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The Legacy of IngenhouszNotes on some Mathematical Translations of

Uncertainty in Living Sciences

Jean-Pierre Aubin21 juillet 2012

The story started 1 in 1785 when Jan Ingenhousz, a Dutch physiologist, biologist andchemist, discovered what was not called the Ingenhouszian movement, but the better knownBrownian movement, rediscovered by the botanist Robert Brown in 1827, however much lessknown than pedesis (from Greek “leaping”).

Jan Ingenhousz 2 described the irregular motion of coal dust particles on the surface ofalcohol, randomly zigzagging as anyone would do in such conditions. He could not forecastthat, centuries later, his discovery would trigger, in part, the development of stochasticdifferential equations ! Quoting him in the title of these notes is an hommage and a way torevive his memory.

A long list of physicists and mathematicians, Pierre de Fermat, Blaise Pascal, Sadi Car-not, Rudolf Clausius, James Maxwell, Ludwig Boltzmann, Thorvald Thiele, Louis Bachelier,Albert Einstein, Paul Langevin, Henri Lebesgue, Rene Gateaux, Norbert Wiener, Paul Levy,Andreı Kolmogorov, Joseph Doob, Viktor Maslov, Ruslan Stratonovitch, Wolfgang Doblin,Kiyoshi Ito, among so many others, devised mathematical metaphors of “uncertainty” mo-tivated by parlor games, thermodynamics and physical problems. However, they all followedsame directions during the xxth century, involving probabilities and stochastic dynamics.It became “THE” quasi unique mathematical framework to translate mathematically theconcept of uncertainty, and “applied” in almost all fields. From physics, the area where itoriginated, through finance, thanks to the staggering mathematical contribution of LouisBachelier in 1900, to living sciences.

However, are living beings 3 behaving like dust particles in an inebriating en-vironment ?

Asking the question is answering it : we suggest that the stochastic translation of uncer-tainty is not always relevant for living systems, and we may attempt to find other ways tocapture the diverse aspects of uncertainty in mathematical terms.

1. Actually, when the Epicurean Lucretius observed “what happens when sunbeams are admitted intoa building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in amultitude of ways... their dancing is an actual indication of underlying movements of matter that are hiddenfrom 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 thecircle of the moon, and reduce heaven under his feet. It is through the vanitie of the same imagination that hedare equall himself to God, that he ascribeth divine conditions unto himself, that he selecteth and separatethhimselfe 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 possibleoutcomes), filtrations Ft of events at time t, the probability P assigning to each event itsprobability (a number between 0 and 1), a Brownian process B(t), a drift γ(S) and a volatilityσ(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, onecan always choose the space of all evolutions or the interval [0, 1] in the proofs of thetheorems). Only the drift and volatility are assumed to be explicitly known. In thefinancial 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 statisticalconsequences (averages, variance, etc.) used for evaluating risk.

Even though in some cases, Monte-Carlo methods provide an approximation of the setof 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 ofevolutions, they do not allow to select the right one whenever, for every time t > 0, theeffective realization ω (which then, depends on time), is known. This excludes a directway 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 evolutionscomputed by Monte-Carlo type of methods.

2 Tychastic Uncertainty

Tychchastic uncertainty is described by

S ′(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 tychesdepending 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 “tychasticuncertainty”.

1. Tyches are identified (velocities or rates of the underlying in the above financialexample) which can then be used in dynamic management systems when the reali-zations 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 insteadof its statistical evaluation) ;

3. required properties are valid for “all” evolutions of tyches t 7→ x(t) ∈ V (t, x(t)) insteadof constant ω’s.

Tyche. Uncertainty without statistical regularity can be translated ma-thematically by parameters on which actors, agents, decision makers,etc. have no controls. These parameters are often perturbations, distur-bances (as in “robust control” or “differential games against nature”) ormore generally, tyches (meaning “chance” in classical Greek, from theGoddess Tyche whose goal was to disrupt the course of events either forgood or for bad.The concept of tychastic uncertainty was introduced by Charles Peircein 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 mostsevere is the insurance against tychastic uncertainty.

3 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(t, x) ∈ U(t, x) independent of the tyches.

Hence, the evolution of the state is governed by a system

x′(t) := f(x(t), u(t, x), v(t)) where v(t) ∈ V (t, x(t)) (2)

The union of the invariance kernels associated with each contingent maps u constitutesthe guaranteed viability kernel, i.e., the set of initial states of a regulated tychastic system forwhich there exists a contingent map under which, for all evolutions of tyches, the evolutionof 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, theleast severe is the insurance against tychastic uncertainty.

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4 Anticipation : Impulse Contingent Uncertainty

Impulse contingent uncertainty involves an “impulse reservoir” composed of a set ofreset feedbacks. Defined on the subset (trap) on which viability is at stake, reset or impulsecontingent maps remedy instanteneously, with infinite velocity (impulse) for restoring thestate of the system outside the trap. Very often, the trap is a subset of the boundary of theenvironment, but not always.

This impulse contingent management method avoids prediction of disasters, but offersopportunities to recover from them when they occur. Instead of seeking an insurance froma tychastic reservoir assumed to be known or predicted (predictive approach), the impulseapproach allows the decision maker to correct the situation whenever the the states reachesthe trap. The viability kernel of a regulated impulse system “evaluates” the subset of initialstates from which discontinuous evolutions satisfy the prescribed properties.

It seems that the strategy to build a reservoir of reset feedbacks is used by living beingsto adapt to their environment before the primates that we are unwisely seek to predict theirfuture while being quite unable to do so. The impulse approach announces the death of theseers and the emergence of a demiurge remedying unforeseen disasters, because most oftenunpredictable.

The larger the impulse map, the larger the guaranteed impulse viabilitykernel, the least severe is the insurance against tychastic uncertainty.

5 Learning from History

Some physical, biological and economic problems motivate the introduction of durationfunctions with variable velocities (representing the “fluidity of time”) offering mathematicalmetaphors of a “subjective fleeting specious time” passing more or less slowly. Durationfunctions are no longer prescribed, but chosen among available ones and regulated : the jointevolution of the duration function and the state is assumed to be governed by a regulatedtychastic and provides, as a by product, the unknown temporal windows on which they evolvetogether. In economics, for instance, the state is a commodity, its velocity a transaction. Ifa cost function is defined on the fluidity of the variable evolution and the transaction ofcommodities, their cumulated cost over the fluidity-transaction pairs could be minimized.Slowing down the fluidity, which widens the investment period (from the milliseconds of high-frequency markets to the centuries of cathedral building) by inventing a shareholder valuetax and decelerating transactions (by implementing the Tobin tax) could be an objective ofsalubrious financial and corporate management.

It may be wiser to understand what happened in the past instead of forecasting whatwill 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 Valery, “Forecasting is a dream from whichreality wakes us up”.

Furthermore, for systems involving living beings, there is not necessarily an actor gover-ning the evolution of regulons according to the above prerequisites.

The choice of criteria is open to question even in static models, even when multicriteriaor several decision makers are involved in the model. The choice (even conditional) of theoptimal controls is made once and for all at some initial time, and thus cannot be changedat each instant so as to take into account possible modifications of the environment of thesystem , thus forbidding adaptation to viability constraints.

However, for evolutionary system of physical and engineering sciences, either throughobservation or experiments, it may happen that the evolution of the state on a translatedtemporal window is equal to the translation of the evolution on the initial time window. Thisis the situation where “the future can be known”.

6 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 withthe 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 forecastinguncertain future evolutions by history dependent (or path dependent, memory dependent,functional) control systems. A each instant, they associate with the history of the evolutionup 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 withmemory

S ′(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 dependentdifferential inclusion.

One can also use history dependent differential equations or inclusions depending onfunctionals on past evolutions, such as their derivatives up to a given order m :

S ′(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 extra-

polating 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 de-pendent al (past) evolutions of upper bounds (HIGH ) and lower bounds (LOW ) of theunderlying 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 ofhistory dependent differential inclusions at the end of the 1970’s.

Since their study, motivated by the evolutionary systems in life sciences, including eco-nomics and finance, is much more involved than the one of differential inclusions, most ofthe viability studies rested on the case of differential inclusions.

Viability Theorems for history dependent dynamics and environment require a specificcalculus of “Clio derivatives” of history dependent maps.

For instance, let a history dependent functional v : ϕ ∈ C(−∞, 0,Rn) 7→ v(ϕ) ∈ R. Theaddition operator ϕ 7→ ϕ + hψ is replaced by the concatenation operator 3h associatingwith each history ϕ ∈ C(−∞, 0; Rn) the function ϕ3hψ ∈ C(−∞, 0; Rn) defined by

(ϕ3hψ)(τ) :=

ϕ(τ + h) if τ ∈]−∞,−h]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 “diffe-rential quotients”

∇hv(ϕ)(ψ) :=v((ϕ3hψ))− v(ϕ)

h∈ X := Rn

for obtainingDv(ϕ)(ψ) := lim

h→0+∇hv(ϕ)(ψ) ∈ X := Rn

if it exists and is linear and continuous on C(−∞, 0,Rn) with respect to ψ.

7 Observing Evolution of Asset Prices

First, we observe the daily evolution of the CAC-40 index and its rate during a 75 dayperiod, short enough for the readability of the graphics.

What we do know for assessing the uncertainty of the evolution of prices are the lowerbounds of prices S[(t) (LOW ) and their upper bounds S](t) (HIGH ) provided by brokeragefirms.

Hence the question arises whether evolutions of prices S(t) governed by “uncertain dy-namical 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, aninterval-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 governedby the “differential inclusion”

∀ t ≥ 0, S ′(t) ∈ DΣ(t, S(t)) (6)

In this very simple case of interval-valued tubes, setting

d[(Σ(t)) := lim infh→0+

S[(t+ h)− S](t)

hand d](Σ(t)) := lim sup

h→0+

S](t+ h)− S[(t)

h(7)

we can prove that under adequate assumptions

DΣ(t, S(t)) ⊂[d[(Σ(t)), d](Σ(t))

](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 :

1 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′(t) ∈ DΣ(t, S(t)) ⊂[d[(Σ(t)), d](Σ(t))

]governing all evolutions S(t) viable in the tube.

8 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 orpolynomial functions, exponentials, periodic functions, etc., among the thousands examplesstudied for many centuries.

3 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, hintsat laws explaining the evolution they govern, providing more information than pattern re-cognition mechanics which may reproduce patterns (such as statistical models, interpolationby spline functions, the VPPI extrapolator, etc.) without providing interpretations of thephenomenon 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 aslong as the recognition of a pattern is possible (such evolutions are called ”heavy”, inthe 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 reproducethe 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 largestwindow over which the time series is recognized by a pattern generated by the patterngenerator. Such instants are regarded as “anomaly dates” ;

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2. the length or duration of the window between two successive anomaly dates, denomi-nated by their “cadence” ;

3. on each window, the restriction of the time series and its recognizing pattern. Thesequence 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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9 Are Stochastic Models Consistent with Observa-

tions ?

The question arises whether the viability property on the price tube t; Σ(t) holds truewhen the data are governed by standard stochastic differential equations : we introduce aspace Ω, filtrations Ft, a probability P, a Brownian process B(t), a drift γ(S) and a volatilityσ(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 thetube Σ(t) on one hand, and that we derive a way to correct the situation : We replace thestochastic differential equation (9), p. 12 by the tychastic system (12), p. 12. This negativeanswer and its cure have been derived from the Strook-Varadhan Support Theorem by HalimDoss.

For that purpose, we introduce the Stratonovitch drift ρ(t)S(t)− σ(t)S2(t)

2and the Stra-

tonovitch tychastic system

S ′(t) = ρ(t)S(t)− σ(t)S2(t)

2+ σ(t)S(t)v(t) where v(t) ∈ R (10)

where the parameters v ∈ R play the role of “tyches” defined below. Indeed, the tyches vconsistent with differential inclusion (6), p. 7 should range over the interval

v(t) ∈ V (t, S(t)) := DΣ(t, S(t))− rho(t)S(t) +σ(t)S2(t)

2− σ(t)S(t)v(t) (11)

since, in this case,

S ′(t) = ρ(t)S(t)− σ(t)S2(t)

2+ σ(t)S(t)v(t) where v(t) ∈ V (t, S(t)) (12)

boils down to the differential inclusion S ′(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 boundon the velocities of the data (which, in the Stratonovich framework, is translated by therequirement that v(t) ∈ R). Knowing that the velocities must belong to the graphical deriveDΣ(t, S) of the tube Σ(t), this amounts to saying that the tyches v range all over the tychastictube V (t, S(t)) instead of R.

Starting with a stochastic differential equation, we assume that the “volatility” σ isknown. This a nightmare since there is not known fiable “volatilimeter”. This questiontriggered 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 knowledgeat our disposal, that the prices must remain in the tube Σ(t) and, consequently, that thevelocities 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 ofstochastic systems because they provide at least the very first requirement that prices shouldrange over the graphical derivative DΣ(t, S(t)) provided by set-valued analysis, which enjoyspractically all properties of usual derivatives of single-valued maps.

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References

[1] Aubin J.-P. (2010) La mort du devin, l’emergence du demiurge. Essai sur la contingence et laviabilite des systemes, Editions Beauchesne

[2] Aubin J.-P., Bayen A. and Saint-Pierre P. (2011) Viability Theory. New Directions, Springer-Verlag http://dx.doi.org/10.1007/978-3-642-16684-6

http://vimades.com/aubin

aubin.jp@gmail.com