Biography – Mathematics
Math Ph.D. • Bell Labs • Courant Institute (NYU) • Duke University • Two Books
Matthias Heymann completed his undergraduate studies in mathematics at the Leibniz University in Hanover, Germany, with a thesis on the topic of fractional powers of operators (functional analysis).
After a one-year internship in the statistics department of Bell Labs (New Jersey, USA), he then completed a Ph.D. program in mathematics at the Courant Institute (New York University, USA), where he focused on both pure and applied aspects of probability theory. In his Ph.D. thesis (in the field of Wentzell–Freidlin theory, a subfield of large deviation theory), he developed a new algorithm for computing the maximum likelihood transition curve between two metastable states in dynamical systems with small noise.
He then accepted a three-year position as an Assistant Research Professor in the Duke University Mathematics Department. His research during this time included both the application of his thesis work to a variety of applied problems, and the development of mathematical criteria under which a maximum likelihood transition curve exists. The latter was published as a 186-page monograph in the Springer Lecture Notes in Mathematics titled “Minimum Action Curves in Degenerate Finsler Metrics“.
After his departure from academia into the world of finance, he continued his mathematical research in the field of interest rate modeling. His book on “The Adaptive Curve Evolution Model for Interest and FX Rates” kept evolving over the years and is now in its third edition (see also the Finance page for details).
Teaching Experience: He has taught a variety of undergraduate and graduate courses at NYU and Duke University, including: Precalculus, Calculus I & II, Linear Algebra, Business Calculus, Scientific Computing, Probability Theory, Ordinary Differential Equations, Complex Analysis (graduate level), and Probability Theory (graduate level).
Publications & Papers
(Turn phone sideways for details)
Year
Title & Info
Fields
Links
2016–24
The Adaptive Curve Evolution Model for Interest & FX Rates
book (sole author), 362 pages, self-published on Amazon
Math. Finance
The ACE model — in its original form developed by Gregory Pelts and now carefully rephrased, refined, extended, and made more accessible by Matthias Heymann in the present book — is the first to combine all of the most desirable analytical properties in one interest rate model.
It is low-dimensional (with the dimension n ∈ 
\ {2} of its state space coinciding with the one of the driving Brownian motion), complete (i.e., it models all tenors), arbitrage-free, highly flexible (it provides 2n+1 discrete parameters in addition to the functional noise parameter σ(x,t)), and time homogeneous if desired, and it imposes a lower bound on rates. Moreover, it has the rare feat of being unspanned (i.e., its bond price function does not depend on σ), which can increase calibration leverage, and which allows the yield curve calibration to be separated from the calibration to caps, swaptions, and other interest rate derivatives.
In addition, we present an extension of the ACE model that covers the interest rates and bond prices of multiple currencies, along with each currency pair’s spot and forward exchange rates.
The book begins with an introduction that compiles a list of all of our desired model features, and that provides a detailed comparison with existing models. Part I of this book (“A Fast Track To ACE”), which is tailored to the reader who merely wishes to understand the ACE model well enough to use it in practice, then lays out all of the results in an easily understandable 
-based formulation, along with some straightforward proofs that require only standard knowledge in analysis, stochastic processes, and mathematical finance. Finally, Part II contains both a new quick derivation of the model equations, and separately the original derivation utilizing a variety of compelling non-standard mathematical techniques — carefully introduced along the way — that may well hold the key also to other financial modeling problems.
See the Finance page for further details.
2015
Minimum Action Curves in Degenerate Finsler Metrics – Existence & Properties
book (sole author), 186 pages, Springer Lecture Notes in Mathematics
Geometry, Probability
Presenting a study of geometric action functionals (i.e., non-negative functionals on the space of unparameterized oriented rectifiable curves), this monograph focuses on the subclass of those functionals whose local action is a degenerate type of Finsler metric that may vanish in certain directions, allowing for curves with positive Euclidean length but with zero action. For such functionals, criteria are developed under which there exists a minimum action curve leading from one given set to another. Then the properties of this curve are studied, and the non-existence of minimizers is established in some settings.
Applied to a geometric reformulation of the quasipotential of Wentzell-Freidlin theory (a subfield of large deviation theory), these results can yield the existence and properties of maximum likelihood transition curves between two metastable states in a stochastic process with small noise.
The book assumes only standard knowledge in graduate-level analysis; all higher-level mathematical concepts are introduced along the way.
2010
Rare Transition Events in Nonequilibrium Systems with State-Dependent Noise: Application to Stochastic Current Switching in Semiconductor Superlattices
with S. Teitsworth and J. Mattingly, unpublished
Theo. Physics, Probability
Using recent mathematical advances, a geometric approach to rare noise-driven transition events in nonequilibrium systems is given, and an algorithm for computing the maximum likelihood transition curve is generalized to the case of state-dependent noise. It is applied to a model of electronic transport in semiconductor superlattices to investigate transitions between metastable electric field distributions. When the applied voltage V is varied near a saddle-node bifurcation at Vth , the mean lifetime ⟨T⟩ of the initial metastable state is shown to scale like log⟨T⟩ ~ |V – Vth|3/2 as V → Vth .
2010
Finding the Sources of Rare Transitions in Continuous-Time Markov Jump Processes
working paper
Probability
For continuous-time Markov jump processes, maximum likelihood paths of rare transitions from one metastable state to another have long been well-understood analytically [1] and can now also be computed efficiently numerically [2]. To experimenalists however who design reaction networks (such as chemical or cell biological systems) and who want to have control over such (wanted or unwanted) switching events, the knowledge of the transition path (the observables during the transition) is not enough: They need to know what is the weak link in the network whose “failure” (atypical behavior) is most likely the reason for the “failure” of the entire system.
In this paper we introduce the concept of maximum likelihood reaction frequencies (MLRFs) which, when compared to the rate functions of the jump process, are the right statistics to look at. They also provide a practical alternative for visualizing transitions paths in higher-dimensional spaces. We prove an analytic formula for the MLRFs and illustrate the concept on an example from synthetic biology, the genetic switch.
2008
The Geometric Minimum Action Method for Computing Minimum Energy Paths
with E. Vanden-Eijnden, Jour. Chem. Phys. 128, 061103
Probability, Chem. Physics
This paper introduces the gMAM algorithm developed in my Ph.D. thesis to the chemistry community.
An algorithm is proposed to calculate the minimum energy path MEP. The algorithm is based on a variational formulation in which the MEP is characterized as the curve minimizing a certain functional. The algorithm performs this minimization using a preconditioned steepest-descent scheme with a reparametrization step to enforce a constraint on the curve parametrization.
2008
The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves
with E. Vanden-Eijnden, Comm. Pure Appl. Math. 61.8, 1052-1117
Probability, Applied Math.
This publication contains the key results of my Ph.D. thesis.
Freidlin–Wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool. Specifically, a numerical algorithm is proposed to compute the quasipotential in the theory, which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous: the equilibrium distribution of the system, the pathways of transition between metastable states and their rate, etc. can all be expressed in terms of the quasipotential. We propose an algorithm to compute these quantities called the geometric minimum action method (gMAM) which is a blend of the original minimum action method (MAM) and the string method. It is based on a reformulation of the large deviations action functional on the space of curves which allows one to easily perform the double minimization of the original action required to compute the quasipotential. The theoretical background of the gMAM in the context of large deviations theory is discussed in detail, as well as the algorithmic aspects of the method. The gMAM is then illustrated on several examples: a finite-dimensional system displaying bistability and modeled by a non-gradient stochastic ordinary differential equation; an infinite-dimensional analogue of this system modeled by a stochastic partial differential equation; and an example of a bistable genetic switch modeled by a Markov jump process.
2007
Pathways of Maximum Likelihood for Rare Events in Nonequilibrium Systems, Application to Nucleation in the Presence of Shear
with E. Vanden-Eijnden, Phys. Rev. Letters 100.14, 140601, 2007
Probability, Applied Math.
This paper introduces the gMAM algorithm developed in my Ph.D. thesis to the physics community.
Even in nonequilibrium systems, the mechanism of rare reactive events caused by small random noise is predictable because they occur with high probability via their maximum likelihood path (MLP). Here a geometric characterization of the MLP is given as the curve minimizing a certain functional under suitable constraints. A general purpose algorithm is also proposed to compute the MLP. This algorithm is applied to predict the pathway of transition in a bistable stochastic reaction-diffusion equation in the presence of a shear flow, and to analyze how the shear intensity influences the mechanism and rate of the transition.
2007
The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves
Ph.D. thesis, advisor: E. Vanden-Eijnden
Probability, Applied Math.
Dynamical systems with small noise (e.g., SDEs or continuous-time Markov chains) allow for rare events that would not be possible without the presence of noise, e.g., for transitions from one stable state into another. Large deviations theory provides the means to analyze both the frequency of these transitions and the maximum likelihood transition path. The key object for the determination of both is the quasipotential,
where 
is the action functional associated to the system, and where the infimum is taken over all 
and all paths 
leading from 
to 
. The numerical evaluation of 
, however, is made difficult by the fact that in most cases of interest no minimizer exists.
Here, this issue is resolved by introducing the action 
on the space of curves (i.e., 
is independent of the parametrization of 
) and proving the alternative geometric formulation of the quasipotential
where the infimum is taken over all curves 
leading from to . In this formulation a minimizer exists, and we use this formulation to build a flexible algorithm (the geometric minimum action method, gMAM) for finding the maximum likelihood transition curve.
We demonstrate on several examples that the gMAM performs well for SDEs, SPDEs and continuous-time Markov chains, and we show how the gMAM can be adjusted to solve also minimization problems with endpoint constraints or endpoint penalties.
Finally, we apply the gMAM to problems from mathematical finance (the valuation of European options) and synthetic biology (the design of reliable standard genetic parts). For the latter, we develop a new tool to identify sources of instability in (genetic) networks that are modelled by continuous-time Markov chains.
2002
A Model for the Online Time of Network Users
with M. Hansen, working paper
Statistics
A model is proposed to describe individual behavior of internet users, and using a maximum likelihood method, the model parameters are fitted to a dataset consisting of two million connections to the Bell Labs network. The model can then be used to generate artificial internet user traffic that simulates human behavior, for example to test new designs of server mechanisms.
2002
A New Set of Sound Commands for R; Sonification of the HMC Algorithm
with M. Hansen, ASA Proceedings 2002, Statistical Computing Section, 1439–1443
Statistics
In this paper, which accompanied my talk at the ASA conference 2002, I present the R sound package that I had developed at Bell Labs to the public:
We describe a new set of commands for the programming language R that we developed to work with sound samples and wav files. The combination of these tools with the large number of statistical commands in the standard R package is an ideal environment for our efforts to sonify datasets.
As an example, we show how we use these commands to tune a parameter in the Hybrid Monte Carlo algorithm. Musicians will also enjoy the great possibilities for creating new sounds and sound effects on a very basic level and without any restrictions.
2002
The R Sound Package: A New Set of Commands for Using Sound in R
with M. Hansen and D. James, code package on the official R project website (CRAN)
Statistics, Comp. Science
The statistical programming and data analysis language R, the popular freeware clone of the commercial language SPLUS, is by itself not suited for using sound files. With this package I provide the R community with a set of additional commands that enables them do load, save, listen to, and manipulate wave files in a variety of ways. The package can be downloaded from CRAN, the official website of the the R development team.
2002
The Stieltjes Convolution and a Functional Calculus for Non-Negative Operators
working paper
Functional Analysis
In this paper we present an approach to the multidimensional distributional Stieltjes transform that allows us to define a convolution operation on our classes of Stieltjes-transformable distributions. As an application, we develop a powerful and intuitive functional calculus for non-negative operators with which one can easily prove a variety of operator equations, even for non-commuting operators and on non-complex Banach spaces. Two representation theorems that identify our classes of distributions as finite sums of derivatives of functions are essential throughout this paper.
2001
Fractional Powers of Operators, and Their Applications
diploma thesis, advisor: U. Schmidt–Westphal
Functional Analysis
In this thesis we develop a powerful operational calculus for operators of class 
, the set of all closed operators 
(where 
is a Banach space fulfilling 
and 
(then 
is called non-negative). With this, operator equations of the form
can be deduced from the corresponding equation related to the multiplication operators 
, without any restrictive conditions, so that only a scalar equation has to be proven.
In the case that 
is densely defined, even equations of the form
(Swipe left on the formula to see the rest.)
can be treated. Here we have 
, for 
, 
, 
a complex Borel measure with 
and in the second case additionally 
and 
as well as 
and 
for 
.
With this operational calculus we extend results of Hövel and Westphal [3] on fractional powers 
, 
, of densely defined operators of class . We provide a complete characterization for all values of 
and demonstrate that one of the basic methods of [3] could not be applied to derive results for 
.
Finally, we illustrate how this operational calculus can be used to generalize results from the literature. We take as our example the characterisation of convergency rates in a generalized Abelian mean ergodic theorem for semigroup generators presented in [9]. We apply the operational calculus to extend this work to all densely defined operators of class .