Floyd B. Hanson
Professor Emeritus
Laboratory for Advanced Computing
Laboratory for Control and Information
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Annotated Publication Preprints in
Computational Stochastic Dynamic Programming
(Most recent papers with preprint links.)

PSViewer=GSview

Some of our important results concern advanced computation for general
stochastic dynamic programming of the optimal control of continuous time
stochastic differential equations, and much of it has been supported by
a series of National Science Foundation Awards (See acknowledgements
below). Preprints of most recent papers, marked by highlighted titles,
are downloadable by clicking on the highlighted title.
31. F. B. Hanson,
Vector multiprocessor implementation for stochastic dynamic
programming, IEEE Distributed Processing Technical Committee
Newsletter, vol. 10, pp. 4448, 1988.
Brief Abstract:
This paper presents some preliminary optimization results for stochastic
dynamic programming on the Alliant FX/8 vector multiprocessor.
33. F. B. Hanson,
Parallel computation for stochastic dynamic programming:
Row versus column code orientation, Proc. 1988 IEEE Conf.
on Parallel Processing, Vol. III Algorithms and Applications,
pp. 117119, August 1988.
Brief Abstract:
In this paper, presented at the 1988 IEEE Conf. on Parallel Processing, St.
Charles, IL, August 1988, it is demonstrated that column oriented
multiplications for dynamic programming is not the best performer in a Fortran
environment of the Alliant FX/8 vector multiprocessor for a 2 state, 2 control
resource application, due to the nonstandard nature of the multiplications in
the general, variable coefficient, dynamic programming problem.
36. F. B. Hanson,
Stochastic Dynamic Programming: advanced Computing Constructs ,
in Proc. 28th IEEE Conf. on Decision and Control, vol. 1,
pp. 901903, December 1989.
Brief Abstract:
In this paper, presented at the 28th CDC, Tampa, Fl, December 1989, a
vector data structure is introduced that paved the way for general state space
implementation and for facilitating supercomputing optimizations.
37. F. B. Hanson,
Parallel and Vector Computation for Optimal Control Applications ,
in Proc. 3rd Annual Conference on Aerospace Computational
Control, JPL Publication 8545, Vol. 1,
D. E. Bernard and G. K. Man, Editors, Jet Propulsion Laboratory,
Pasadena, pp. 184197, December 1989.
Brief Abstract:
In this paper, presented at the 3rd Annual Conf. Aerospace Computational
Control, Williamsburg, VA, July 1989, the block decomposition of vector data
structure loops is introduced to further enhance performance for 3 state space
resource applications.
40. S.L. Chung and F. B. Hanson,
Parallel Optimizations for Computational Stochastic Dynamic
Programming , in Proc. 1990 Int. Conf. on Parallel Processing,
Vol. III Algorithms and Applications, pp. 254260, August 1990.
Brief Abstract:
In this paper, presented at the 1990 Int. Conf. on Parallel Processing, St.
Charles, IL, Aug. 1990, more performance is attained by modified blocking,
with more work transformed to inner loops and even greater performance by
collapsing loops into vectorconcurrent form.
42. S.L. Chung and F. B. Hanson,
Optimization Techniques for Stochastic Dynamic Programming ,
in Proc. 29th IEEE Conf. on Decision and Control, vol. 4,
pp. 24502455, December 1990.
Brief Abstract:
In this paper was presented at the 29th CDC, Honolulu, HI, December, 1990.
More improvements are obtained by more efficient handling of boundary nodes on
the Cray X/MP/48 vector supercomputer, while use of the massive memory Cray
2S/4128 supercomputer permitted the solution of the 4 state space problem.
44. S.L. Chung, F. B. Hanson and H. Xu,
Supercomputer Optimizations for Stochastic Optimal Control
Applications , in Proc. 4th NASA Workshop on Computational
Control of Flexible Aerospace Systems, NASA Conf. Publ. 10065, Part 1,
L. W. Taylor, Editor, NASA Langley Research Center, pp. 5770, March 1991.
Brief Abstract:
In this paper, presented during July 1990 in Williamsberg, multitasking (Cray
parallelization) is used on the Cray 2S/4128 to gain additional performance
for the 4 state problem, and the massively data parallel Connection Machine
CM2 is applied to show that the performance stochastic dynamic programming
code, in spite of significant conditional dependencies, scales up almost
linearly with the number of state dimensions or mesh points per state.
45. F. B. Hanson,
Computational Dynamic Programming on a Vector Multiprocessor ,
IEEE Trans. Automatic Control, vol. 36(4), pp. 507511, April
1991.
Brief Abstract:
In this paper, presented at the First Int. Conf. on Industrial and Applied
Math., Paris, July 1987, the fundamental numerical method, with its
parallelvector optimizations, is described and high performance is
demonstrated.
46. H. H. Xu, F. B. Hanson and S.L. Chung,
Optimal Data Parallel Methods for Stochastic Dynamic
Programming , in Proc. 1991 Int. Conf. on Parallel Processing,
Vol. III Algorithms and Applications, pp. 142146, August 1991.
Brief Abstract:
In this paper, presented at the 1991 ICPP meeting, operator decomposition
and 1tomany broadcasting techniques were implemented, helping to
significantly improve the mapping of the dynamic programming problem to the
CM2.
47. H. H. Xu, F. B. Hanson and S.L. Chung,
Data Parallel Solutions of Dimensionality Problems in Stochastic
Dynamic Programming , in Proc. 30th IEEE Conf. on Decision and
Control, vol. 2, pp. 17171723, December 1991.
Brief Abstract:
In this invited paper, presented at the 1991 CDC meeting, Brighton, UK,
December 1991. The broadcasting and decomposition data structures are refined
to make additional gains on the Bellman dimensionality problem.
48. F. B. Hanson, D. J. Jarvis and H. H. Xu,
Applications of FORALLFormed Computations in Large Scale
Stochastic Dynamic Programming , Proc. Scalable High Perf.
Comp. Conf.: SHPCC92, IEEE Computer Society, pp. 182185, April 1992.
Brief Abstract:
In this paper, for the 1992 Scalable High Performance Computing Conference,
the definition and structure of forallformed computations is
introduced to promote the transformation of the computations into optimizing
data parallel forall loops.
49. H. H. Xu, D. J. Jarvis and F. B. Hanson,
Parallel Data Vault Methods for Larger Scale Stochastic Dynamic
Programming , in Proc. 1992 Amer. Control Conf.,
vol. 1, pp. 142146, June 1992.
Brief Abstract:
In this invited paper, presented at the 1992 ACC meeting, the preliminary
development using the operator decomposition to map the problem onto the
CM2 mass storage data vaults was discussed, with the projection that
this would enable the computation of six state problems.
50. S.L. Chung, F. B. Hanson and H. H. Xu,
Parallel Stochastic Dynamic Programming: Finite Element Methods
,
Lin. Alg. Applic., vol. 172, pp. 197218, July 1992.
Brief Abstract:
In this paper, presented in an invited talk at the Second NIU Conference on
Linear Algebra, Numerical Linear Algebra and Applications, DeKalb, IL, May
1991, the finite element version of the stochastic dynamic programming code is
developed and the storage efficiency advantages over the finite difference
version are calculated.
51. C. J. Pratico, F. B. Hanson, H. H. Xu, D. J. Jarvis and M. S. Vetter,
Visualization for the Management of Renewable Resources
in an Uncertain Environment , Proc. Supercomputing '92 ,
pp. 258266, color plates p. 843, November 1992.
Brief Abstract:
This paper was presented at the Supercomputing '92 meeting in Minneapolis,
MN, November 1992 with actual workstation demonstration was at SIGGRAPH '92
Showcase. Our multidimensional innerouter world visualization of a three
state resource management application in uncertain environments is described,
along with state and vector control law views, as well as parameter sensitivity
simulated animation.
52. F. B. Hanson, C. J. Pratico, M. S. Vetter, and H. H. Xu,
Multidimensional Visualization Applied to Renewable Resource
Management, Proc. Sixth SIAM Conference on Parallel Processing
for Scientific Computing, vol. 2, pp. 10331036, March 1993.
Brief Abstract:
This paper was presented at the Sixth SIAM Conference on Parallel
Processing for Scientific Computing, Norfork, VA, March 1993, with a more
advanced version presented at the 3rd SIAM Conference on Linear Algebra,
Signals, Systems, and Control, Seattle, WA, August 1993. This paper describes
the port of our graphical visualization system to the NeXT computer and
compares the results to the original Silicon Graphics implementation.
53 [D&C93]. K. Naimipour and F. B. Hanson,
Convergence of a Numerical Method for the Bellman Equation of
Stochastic Optimal Control with Quadratic Costs and Constrained Control ,
Dynamics and Control, vol. 3(3), pp.237259, July 1993.
Brief Abstract:
In this paper, the uniform numerical convergence mesh ratio criteria, that
is so critical to the robust computation with the stochastic dynamic
programming approach, is presented.
54 [IFAC93]. F. B. Hanson and K. Naimipour,
"
Convergence of Numerical Method
for Multistate Stochastic Dynamic Programming ,"
Proc. Int. Federation of Automatic Control 12th World Congress,
vol. 9,
pp. 501504, July 1993.
Brief Abstract:
This invited paper was presented at the International Federation of
Automatic Control 12th World Congress, Sydney, Australia, July 1993. This
presents additional results for the multidimensional stochastic dynamics
programming corrected convergence problem and robust mesh ratio criterion.
56 [CDC94]. F. B. Hanson and J. J. Westman,
"
Computational Stochastic Dynamic
Programming Problems: Groundwater Quality Remediation,
"
in Proc. 33th IEEE Conf. on Decision and Control,
vol. 1, pp. 455460, December 1994.
Brief Abstract:
This paper will be presented at the 33rd CDC in December at Buena Vista,
FL. This is an introduction to the stochastic control model for ground water
modeling using stochastic dynamic programming with both state and space
dependence.
57 [ICCI95]. F. B. Hanson and J. J. Westman,
"
Parallel Stochastic Dynamic
Programming Applications: Spatial and State Finite Elements,
"
Proceedings of
International Conference on Control and Information, The Chinese
University Press, Chinese University of Hong Kong, Sha Tin, Hong Kong,
pp. 145150, June 1995.
Brief Abstract:
This paper covers more of the finite element
techniques used in the computational stochastic dynamic programming for
groundwater applications.
58 [CADS96]. F. B. Hanson,
"
Techniques in Computational Stochastic Dynamic Programming,
"
in Stochastic Digital Control System Techniques,
within series Control and Dynamical Systems: Advances in Theory and
Applications,
vol. 76, (C. T. Leondes, Editor), Academic Press, New York, NY,
pp. 103162, Invited Book Chapter, April 1996.
Brief Abstract:
This invited book chapter summarizes much of the stochastic dynamic
programming accomplishments of my scientific supercomputing group.
Our methods are compared to the Markov Chain Approximation methods of Kushner
and the Differential Dynamic Programming Method for deterministic problems.
(
Click for Technical Abstract and Table of Contents?)
61 [ACC97]. John J. Westman and Floyd B. Hanson,
The LQGP problem: a manufacturing application,
in Proceedings of 1997 American Control Conference,
vol. 1, American Automatic Control Council, pp. 566570, June 1997.
(paper presented in Session WM06 on 04 June 1997; Westman won a Finalist
Award of $700 in the Best Student Paper Competition;
for color copy of preprint or slides
click
here).
Abstract:
The Linear Quadratic Gaussian Poisson (LQGP) problem denotes an optimal
control problem with linear dynamics and quadratic costs with both Gaussian
and Poisson noise disturbances. The LQGP problem provides a benchmark
model with sufficient complexity while permitting formal solutions
for testing both theoretical and computational methods. A numerical example
for a multistage manufacturing system application of the LQGP problem is
presented for the regular or unconstrained control problem.
62 [D&C97]
R. M. AbuSaris and F. B. Hanson,
"
Computational Efficient Suboptimal Filter for a Class of WienerPoisson
Driven Stochastic Processes,
"
Dynamics and Control,
vol. 7, no. 3, pp. 279292, July 1997.
Abstract:
The minimum mean square estimate (MMSE) for a
stochastic process driven simultaneously by Wiener and Poisson processes is
characterized by an infinite number of stochastic differential equations
(even in the simplest linear case), and so is not practically implementable.
In this article, a practical approximation to the solution is developed
in terms of a computationally suboptimal filter for the estimation
problem. Basically, it detects and estimates the Poisson driving process
using a Maximum A Posteriori (MAP) criterion, and then reconstructs the entire
system state using MMSE applied to a system approximating the original one.
65 [SIAMCCIA98poster].
J. J. Westman and Floyd B. Hanson,
"Stochastic Optimal Control for Palliative Treatment of Cancer Using
Chemotherapy and Radiotherepy", awarded the Best Poster Presentation
at the Fourth SIAM Conference on Control and Its Applications, held May 79,
1998 in Jacksonville, Florida; Poster Presentation by Westman.
65 [ACC98].
J. J. Westman and Floyd B. Hanson,
"
The NLQGP Problem: Application to a Multistage Manufacturing System
,"
in Proceedings of 1998 American Control Conference,
American Automatic Control Council, vol. 2, pp. 11041108, June 1998.
(paper presented in Session W08 on 24 June 1998; again Westman won a
Finalist Award of $700 in the Best Student Paper Competition).
Abstract:
The Nonlinear Quadratic Gaussian Poisson (NLQGP) problem denotes an optimal
control problem with nonlinear dynamics and quadratic costs with both Gaussian
and Poisson noise disturbances. The NLQGP problem provides a comprehensive
model for many applications since the noises considered are quite robust
and add extra realism to physical models. The problem is
examined and is illustrated with an application to
a multistage manufacturing system (MMS) in an uncertain environment.
66 [CIMASI98].
Floyd B. Hanson and J. J. Westman,
"
Nonlinear State Dynamics, Computational Approximations and
Multistage Manufacturing System Application ,"
in Proc. Deuxième Conférence Internationale Sur les
Mathématiques Appliquées et les Sciences de l'Ingénieur
(CIMASI'98), Ecole Supérierure de Technologie de Casablanca,
Morocco, vol. 2, pp. 736739, 27 October 1998.
Abstract:
In this paper, we treat stochastic optimal control problems that
are nonlinear in the state, but otherwise are a
LinearQuadraticGaussianPoisson problem in the control (LQGP/U),
such that the dynamics are linear in the control with quadratic
costs in control. The uncertainty in the environment is modeled
by Gaussian noise for continuous background fluctuations and discrete
random jumps by Poisson noise. Computational approximations in
the dynamic programming approach include least squares equivalent
LQGP problems in the state for a state independent approximation
to solutions, modified Gauss quadrature to handle arbitrary jump
distributions, and multidimensional linear interpolation for Poisson
functional terms. The approximations are applied to a multistage
manufacturing system (MMS) in an uncertain environment, along with
implementation procedures needed to modify the formal theory for
hybrid systems.
67 [CDC98]. Daniel L. Kern and Floyd B. Hanson,
"Systematic Perturbations of
DiscreteTime Stochastic Dynamical Systems,"
in Proc. 37th IEEE Conference on Decision and Control,
pp. 18711876, December 1998.
Abstract:
The discretetime stochastic optimal control problem is approximated
by a variation of differential dynamic programming with systematic
calculations of the perturbations due to small stochastic noise.
This problem is related to the dual control aspects of stochastic
optimal control problems. The motivation is to correct prior
calculations for missing terms and to examine the foundations of
the method. The state vector is properly expanded asymptotically,
in addition to the control vector, in contrast to previous solutions.
Corrections are given for the small noise expansions of the solution.
68 [ACC99k].
Daniel L. Kern and Floyd B. Hanson,
"
Filtering Approximation Using Systematic Perturbations of
a DiscreteTime Stochastic Dynamical System,"
Proceedings of 1999 American Control Conference,
Paper no. ACC99IEEE0116, pp. 445449, June 1999.
Abstract:
The standard problem of groundwater pollutant remediation by well
pumping is modeled as a discretetime LQG stochastic optimal control
problem. The control is approximated by using a variation of
differential dynamic programming (DDP) that includes systematic
perturbations. Kalman filtering is used to estimate the partially
observed state variables in a tractable format. This is a filtering
application of the DDP method used by the authors in an earlier
perturbation paper.
69 [ACC99w].
J. J. Westman and Floyd B. Hanson,
"
Computational Method for Nonlinear Stochastic Optimal Control,
"
Proceedings of 1999 American Control Conference,
Paper no. ACC99IEEE0038, pp. 27982802, June 1999.
Abstract:
Nonlinear stochastic optimal control problems are treated that are
nonlinear in the state dynamics, but are linear in the control. The
cost functional is a general function of the state, but the costs are
quadratic in the control. The system is subject random fluctuations
due to discontinuous Poisson noise that depends on both the state and
control, as well as due to discontinuous Gaussian noise. This general
framework provides a comprehensive model for numerous applications that
are subject to random environments. A stochastic dynamic programming
approach is used and the theory for an iterative algorithm is
formulated utilizing a least squares equivalent of a genuine LQGP
problem to approximate the nonlinear state space dependence of the LQGP
problem in control only in order to accelerate the convergence of the
nonlinear control problem. A particular contribution of this paper is
the treatment of a Poisson jump process that is linear in the control
vector, within the context of a nonlinear control problem.
70 [CDC99].
J. J. Westman and Floyd B. Hanson,
"
State Dependent Jump Models in Optimal Control,"
in Proceedings of 38th IEEE Conference on Decision and Control,
pp. 23782384, 07 December 1999.
Abstract:
Models of linear and nonlinear optimal control applications are
considered in which random discrete jumps in the system are state
dependent in both rate and amplitude. These discrete jumps are
treated as a Poisson processes in continuous time. This type of
random noise allows for greater realism while modeling industrial
and natural phenomena in which important changes occur with jumps.
Modeling concerns are described and the appropriate modifications
are indicated for numerically solving the resulting optimal control
problems. Applications to a multistage manufacturing system and
to the management of a natural resource under stochastic price
fluctuations are used to illustrate this type of dynamical formulation.
71 [IJC00].
J. J. Westman
and F. B. Hanson,
"
Nonlinear State Dynamics: Computational Methods and Manufacturing
Application,"
International Journal of Control, pp. 464480, April 2000.
Abstract:
Stochastic optimal control problems are considered that are nonlinear
in the state dynamics, but otherwise are an LQGP problem in the
control, i.e., the dynamics are linear in the control vector and
the costs are quadratic in the control. In addition the system is
randomly perturbed by both continuous Gaussian (G) and discontinuous
Poisson (P) noise. The approach to the solution is by way of
computational stochastic dynamic programming using a least squares
equivalent LQGP problem in the state to accelerate the iterative
convergence. Methods are discussed for numerically handling the
Poisson integral terms. The methods are illustrated for a multistage
manufacturing system (MMS) in an uncertain environment, together
with implementation procedures needed to modify the formal general
theory.
72 [MTNS00].
F. B. Hanson and
J. J. Westman,
"
Computational Stochastic Multistage Manufacturing Systems with Strikes
and Other Adverse Random Events," Proceedings of Mathematical
Theory of Networks and Systems, 2000, 10 pages,
19 June 2000.
Abstract:
Multistage manufacturing systems (MMS) are models for
the assembly of consumable goods. In the simple case, a linear
assembly line of workstations, components, or value, are added to
the product. Some examples assembly line products are automobiles
or printed circuit boards. Production scheduling typically takes
in to account workstation repair, failure, and defective pieces as
stochastic events, effecting the workstation production rates.
The supply routing problem of raw materials is not usually taken
into account. However, in this treatment, the effects of strikes
and natural disasters, which may affect the routing of raw materials,
are considered for the MMS. Numerical results illustrate the
optimal control of MMS undergoing strikes, as well as workstation
repair and failure.
73 [ACC00].
J. J. Westman
and F. B. Hanson,
"
MMS Production Scheduling Subject to Strikes in Random
Environments
," Proceedings of 2000 American Control Conference,
Paper no. ACC00IEEE1327, pp. 21942198, 28 June 2000.
Abstract:
A multistage manufacturing system (MMS) is the normal paradigm used
for the final assembly of a consumable good, for example a car or
printed circuit board. A simple MMS consists normally of a linear
sequence of workstations at which components or value is added to
a product, this is essentially an assembly line. When planning
the production scheduling, normally only workstation repair, failure,
and defective pieces are considered as stochastic events that can
affect the production rates for the various workstations.
Additionally, it is assumed that all of the raw materials necessary
to assemble the finished product are available, and therefore the
supply routing problem of raw materials is not considered. In this
treatment of the problem, we consider the effects of strikes and
of natural disasters. A strike or natural disaster can affect the
MMS itself and/or the way in which raw materials are introduced
into the MMS that is the supply routing problem is treated as well.
A numerical example is presented to illustrate the model that
includes a strike as well as workstation repair and failure.
74 [CCA00].
J. J. Westman
and Floyd B. Hanson,
"
Manufacturing Production Scheduling with Preventive Maintenance
in Random Environments ,"
Proceedings of IEEE International Conference on Control
and Applications, pp. 582587, 25 Septmber 2000.
Abstract:
Consider the optimal control of a manufacturing system consisting
of k stages in which a single consumable good is produced in a
random jump environment. At each stage of the manufacturing process
there are n workstations that can fail and be repaired. The
workstations are assumed to have different operating parameters
for a given stage. The mean time to failure for a given workstation,
on a given stage, is modeled as a function of the uptime of the
workstation. The uptime of the workstations is a monotone increasing
function, which can be reset to a lower level by preventive
maintenance. This formulation combines features of flexible and
multistage manufacturing systems. The goal is to schedule the
production of the consumable good subject to random effects and
preventive maintenance.
75 [ACC01mfg].
J. J. Westman,
F. B. Hanson
and
E. K. Boukas,
"
Optimal Production Scheduling for Manufacturing Systems
with Preventive Maintenance in an Uncertain Environment
,"
Proceedings of 2001 American Control Conference,
pp. 13751380, 25 June 2001.
Abstract:
Consider a manufacturing system in which a single consumable good is
fabricated in a process that consists of $k$ stages in an uncertain
environment. On each stage, there are a number of workstations that
are assumed to have different operating parameters that are subject to
failure, repair, and preventive maintenance which generate discrete
jumps in the value of the state. A JustInTime manufacturing
discipline is assumed for the workstations with running costs that
include penalties for shortfall and surplus production. The
formulation presented here for the optimal production scheduling for
the manufacturing system requires extensions to the results of the LQGP
problem with state dependent Poisson processes (SDPP) by the inclusion
of coefficients for the dynamics and the costs that are parameterized
by the value of the state. The cost functional used is fully quadratic
which is an enhancement for the LQGP problem. The functionality of
this canonical model is demonstrated with a numerical example.
76 [ACC01fin].
F. B. Hanson
and
J. J. Westman,
"
Optimal Consumption and Portfolio Policies
for Important Jump Events: Modeling and Computational Considerations
,"
Proceedings of 2001 American Control Conference,
pp.44564661, 25 June 2001.
Abstract:
While the volatility of portfolios are often modeled by continuous
Brownian motion processes, discontinuous jump processes are more
appropriate for modeling important external events that significantly
affect the prices of financial assets. Here the discontinuous jump
processes are modeled by state and control dependent compound Poisson
processes, such that the random jumps come at the times of a pure
Poisson process with jump amplitudes that are randomly distributed.
The optimal consumption and investment portfolio policy formulation is
in terms of stochastic differential equations with optimal discounted
utility objectives. This paper was motivated by a recent paper of
Rishel (1999) concerning portfolio optimization when prices are
dependent on external events. However, the model has been
significantly generalized for realistic computational considerations.
77 [CDC01].
J. J. Westman,
E. K. Boukas
and
F. B. Hanson,
"
Optimal Production Scheduling for Manufacturing Systems with Preventive
Maintenance and Finite Buffers,"
Proceedings of 2001 Conference on Decision and Control,
pp. 23432348, 04 December 2001.
Abstract:
Consider the production of a single consumable product that is
fabricated in a process of $k$ stages that is subject to an
uncertain environment. There are a number of workstations on
each stage that have different operating parameters. The
workstations are subject to the discrete events of repair,
failure, and preventive maintenance that generate a jump in the
state of the system. Between each stage of the manufacturing
process is a finite buffer that holds pieces before they can
be processed by the next stage. If a buffer is full, then the
preceding stage cannot produce pieces since there will be no
place for them to go. This formulation of a manufacturing system
is a hybrid system consisting of the meeting the global production
demand while locally managing the operational status of the
workstations. This formulation of the optimal scheduling of
production is a quasiLQGP problem whose jumps are generated by
State Dependent Poisson Processes (SDPP). A numerical example is
presented to illustrate the model.
78 [ACC02].
F. B. Hanson
and
J. J. Westman,
"
Optimal Consumption and Portfolio Control
for JumpDiffusion Stock Process with LogNormal Jumps (corrected),
"
Proceedings of 2002 American Control Conference,
pp. 42564261, 8 May 2002.
Abstract:
A jumpdiffusion with lognormal jumps is presented. The distribution
of the changes in the logarithm of the price of the financial
instrument has heavier tails and a higher mode than the usual geometric
Brownian motion. Such a distribution is called leptokurtic and is more
characteristic of real market distributions than the corresponding
lognormal distribution. In addition, this jumpdiffusion return
distribution can be negatively skewed as is real market returns
distribution. The geometric Gaussian mark jumpdiffusion process is
modeled, basic properties are derived and applied to the problem of
portfolio and consumption control optimization.
79 [CMS02].
Daniel L. Kern,
J. J. Westman
and
Floyd B. Hanson,
"
Locally Optimal Pumping and Treatment Rates in Uncertain
Environments
,"
in Fluid Flow and Transport in Porous Media: Mathematical and
Numerical Treatment, AMS Contempory Mathematics, vol. 295,
edited by Zhangxin Chen and Richard E. Ewing,
pp. 305315, July 2002.
Abstract:
Stochastic control theory can be applied to the problem of groundwater
remediation in order to determine optimal pumping and treatment rates
in an uncertain environment. The approach utilizes stochastic dynamic
programming that optimizes a cost functional for treatment subject to
the constraints of the contaminant transport equations via a search
over the statecontrol space. However, this approach for even small
problems requires huge amounts of memory to solve numerically and is
exceptionally prohibitive even using finite elements for the
statecontrol space and time large jumps present. The approach here is
to reduce the computational complexity by casting the contaminant flow
equations to be linear locally while allowing for jumps and small
background fluctuations in the dynamical system. The quasiLQGP
problem can then be used to determine the local optimal rates for
pumping and treating, which does not require prohibitive amounts of
memory. This linearized model can then be used as the starting iterate
to hopefully accelerate the solution to the true nonlinear system for
the contaminant transport equations.
80 [KU02FM].
Floyd B. Hanson and J. J. Westman,
"
Stochastic Analysis of JumpDiffusions for Financial LogReturn
Processes (corrected),"
in Stochastic Theory and Control, Proceedings of a Workshop held in
Lawrence, Kansas, October 1820, 2001, Lecture Notes in Control and
Information Sciences, B.PasikDuncan (Editor), SpringerVerlag,
New York, pp. 169184, 24 July 2002 (invited paper).
Abstract:
A jumpdiffusion logreturn process with lognormal jump amplitudes is
presented. The probability density and other properties of the theoretical
model are rigorously derived.
This theoretical density is fit to empirical logreturns of
Standard & Poor's 500 stock index data.
The model repairs some failures found from the lognormal distribution
of geometric Brownian motion to model features of realistic financial
instruments:
(1) No large jumps or extreme outliers,
(2) Not negatively skewed
such that the negative tail is thicker than
the positive tail, and
(3) Nonleptokurtic due to the lack of thicker tails
and higher mode.
83 [MTNS02FMB].
F. B. Hanson
and
J. J. Westman,
"
Computational Methods for Portfolio and Consumption Optimization
in LogNormal Diffusion, LogUniform Jump Environments,"
Proceedings of the 15th International Symposium on Mathematical
Theory of Networks and Systems, 9 pages, 12 August 2002 (invited paper).
Abstract:
Computational methods for a jumpdiffusion portfolio optimization
application using a loguniform jump distribution are considered.
In contrast to the usual geometric Brownian motion problem based
upon two parameters, mean appreciation and diffusive volatility,
the jumpdiffusion model will have at least five, since jump process
needs at least a rate, a mean and a variance, depending on the
jumpamplitude distribution. As the number number of parameters
increases, the computational complexity of the problem of determining
the parameter set of the underlying model becomes greater. In a
companion stochastic parameter estimation paper, real market data,
here a decade of logreturns for Standard and Poor's 500 index
closings, is used to fit the jumpdiffusion parameters, with
constraints based on matching the data mean and variance to keep
the unconstrained parameter space to 3 dimensions. A weighted
least squares method has been used. The jumpdiffusion theoretical
distribution and weights has been derived. In this computational
paper, the computational features of a new multidimensional,
derivativeless global search method used in the companion paper
are discussed. The main part of this paper is to discuss the
computational solution of an optimal portfolio and consumption
finance application with these more realistic parameter results.
The constant relative risk aversion (CRRA) canonical model is used
to reduce the high dimensionality of the PDE of stochastic dynamic
programming problem to something more reasonable. Many computational
issues arise due to the jump process part of the model, since
several jump integrals arise which are not present in the pure
diffusion with drift model. The loguniformly distributed jumps
allow a wider range of portfolio policies than does previous work
with normally distributed jumps.
84 [MTNS02FMT].
F. B. Hanson
and
J. J. Westman,
"
JumpDiffusion Stock Return Models in Finance:
Stochastic Process Density with UniformJump Amplitude,"
Proceedings of the 15th International Symposium on Mathematical
Theory of Networks and Systems, 7 pages, 12 August 2002 (invited paper).
Abstract:
The stochastic analysis is presented for the parameter estimation
problem for fitting a theoretical jumpdiffusion model to the
logreturns from closing data of the Standard and Poor's 500
(S\&P500) stock index during the prior decade 19922001. The
jumpdiffusion model combines a the usual geometric Brownian motion
for the diffusion and a spacetime Poisson process for the jumps
such that the jump amplitudes are uniformly distributed. The
uniform jump distribution accounts for the rare large outlying
logreturns, both negative and positive in magnitude. The lognormal,
loguniform jumpdiffusion density is derived, leading to a
jumpdiffusion simulator approximation for the case the the logreturn
time is a small fraction of a year. There are five jumpdiffusion
parameters that need to be determined, the means and variances for
both diffusion and jumps, as well as the jump rate, given the
average logreturn time. A weighted least squares is used to fit
the theoretical jumpdiffusion model to the S\&P500 data optimizing
with respect to three free parameters, with the two other parameters
constrained by the mean and variance of the S\&P500 data. The
weight distribution derives from stochastic methods. The ideal
fitted model determines the three free parameters, but the
corresponding simulated results resemble the original S\&P500 data
better. This stochastic analysis paper is a companion to a
computational methods and portfolio optimization paper at this
conference
85 [CDC02].
F. B. Hanson
and
J. J. Westman,
"
Portfolio Optimization with JumpDiffusions:
Estimation of TimeDependent Parameters and Application,"
Proceedings of 2002 Conference on Decision and Control,
pp. 377382, 0913 December 2002, invited paper CDC02INV0302.
Abstract:
This paper treats jumpdiffusion processes in continuous time, with
emphasis on the jumpamplitude distributions, developing more
appropriate models using parameter estimation for the market in one
phase and then applying the resulting model to a stochastic optimal
portfolio application in a second phase. The new developments are
the use of uniform jumpamplitude distributions and timevarying
market parameters, introducing more realism into the application model,
a LogNormalDiffusion, LogUniformJump model.
86 [ACC03].
F. B. Hanson
and
J. J. Westman,
"
JumpDiffusion StockReturn Model with
Weighted Fitting of TimeDependent Parameters,"
Proceedings of 2003 American Control Conference,
pp. 48694874, 04 June 2003.
Abstract:
This paper treats jumpdiffusion processes in continuous time, with
emphasis on the jumpamplitude distributions, developing more
appropriate models using parameter estimation for the market.
The proposed method of parameter estimation is weighted least squares
of the difference between theoretical and experimental bin frequencies,
where the weights or reciprocal variances are chosen as by the
theory of jumpdiffusion simulation applied to bin frequencies.
The empirical data is taken from a decade of
Standard & Poor 500 index of stock closings and are viewed
as one moderately large simulation.
The new developments are the combined use of uniform jumpamplitude
distributions, least squares weights and timevarying market parameters,
introducing more realism into the model,
a LogNormalDiffusion, LogUniformJump financial market model.
The optimal parameter estimation is highly nonlinear, computationally
intensive, and the optimization is with respect to the three parameters
of the loguniform jump distribution, while the diffusion parameters
are constrained by the first two moments of the Standard & Poor's data.
88 [CDC03].
Floyd B. Hanson,
Computational Stochastic Control: Basic Foundations, Complexity and
Techniques,
Proceedings of 2003 Conference on Decision and Control,
Invited Poster/Interactive Paper in a Control Education
Session, pp. 30243029, December 2003.
Longer 9 page Submitted Version.
Abstract:
Much research in control systems is purely mathematical, but
advances in stochastic control problem solving can be used
beyond the limits of where theoretical mathematics can help.
Theoretical and computational mathematics are complementary.
Computation is important where the problem
is mathematically intractable, of high dimension as in stochastic
dynamic programming or solving the problem is urgent as in
competitive financial engineering predictions. Many advances in
solving large scale control problems have been gained through
technical improvements in computing hardware, but as many advances
have been made in the development of new and better algorithms,
the theoretical side of computation. Both analysis and
computation are important in solving problems. Both rely on
mathematics, but rely on them in different ways. An important
part of educational training is general preparation for problem solving
since the postgraduate job is uncertain in the current
world.
In this expository paper, a selection of basic computational
considerations, high performance computers and some useful algorithms
are surveyed. Some of the computational methodology in both
algorithms and advanced computers arose from the author's own
research. Much of the knowledge has been transferred to classes
in computation and control, so that student instruction is at the
leading edge, a buffer against obsolescence. An important general
lesson in computational education is that the computation, if done
properly, forms the other half of mathematics, beyond the topics
of regular or traditional mathematics courses. Computation has its
own algebra, oriented to finite precision arithmetic, and its own
analysis that is numerically oriented. The view is that of an
applied analyst and computational control scientist explaining the
field to those with a regular mathematics background.
89 [TAC04].
Floyd B. Hanson and John J. Westman,
"Optimal Portfolio and Consumption Policies
Subject to Rishel's Important Jump Events Model:
Computational Methods,"
Trans. Automatic Control,
vol. 49, no. 3, Special Issue on Stochastic Control Methods in
Financial Engineering, pp. 326337, March 2004.
Abstract:
At important events or announcements, there can be large changes
in the value of financial portfolios. Events and their corresponding
jumps can occur at random or scheduled times. However, the amplitude
of the response in either case can be unpredictable or random.
While the volatility of portfolios are often modeled by continuous
Brownian motion processes, discontinuous jump processes are more
appropriate for modeling the response to important external events
that significantly affect the prices of financial assets.
Discontinuous jump processes are modeled by compound Poisson
processes for random events or by quasideterministic jump processes
for scheduled events. In both cases, the responses are randomly
distributed and are modeled in a stochastic differential equation
formulation. The objective is the maximal, expected total discounted
utility of terminal wealth and instantaneous consumption. This paper
was motivated by a paper of Rishel (1999) concerning portfolio
optimization when prices are dependent on external events. However,
the model has been significantly generalized for more realistic
computational considerations with constraints and parameter values.
The problem is illustrated for a canonical riskadverse power
utility model, but the usual explicit canonical solution is not
strictly valid. Fortunately, iterations about the canonical solution
result in computationally feasible approximations.
90 [CSM04]. Molly H. Shor and Floyd B. Hanson,
"
Bringing Control to Students and Teachers,"
Control Systems Magazine, vol. 24, no. 3, pp. 2030, June 2004.
Abstract:
CDC03 Conference Report on ``Ideas and Technology of Control Systems:
NSF Workshop for Middle School and High School Students and Teachers''
in Maui, Hawaii in December 2003. See CSM issue for figures and
sidebars.
91 [SB03/CM04].
Floyd B. Hanson, John J. Westman and Zongwu Zhu,
"
Maximum Multinomial Likelihood Estimation of Market Parameters
for Stock JumpDiffusion Models,"
in Proc. 2003 AMS/IMS/SIAM Summer
Research
Conference on Mathematics of Finance, AMS Contemporary Mathematics,
pp.~115, December 2003, invited paper to appear 2004.
Abstract:
The estimated parameters of the logreturn density for
lognormaldiffusion,
loguniform jump process are found for an observed financial market
distribution. When the observed data is collected into bins, it
is shown that the appropriate parameter estimation method is the
multinomial maximum likelihood estimation. This result is independent
of the theoretical distribution, since it is only assumed that the
observed distribution is the simulation of independent, identically
distributed random variables. For the application to the theoretical
jumpdiffusion distribution, the estimation procedure is constrained
by forcing the first two moments of the theoretical distribution to be
the same as that for observed market distribution. The Standard and
Poor's
500 stock index for the 19922001 decade is used as the observed market
data.
Numerically, the classical NelderMead and our own direct search method
are used to find the maximum likelihood estimation of the parameters.
The results and performance of these numerical methods are compared
along
with the our older weighted least squares estimation method. The
results
of the two numerical approximations for the multinomial estimation
methods were similar, but the weighted least squares results are not as
good.
In the severe test on the third and fourth moment measures, the
multinomial
based methods differed significantly from the same measures on the
observed data, but did much better than the normal distribution
based weighted least squares.
92 [CDC04].
Floyd B. Hanson
and
Zongwu Zhu,
"
Comparison of Market Parameters for JumpDiffusion Distributions
Using Multinomial Maximum Likelihood Estimation,"
Proceedings of 43th IEEE Conference on Decision
and Control, pp. 16, invited paper, 13 December 2004.
Abstract:
Previously, we have shown that the proper method for estimating parameters
from discrete, binned stock log returns is the multinomial maximum
likelihood estimation, and its performance is superior to the method of
least squares. Useful formulas have been derived for the jumpdiffusion
distributions. Numerically, the parameter estimation can be a large
scale nonlinear optimization, but we have used techniques to reduce the
computation demands of multidimensional direct search. In this paper,
three jumpdiffusion models using different jumpamplitude distributions
are compared. These are the normal, uniform and doubleexponential.
The parameters of all three models are fit to the Standard and Poor's 500
logreturn market data, constrained by the data first and second moments.
While the results for the skew and kurtosis moments are mixed, the
uniform jump distribution has superior qualitative performance since it
produces genuine fat tails that are typical of market data, whereas the
others have exponentially thin tails. However, the lognormal model has
a big advantage in computational time of parameter estimation compared
with the others, while the doubleexponential is most costly due to
having one more model parameter.
93 [ACC05]
Siddhartha P. Chakrabarty and Floyd B. Hanson,
"
Optimal Control of Drug Delivery to Brain Tumors for a
Distributed Parameters Model,"
Proceedings of 2005 American Control Conference,
pp. 973978, June 2004.
Abstract:
The growth and treatment of brain tumors is mathematically examined
using a distributed parameters model. The model is a system of three
coupled reaction diffusion equations involving the tumor cells,
normal tissue and the drug concentration. An optimal control problem
is designed, with the drug delivery rate as the control and solved
to obtain the state and costate equations as well as the regular
control using a modified double shot forwardbackward method.
This gives rise to a
coupled system of equations with a forward state equation and a
backward costate equation, which is solved using a double shot method.
A numerical procedure based upon the CrankNicolson method is
used to solve the coupled system of two threedimension partial
differential equations.
94 [CDC05hc]
Floyd B. Hanson and Siddhartha P. Chakrabarty,
"
Optimal Control of Drug Delivery to Brain Tumors for a
PDE Driven Model Using the Galerkin Finite Element Method
,"
Proceedings of Joint 44th IEEE Conference on Decision
and Control and European Control Confence,
pp. 16131618, 12 December 2005.
Abstract:
The Galerkin finite element method is used to examine the optimal
drug delivery to brain tumors. The PDE driven mathematical model is
a system of three coupled reaction diffusion equations involving the
tumor cells, the normal tissue and the drug concentration. An optimal
control problem is formulated keeping in mind the primary goals of the
treatment, i.e., minimizing the tumor cell density and reducing the side
effects of drugs. A distributed parameter method based on application
of variational calculus to a pseudoHamiltonian, is used to obtain a
coupled system of forward state equations and backward costate
equations.
The Galerkin form of the finite element method is used due to its
greater
facility in numerically representing complex structures such as those
in the brain. Finally, a twodimensional circular disk test case is
considered and partitioned into a set of rectangular finite elements in
polar coordinates, with bilinear basis functions in the interior, but
linearquadratic basis function for elements adjacent to the boundary
to exactly satisfy the noflux boundary conditions.
95 [CDC05zh].
Zongwu Zhu
and
Floyd B. Hanson,
"
A MonteCarlo OptionPricing Algorithm for LogUniform JumpDiffusion
Model
,
Proceedings of Joint 44th IEEE Conference on Decision
and Control and European Control Conference,
pp. 52215226, 12 December 2005.
Abstract:
A reduced European call option pricing formula by riskneutral
valuation is given. It is shown that the European call and put
options for jumpdiffusion models are worth more than that for the
BlackScholes (diffusion) model with the common parameters. Due to
the complexity of the jumpdiffusion models, obtaining a closed option
pricing formula like that of BlackScholes is not viable. Instead,
a Monte Carlo algorithm is used to compute European option prices.
Monte Carlo variance reduction techniques such as both antithetic and
control variates are used. The numerical results show that this is a
practical, efficient and easily implementable algorithm.
97 [Sethi06]
Zongwu Zhu and Floyd B. Hanson,
"
Optimal Portfolio Application with DoubleUniform Jump
Model,"
Stochastic Processes, Optimization, and Control Theory:Applications
in Financial Engineering, Queueing Networks
and Manufacturing Systems/A Volume in Honor of Suresh Sethi,
International Series in Operations Research & Management Sciences,
Vol. 94, H. Yan, G. Yin, Q. Zhang (Eds.), Springer Verlag, New York,
pp. 331358, Invited chapter, June 2006.
Abstract:
This paper treats jumpdiffusion processes in continuous time,
with emphasis on the jumpamplitude distributions, developing more
appropriate models using parameter estimation for the market in one
phase and then applying the resulting model to a stochastic optimal
portfolio application in a second phase. The new developments are the
use of doubleuniform jumpamplitude distributions and timevarying
market parameters, introducing more realism into the application model
 a lognormal diffusion, logdoubleuniform jumpamplitude model.
Although unlimited borrowing and shortselling play an important role
in pure diffusion models, it is shown that borrowing and shorting is
limited for jumpdiffusions, but finite jumpamplitude models can
allow very large limits in contrast to infinite range models which
severely restrict the instant stock fraction to [0,1]. Among all the
timedependent parameters modeled, it appears that the interest and
discount rate have the strongest effects.
98 [CDC06]
Siddhartha P. Chakrabarty
and
Floyd B. Hanson,
"
Cancer Drug Delivery in Three Dimensions For a
Distributed Parameter Control Model Using Finite Elements
,"
Proceedings of 45th Conference on Decision and Control
, pp.~16, San Diego, CA, 1315 December 2006.
Abstract:
Optimal distributed parameter control and a Galerkin finite element
method are used to develop procedures for an ideal model of optimal drug
delivery to brain tumors. The mathematical model comprises of a system
of three coupled reaction diffusion models, involving the density of
tumor cells, normal tissue and also the drug concentration. An optimal
control problem is formulated with the goal of minimizing the tumor cell
density and reducing the side effects of the drug. A distributed
parameter method based on a simple application of variational calculus
is used on a pseudoHamiltonian, which is then used to obtain a coupled
system of forward state equations and backward costate equations. The
Galerkin finite element method is used to realistically represent the
brain structure. Finally, an ideal three dimensional test case is
considered and partitioned into a set of brick finite elements in
spherical coordinates, with trilinear basis functions. Nonuniqueness
of nodes in spherical coordinates is removed by combining like nodes,
such as at the origin, at the poles and at the polar angle
discontinuity. The Galerkin ODEs are solved by a combination of
CrankNicolson and predictorcorrector methods.
100 [SIAMbook].
Floyd B. Hanson,
Applied Stochastic Processes and Control for JumpDiffusions:
Modeling, Analysis and Computation,
SIAM Books: Advances in Design and Control Series, Order Code DC13,
28 + 441 pages, published 03 October 2007.
101 [MBS09].
Siddhartha P. Chakrabarty
and
Floyd B. Hanson,
"Distributed Parameters Deterministic Model for Treatment of
Brain Tumors Using Galerkin Finite Element Method
,"
Mathematical Biosciences, pp. 154,
accepted for publication, 27 March 2009.
Abstract:
In this paper we present a distributed parameters deterministic model
for
treatment of brain tumors using Galerkin finite element method. The
dynamic
model comprises of system of three coupled reaction diffusion models,
involving the tumor cells, the normal tissues and the drug
concentration.
An optimal control problem is formulated
with the goal of minimizing the tumor cell density and reducing the
side effects of the drug. A distributed parameters method based on
the application of variational calculus is used on an
integralHamiltonian,
which is then used to obtain an optimal
coupled system of forward state equations
and backward costate equations. The Galerkin finite element method is
used to
realistically represent the brain structure as well as to facilitate
computation. Finally a three dimensional test case
is considered and partitioned into a set of
spherical finite elements, using trilinear basis functions, except for
the
elements effected by singularities of polar and azimuthal angles, as
well the
origin.
[WEORMS10 (Encylopedia Chapter)]
Floyd B. Hanson,
Computation in Dynamic Programming,
in Wiley Encyclopedia of Operations Research and Management
Sciencee, edited by James J. Cochran, John Wiley & Sons, Inc.,
New York, NY, 7 pages, In Press, 27 May 2010.
Abstract:
The presentation in this chapter is in the formal manner of classical
applied mathematics and probability
in order to focus on the methods and their implementation.
In Section 1, a fairly general model of
stochastic dynamic programming in continuous time is outlined.
In Section 2, canonical forms, such as a linear dynamics and quadratic
cost model in control, that lead to a large reduction in computational
effort are given or discussed.
In Section 3, finite difference partial differential equation methods
are given that are suitable for approximately solving
nonlinear Bellman dynamic programming equations.
Alternatively, Markov chain approximation probabilistic methods which
systematically justify the stability and weak convergence of the
approximating Markov chain are summarized in Section 4.
In the last section, there is a brief summary and directions to some
other approaches.
103 [IISC07].
Floyd B. Hanson,
"Stochastic Processes and Control for
JumpDiffusions,"
under revision, 44 pages,
22 October 2007.
Abstract:
An applied compact introductory survey of Markov stochastic processes and
control in continuous time is presented. The presentation is in tutorial
stages, beginning with deterministic dynamical systems for contrast
and continuing on to perturbing the deterministic model with diffusions
using Wiener processes. Then jump perturbations are added using simple
Poisson processes constructing the theory of simple jumpdiffusions.
Next, markedjumpdiffusions are treated using compound Poisson processes
to include random marked jumpamplitudes in parallel with the equivalent
Poisson random measure formulation. Otherwise, the approach is quite
applied, using basic principles with no abstractions beyond Poisson
random measure. This treatment is suitable for those in classical applied
mathematics, physical sciences, quantitative finance and engineering,
but have trouble getting started with the abstract measuretheoretic
literature. The approach here builds upon the treatment of continuous
functions in the regular calculus and associated ordinary differential
equations by adding nonsmooth and jump discontinuities to the model.
Finally, the stochastic optimal control of markedjumpdiffusions is
developed, emphasizing the underlying assumptions. The survey concludes
with applications in biology and finance, some of which are canonical,
dimension reducible problems and others are genuine nonlinear problems.
104 [BFS08].
Floyd B. Hanson,
"Optimal Portfolio Problem for StochasticVolatility,
JumpDiffusion Models with JumpBankruptcy Condition:
Practical Theory and Computation,"
Fifth World Congress of Bachelier Finance Society, 2008,
27 pages, revised 11 July 2008.
Abstract:
This paper treats the riskaverse optimal portfolio problem
with consumption in continuous time with a stochasticvolatility,
jumpdiffusion (SVJD) model of the underlying risky asset and the
volatility.
The new developments are the use of the SVJD model with
doubleuniform jumpamplitude distributions and
timevarying market parameters for the optimal portfolio problem.
Although unlimited borrowing and shortselling play an important
role in pure diffusion models, it is shown that borrowing and
shortselling are constrained for jumpdiffusions.
Finite range jumpamplitude models can allow constraints to be very
large in contrast to infinite range models which severely
restrict the optimal instantaneous stockfraction to [0,1].
The reasonable constraints in the optimal stockfraction
due to jumps in the wealth argument for stochastic dynamic
programming jump integrals remove a singularity in the
stockfraction due to vanishing volatility.
Main modifications for the usual constant relative risk aversion (CRRA)
power utility model are for handling the partial integrodifferential
equation (PIDE) resulting from the additional variance independent
variable, instead of the ordinary integrodifferential equation (OIDE)
found for the pure jumpdiffusion model of the wealth process.
In addition to natural constraints due to jumps when enforcing
the positivity of wealth condition, other constraints are considered
for all practical purposes under finite market conditions.
Computational, result are presented for optimal portfolio values,
stock fraction and consumption policies.
Also, a computationally practical solution of Heston's (1993)
squarerootdiffusion model for the underlying asset variance
is derived.
This shows that the non\negativity of the variance
is preserved through the proper singular limit of a simple
perfectsquare form.
An exact, non\singular solution is found for a special combination of
the
Heston stochastic volatility parameters.
Summary
In summary, we have made significant advances in alleviating Bellman's
curse of dimensionality for stochastic dynamic programming
applications through the use of massive memories, massively parallel
processors, optimal data structures, data parallel methods and scientific
visualization. In addition, we have produced solutions to many applications
both computationally and theoretically.
Acknowledgements

This material is based upon work supported by awards from the National Science
Foundation under Grants Numbers NSF 880699, 9102343, 9301107, 9626692, 9973231,
0207081.

Supercomputing research support was provided by
 San Diego Supercomputing Center NPACI Account UIL203;

National Center for Supercomputing Applications
Grant Numbers DCR860001N, DMS890009N, DMS900016N, DMS920003N,
DMS960002N;
 Pittsburgh Supercomputing Center Grant Numbers DMS940001P and DMS9400011P;
 Argonne National Laboratory's Advanced Computing Research Facility;
 Los Alamos National Laboratory's Advanced Computing Laboratory
ACLAccounts Z803835 and gwqfbhd.

Any opinions, findings, and conclusions or recommendations expressed in
this material are those of the author(s) and do not necessarily reflect
the views of the National Science Foundation or other agency.
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