jumps: the whole nine yards
This paper is devoted to obtaining a wellposedness result for
multidimensional BSDEs with possibly unbounded random time horizon and driven
by a general martingale in a filtration only assumed to satisfy the usual
hypotheses, i.e. the filtration may be stochastically discontinuous. We show
that for stochastic Lipschitz generators and unbounded, possibly infinite, time
horizon, these equations admit a unique solution in appropriately weighted
spaces. Our result allows in particular to obtain a wellposedness result for
BSDEs driven by discrete--time approximations of general martingales.Comment: 48 pages, final version, forthcoming in the Electronic Journal of
Probabilit
Modeling and computation of an integral operator Riccati equation for an infinite-dimensional stochastic differential equation governing streamflow discharge
We propose a linear-quadratic (LQ) control problem of streamflow discharge by
optimizing an infinite-dimensional jump-driven stochastic differential equation
(SDE). Our SDE is a superposition of Ornstein-Uhlenbeck processes (supOU
process), generating a sub-exponential autocorrelation function observed in
actual data. The integral operator Riccati equation is heuristically derived to
determine the optimal control of the infinite-dimensional system. In addition,
its finite-dimensional version is derived with a discretized distribution of
the reversion speed and computed by a finite difference scheme. The optimality
of the Riccati equation is analyzed by a verification argument. The supOU
process is parameterized based on the actual data of a perennial river. The
convergence of the numerical scheme is analyzed through computational
experiments. Finally, we demonstrate the application of the proposed model to
realistic problems along with the Kolmogorov backward equation for the
performance evaluation of controls
A Jump Ornstein-Uhlenbeck Bridge Based on Energy-optimal Control and Its Self-exciting Extension
We study a version of the Ornstein-Uhlenbeck bridge driven by a
spectrally-positive subordinator. Our formulation is based on a
Linear-Quadratic control subject to a singular terminal condition. The
Ornstein-Uhlenbeck bridge, we develop, is written as a limit of the obtained
optimally controlled processes, and is shown to admit an explicit expression.
Its extension with self-excitement is also considered. The terminal condition
is confirmed to be satisfied by the obtained process both analytically and
numerically. The methods are also applied to a streamflow regulation problem
using a real-life dataset.Comment: This is a revised versio
- âŠ