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The Legacy of Ingenhousz
Notes on some Mathematical Translations of
Uncertainty in Living Sciences
Jean-Pierre Aubin
21 juillet 2012
The story started 1 in 1785 when Jan Ingenhousz, a Dutch physiologist, biologist and
chemist, discovered what was not called the Ingenhouszian movement, but the better known
Brownian movement, rediscovered by the botanist Robert Brown in 1827, however much less
known than pedesis (from Greek “leaping”).
Jan Ingenhousz 2 described the irregular motion of coal dust particles on the surface of
alcohol, randomly zigzagging as anyone would do in such conditions. He could not forecast
that, centuries later, his discovery would trigger, in part, the development of stochastic
differential equations ! Quoting him in the title of these notes is an hommage and a way to
revive his memory.
A long list of physicists and mathematicians, Pierre de Fermat, Blaise Pascal, Sadi Carnot, Rudolf Clausius, James Maxwell, Ludwig Boltzmann, Thorvald Thiele, Louis Bachelier,
Albert Einstein, Paul Langevin, Henri Lebesgue, René Gâteaux, Norbert Wiener, Paul Lévy,
Andreı̈ Kolmogorov, Joseph Doob, Viktor Maslov, Ruslan Stratonovitch, Wolfgang Döblin,
Kiyoshi Ito, among so many others, devised mathematical metaphors of “uncertainty” motivated by parlor games, thermodynamics and physical problems. However, they all followed
same directions during the xxth century, involving probabilities and stochastic dynamics.
It became “THE” quasi unique mathematical framework to translate mathematically the
concept of uncertainty, and “applied” in almost all fields. From physics, the area where it
originated, through finance, thanks to the staggering mathematical contribution of Louis
Bachelier in 1900, to living sciences.
However, are living beings 3 behaving like dust particles in an inebriating environment ?
Asking the question is answering it : we suggest that the stochastic translation of uncertainty is not always relevant for living systems, and we may attempt to find other ways to
capture the diverse aspects of uncertainty in mathematical terms.
1. Actually, when the Epicurean Lucretius observed “what happens when sunbeams are admitted into
a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a
multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden
from our sight” in De rerum natura.
2. Who also discovered photosynthesis and cellular respiration.
3. and, among them, human beings and speculators : Of all creatures man is the most miserable and
fraile, and therewithall the proudest and disdainfullest [...] and yet dareth imaginarily place himself above the
circle of the moon, and reduce heaven under his feet. It is through the vanitie of the same imagination that he
dare equall himself to God, that he ascribeth divine conditions unto himself, that he selecteth and separateth
himselfe from out the ranke of other creatures. according to Montaigne, in his Apology for Raymond Sebond.
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1
Stochastic Uncertainty
In the general case, stochastic uncertainty is described by a sample space Ω (of all possible
outcomes), filtrations Ft of events at time t, the probability P assigning to each event its
probability (a number between 0 and 1), a Brownian process B(t), a drift γ(S) and a volatility
σ(S) : dS(t) = ρ(S(t))dt + σ(S(t))dB(t).
1. The sample sets and the random events are not explicitly identified (in practice, one
can always choose the space of all evolutions or the interval [0, 1] in the proofs of the
theorems). Only the drift and volatility are assumed to be explicitly known. In the
financial example, the set Ω is known (the velocities or the rates) of the prices ;
2. Stochastic uncertainty does not study the “package of evolutions” (depending on ω ∈
Ω), but functionals over this package, such as the different moments and their statistical
consequences (averages, variance, etc.) used for evaluating risk.
Even though in some cases, Monte-Carlo methods provide an approximation of the set
of evolutions (for constant ω only), there is no mechanism used for selecting the one(s)
satisfying such or such prescribed property ;
3. Required properties are valid for “almost all” constant ω.
4. Stochastic differential equations providing only measure functionals on the package of
evolutions, they do not allow to select the right one whenever, for every time t > 0, the
effective realization ω (which then, depends on time), is known. This excludes a direct
way to regulate the system by assigning to each state the proper ω, which, in this case,
would depend on t, and thus, may not belong to an approximated set of evolutions
computed by Monte-Carlo type of methods.
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Tychastic Uncertainty
Tychchastic uncertainty is described by
S 0 (t) := f (S(t), v(t)) where v(t) ∈ V (t, S(t))
(1)
The set-valued map (t, S) ; V (t, S), called the tychastic map, describes the set of tyches
depending on time and price independently of the decision maker.
As we saw in the financial example, we do know, for each time t, the range Σ(t) :=
[S [ (t), S ] (t)] in which the price S(t) evolves. This is an observable measure of “tychastic
uncertainty”.
1. Tyches are identified (velocities or rates of the underlying in the above financial
example) which can then be used in dynamic management systems when the realizations of events are actually observed and known at each date during the evolution ;
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2. For this reason, the results are computed in the worst case (eradication of risk instead
of its statistical evaluation) ;
3. required properties are valid for “all” evolutions of tyches t 7→ x(t) ∈ V (t, x(t)) instead
of constant ω’s.
Tyche. Uncertainty without statistical regularity can be translated mathematically by parameters on which actors, agents, decision makers,
etc. have no controls. These parameters are often perturbations, disturbances (as in “robust control” or “differential games against nature”) or
more generally, tyches (meaning “chance” in classical Greek, from the
Goddess Tyche whose goal was to disrupt the course of events either for
good or for bad.
The concept of tychastic uncertainty was introduced by Charles Peirce
in 1893 in Evolutionary Love.
Tyche became “Fortuna” in Latin, “rizikon” in Byzantine Greek, “rizq”
†PP
in Ara-
,
bic (with a positive connotation in these three cases). “ Reaction, change”,
translates the concept of tychasticity.
The invariance kernel under a of a tychastic system is the set of initial states such that,
for all evolutions of tyches, the evolution of the state satisfies the required properties.
The larger the tychastic map, the smaller the invariance kernel, the most
severe is the insurance against tychastic uncertainty.
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Contingent Uncertainty and its Redundancy
How to offset tychastic uncertainty ?
1. By introducing a reservoir of controls or regulons (contingent map x ; U (t, x)) ;
2. building a contingent map (t, x) 7→ u
e(t, x) ∈ U (t, x) independent of the tyches.
Hence, the evolution of the state is governed by a system
x0 (t) := f (x(t), u
e(t, x), v(t)) where v(t) ∈ V (t, x(t))
(2)
The union of the invariance kernels associated with each contingent maps u
e constitutes
the guaranteed viability kernel, i.e., the set of initial states of a regulated tychastic system for
which there exists a contingent map under which, for all evolutions of tyches, the evolution
of the state satisfies the required properties.
The size of the contingent map describes the redundancy :
The larger the contingent map, the larger the guaranteed viability kernel, the
least severe is the insurance against tychastic uncertainty.
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Anticipation : Impulse Contingent Uncertainty
Impulse contingent uncertainty involves an “impulse reservoir” composed of a set of
reset feedbacks. Defined on the subset (trap) on which viability is at stake, reset or impulse
contingent maps remedy instanteneously, with infinite velocity (impulse) for restoring the
state of the system outside the trap. Very often, the trap is a subset of the boundary of the
environment, but not always.
This impulse contingent management method avoids prediction of disasters, but offers
opportunities to recover from them when they occur. Instead of seeking an insurance from
a tychastic reservoir assumed to be known or predicted (predictive approach), the impulse
approach allows the decision maker to correct the situation whenever the the states reaches
the trap. The viability kernel of a regulated impulse system “evaluates” the subset of initial
states from which discontinuous evolutions satisfy the prescribed properties.
It seems that the strategy to build a reservoir of reset feedbacks is used by living beings
to adapt to their environment before the primates that we are unwisely seek to predict their
future while being quite unable to do so. The impulse approach announces the death of the
seers and the emergence of a demiurge remedying unforeseen disasters, because most often
unpredictable.
The larger the impulse map, the larger the guaranteed impulse viability
kernel, the least severe is the insurance against tychastic uncertainty.
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Learning from History
Some physical, biological and economic problems motivate the introduction of duration
functions with variable velocities (representing the “fluidity of time”) offering mathematical
metaphors of a “subjective fleeting specious time” passing more or less slowly. Duration
functions are no longer prescribed, but chosen among available ones and regulated : the joint
evolution of the duration function and the state is assumed to be governed by a regulated
tychastic and provides, as a by product, the unknown temporal windows on which they evolve
together. In economics, for instance, the state is a commodity, its velocity a transaction. If
a cost function is defined on the fluidity of the variable evolution and the transaction of
commodities, their cumulated cost over the fluidity-transaction pairs could be minimized.
Slowing down the fluidity, which widens the investment period (from the milliseconds of highfrequency markets to the centuries of cathedral building) by inventing a shareholder value
tax and decelerating transactions (by implementing the Tobin tax) could be an objective of
salubrious financial and corporate management.
It may be wiser to understand what happened in the past instead of forecasting what
will happen in the future. This (non Popperian) viewpoint seems to be more adequate for
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studying life science systems. To quote Paul Valéry, “Forecasting is a dream from which
reality wakes us up”.
Furthermore, for systems involving living beings, there is not necessarily an actor governing the evolution of regulons according to the above prerequisites.
The choice of criteria is open to question even in static models, even when multicriteria
or several decision makers are involved in the model. The choice (even conditional) of the
optimal controls is made once and for all at some initial time, and thus cannot be changed
at each instant so as to take into account possible modifications of the environment of the
system , thus forbidding adaptation to viability constraints.
However, for evolutionary system of physical and engineering sciences, either through
observation or experiments, it may happen that the evolution of the state on a translated
temporal window is equal to the translation of the evolution on the initial time window. This
is the situation where “the future can be known”.
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Prediction : Historic Differential Inclusions
The knowledge of the past may allow us to extrapolate it by adequate history dependent
(or path dependent, memory dependent, functional) differential inclusions associating with
the history of the evolution up to each time t a set of velocities.
One can propose to replace the use of stochastic differential equations for forecasting
uncertain future evolutions by history dependent (or path dependent, memory dependent,
functional) control systems. A each instant, they associate with the history of the evolution
up to each time t a set of velocities.
“Histories” are evolutions ϕ ∈ C(−∞, 0, Rn ) defined for negative times, a “storage”
space which plays the role of a state space. They are the inputs of differential inclusions with
memory
S 0 (t) ∈ F (κ(−t)S(·))
(3)
where
∀ τ ≤ 0, (κ(−t)S(·))(τ ) := S(t + τ )
and F : C(−∞, 0; Rn ) ; Rn is a set-valued map defining the dynamics of history dependent
differential inclusion.
One can also use history dependent differential equations or inclusions depending on
functionals on past evolutions, such as their derivatives up to a given order m :
S 0 (t) ∈ F (Dp (κ(−t)S(·)))|p|≤m
(4)
in order to take into account not only the history of an evolution, but its “trends”.
For instance, these history dependent differential inclusions have been be used for extrapolating the asset prices.
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The aforementioned VIMADES Extrapolator (based on Laurent Schwartz distributions)
is an example of history dependent differential inclusion by extrapolating each history dependent al (past) evolutions of upper bounds (HIGH ) and lower bounds (LOW ) of the
underlying prices provided by brokerage firms.
The history dependent environments are subsets K ⊂ C(−∞, 0; Rn ) of histories. Actually,
the first “general” viability theorem was proved by Georges Haddad in the framework of
history dependent differential inclusions at the end of the 1970’s.
Since their study, motivated by the evolutionary systems in life sciences, including economics and finance, is much more involved than the one of differential inclusions, most of
the viability studies rested on the case of differential inclusions.
Viability Theorems for history dependent dynamics and environment require a specific
calculus of “Clio derivatives” of history dependent maps.
For instance, let a history dependent functional v : ϕ ∈ C(−∞, 0, Rn ) 7→ v(ϕ) ∈ R. The
addition operator ϕ 7→ ϕ + hψ is replaced by the concatenation operator 3h associating
with each history ϕ ∈ C(−∞, 0; Rn ) the function ϕ3h ψ ∈ C(−∞, 0; Rn ) defined by
ϕ(τ + h)
if τ ∈] − ∞, −h]
(ϕ3h ψ)(τ ) :=
ϕ(0) + ψ(τ + h) if τ ∈ [−h, 0]
This allows us to define the concept of Clio 4 derivatives by taking the limits of “differential quotients”
∇h v(ϕ)(ψ) :=
v((ϕ3h ψ)) − v(ϕ)
∈ X := Rn
h
for obtaining
Dv(ϕ)(ψ) := lim ∇h v(ϕ)(ψ) ∈ X := Rn
h→0+
if it exists and is linear and continuous on C(−∞, 0, Rn ) with respect to ψ.
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Observing Evolution of Asset Prices
First, we observe the daily evolution of the CAC-40 index and its rate during a 75 day
period, short enough for the readability of the graphics.
What we do know for assessing the uncertainty of the evolution of prices are the lower
bounds of prices S [ (t) (LOW ) and their upper bounds S ] (t) (HIGH ) provided by brokerage
firms.
Hence the question arises whether evolutions of prices S(t) governed by “uncertain dynamical systems” are “viable” in the sense that
4. Clio, muse of history, was born as the other muses out of the love between Zeus and Mnenosyne,
Goddess of memory.
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∀ t ∈ [T1 , T2 ], S(t) ∈ Σ(t) := [S [ (t), S ] (t)]
(5)
The “tube” map t ; Σ(t) is an example of “tychastic tube”, i.e., a set-valued (here, an
interval-valued) map from R to R in this simple case.
Introducing the (graphical) derivative DΣ(t, S) of this set-valued map Σ defined at (t, S)
where S ∈ Σ(t), we derive from the viability theorem that the viable evolutions are governed
by the “differential inclusion”
∀ t ≥ 0, S 0 (t) ∈ DΣ(t, S(t))
(6)
In this very simple case of interval-valued tubes, setting
d[ (Σ(t)) := lim inf
h→0+
S ] (t + h) − S [ (t)
S [ (t + h) − S ] (t)
and d] (Σ(t)) := lim sup
h
h
h→0+
we can prove that under adequate assumptions
DΣ(t, S(t)) ⊂ d[ (Σ(t)), d] (Σ(t))
(7)
(8)
We can also provide the tubes in which the rates and the acceleration of the data are viable.
In the discrete version, they are provided in Figure 1, p. 7 :
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Tubes of data and of their velocities.
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2
A tube and its Derivative. The first figure displays the (discrete time) evolution of
CAC-40 indexes (high, last and low prices). The second one displays the derivative of a tube, which
is itself a tube in which evolves the derivative of the last price evolution : S 0 (t) ∈ DΣ(t, S(t)) ⊂
[
d (Σ(t)), d] (Σ(t)) governing all evolutions S(t) viable in the tube.
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Detecting Patterns of Evolution
The question arises to single out dynamical systems regarded as ‘‘pattern generators” :
they govern well identified time series regarded as patterns of interest. For instance, linear or
polynomial functions, exponentials, periodic functions, etc., among the thousands examples
studied for many centuries.
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VIMADES extrapolator. The VIMADES extrapolator takes into account the velocity,
the acceleration and the jerk of the history of an evolution for capturing its trend. Even though
the extrapolation is good, it does not provide any explanation on the possible existence of patterns
provided by differential equations. The figure displays an example of sliding extrapolation of the
CAC 40 indexes, from 2009-11-18 to 2011-10-28, using the VIMADES extrapolator
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Delivering a differential equation, if any, which provides evolutions viable in a tube, hints
at laws explaining the evolution they govern, providing more information than pattern recognition mechanics which may reproduce patterns (such as statistical models, interpolation
by spline functions, the VPPI extrapolator, etc.) without providing interpretations of the
phenomenon involved, if any.
A generator of detectors of patterns should provide
1. a viable pattern generator in a given class of dynamical systems ;
2. the pattern regulator providing at each time the adequate parameters kept constant as
long as the recognition of a pattern is possible (such evolutions are called ”heavy”, in
the sense of heavy trends).
3. the largest window on which pattern recognition occurs ;
4. the detected pattern.
Once detected, the pattern generator and regulator may allow us to explain and reproduce
the underlying dynamics concealed in the time series as a prediction mechanism.
Hence, it is relevant to design generators of detectors which provide
1. the sequence of impulse or punctuation dates providing the ending date of the largest
window over which the time series is recognized by a pattern generated by the pattern
generator. Such instants are regarded as “anomaly dates” ;
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2. the length or duration of the window between two successive anomaly dates, denominated by their “cadence” ;
3. on each window, the restriction of the time series and its recognizing pattern. The
sequence of patterns on the successive windows constitute the “punctuated evolution”
generated by the impulse differential inclusions describing the pattern generator ;
4 Quadratic and Exponential Detectors. These figures display the detection of piecewise
quadratic patterns in a tube surrounding a temporal series of CAC-40 indexes. The anomaly dates
are represented by bars. The upper figure displays the tube, the series and its detection by quadratic
patterns. The middle one displays the relative errors of this detection process. The lower one
provides an example indicating that exponential financial growth provided by compounded interests
does not fit the evolution of the CAC 40 indexes.
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Are Stochastic Models Consistent with Observations ?
The question arises whether the viability property on the price tube t ; Σ(t) holds true
when the data are governed by standard stochastic differential equations : we introduce a
space Ω, filtrations Ft , a probability P, a Brownian process B(t), a drift γ(S) and a volatility
σ(S), allowing us to define the Ito stochastic differential equation
dS(t) = ρ(t)S(t)dt + σ(t)S(t)dB(t)
(9)
We observe that all realizations Sω (t) of the stochastic process S cannot be viable in the
tube Σ(t) on one hand, and that we derive a way to correct the situation : We replace the
stochastic differential equation (9), p. 12 by the tychastic system (12), p. 12. This negative
answer and its cure have been derived from the Strook-Varadhan Support Theorem by Halim
Doss.
σ(t)S 2 (t)
and the StraFor that purpose, we introduce the Stratonovitch drift ρ(t)S(t) −
2
tonovitch tychastic system
S 0 (t) = ρ(t)S(t) −
σ(t)S 2 (t)
+ σ(t)S(t)v(t) where v(t) ∈ R
2
(10)
where the parameters v ∈ R play the role of “tyches” defined below. Indeed, the tyches v
consistent with differential inclusion (6), p. 7 should range over the interval
σ(t)S 2 (t)
− σ(t)S(t)v(t)
v(t) ∈ V (t, S(t)) := DΣ(t, S(t)) − rho(t)S(t) +
2
(11)
since, in this case,
S 0 (t) = ρ(t)S(t) −
σ(t)S 2 (t)
+ σ(t)S(t)v(t) where v(t) ∈ V (t, S(t))
2
(12)
boils down to the differential inclusion S 0 (t) ∈ DΣ(t, S(t)) under which the price tube Σ(t)
is viable
The assumption underlying the use of the Brownian movement is that there is no bound
on the velocities of the data (which, in the Stratonovich framework, is translated by the
requirement that v(t) ∈ R). Knowing that the velocities must belong to the graphical derive
DΣ(t, S) of the tube Σ(t), this amounts to saying that the tyches v range all over the tychastic
tube V (t, S(t)) instead of R.
Starting with a stochastic differential equation, we assume that the “volatility” σ is
known. This a nightmare since there is not known fiable “volatilimeter”. This question
triggered a thousand of studies to determine the volatilities (implicit viability, for instance).
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So, it may be more efficient to use an inverse approach starting with the only knowledge
at our disposal, that the prices must remain in the tube Σ(t) and, consequently, that the
velocities have to be chosen in DΣ(t, S(t)), bypassing the ineffective use of volatilities.
This is one of the reasons why we advocate the use of tychastic systems instead of
stochastic systems because they provide at least the very first requirement that prices should
range over the graphical derivative DΣ(t, S(t)) provided by set-valued analysis, which enjoys
practically all properties of usual derivatives of single-valued maps.
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Références
[1] Aubin J.-P. (2010) La mort du devin, l’émergence du démiurge. Essai sur la contingence et la
viabilité des systèmes, Éditions Beauchesne
[2] Aubin J.-P., Bayen A. and Saint-Pierre P. (2011) Viability Theory. New Directions, SpringerVerlag http://dx.doi.org/10.1007/978-3-642-16684-6
http://vimades.com/aubin
[email protected]