Stochastic Controledited Part 3 ppt

conditional probabilities defined by Hausdorff outer and inner measures 87Stochastic independence with respect to upper and lower conditional probabilities defined by Hausdorff outer and

Fig 10 Mean density ρ vs p for different rules evolving under the third self-synchronization

method The density of the system decreases linearly with p.

that the behavior reported in the first self-synchronization method is newly obtained in this

case Rule 18 undergoes a phase transition for a critical value of ˜p For ˜p greater than the

critical value, the method is able to find the stable structure of the system (Sanchez &

Lopez-Ruiz, 2006) For the rest of the rules the freezing phase is not found The dynamics generates

patterns where the different marginally stable structures randomly compete Hence the DA

density decays linearly with ˜p (see Fig 8).

4.3 Third Self-Synchronization Method

At last, we introduce another type of stochastic element in the application of the rule Φ Given

an integer number L, the surrounding of site i at each time step is redefined A site i l is

randomly chosen among the L neighbors of site i to the left,(i−L, , i−1) Analogously,

a site i r is randomly chosen among the L neighbors of site i to the right, (i+1, , i+L)

The rule Φ is now applied on the site i using the triplet(i l , i, i r)instead of the usual nearest

neighbors of the site This new version of the rule is called ΦL, being ΦL=1= Φ Later the

operator Γpacts in identical way as in the first method Therefore, the dynamical evolution

law is:

σ(t+1) =Γp[(σ1(t), σ2(t))] =Γp[(σ(t), ΦL[σ(t)])] (13)

The DA density as a function of p is plotted in Fig 9 for the rule 18 and in Fig 10 for other

rules It can be observed again that the rule 18 is a singular case that, even for different L,

maintains the memory and continues to self-synchronize It means that the influence of the

rule is even more important than the randomness in the election of the surrounding sites The

system self-synchronizes and decays to the corresponding stable structure Contrary, for the

rest of the rules, the DA density decreases linearly with p even for L=1 as shown in Fig 10

Fig 11 Space-time configurations of automata with N = 100 sites iterated during T=400

time steps evolving under rules 18 and 150 for p  p c Left panels show the automatonevolution in time (increasing from top to bottom) and the right panels display the evolution

of the corresponding DA

The systems oscillate randomly among their different marginally stable structures as in theprevious methods (Sanchez & Lopez-Ruiz, 2006)

5 Symmetry Pattern Transition in Cellular Automata with Complex Behavior

In this section, the stochastic synchronization method introduced in the former sections(Morelli & Zanette, 1998) for two CA is specifically used to find symmetrical patterns in theevolution of a single automaton To achieve this goal the stochastic operator, below described,

is applied to sites symmetrically located from the center of the lattice It is shown that a metry transition take place in the spatio-temporal pattern The transition forces the automaton

sym-to evolve sym-toward complex patterns that have mirror symmetry respect sym-to the central axe ofthe pattern In consequence, this synchronization method can also be interpreted as a controltechnique for stabilizing complex symmetrical patterns

Cellular automata are extended systems, in our case one-dimensional strings composed of N sites or cells Each site is labeled by an index i=1, , N, with a local variable s icarrying a

binary value, either 0 or 1 The set of sites values at time t represents a configuration (state

or pattern) σ t of the automaton During the automaton evolution, a new configuration σ t+1at

time t+1 is obtained by the application of a rule or operator Φ to the present configuration(see former section):

Fig 10 Mean density ρ vs p for different rules evolving under the third self-synchronization

method The density of the system decreases linearly with p.

that the behavior reported in the first self-synchronization method is newly obtained in this

case Rule 18 undergoes a phase transition for a critical value of ˜p For ˜p greater than the

critical value, the method is able to find the stable structure of the system (Sanchez &

Lopez-Ruiz, 2006) For the rest of the rules the freezing phase is not found The dynamics generates

patterns where the different marginally stable structures randomly compete Hence the DA

density decays linearly with ˜p (see Fig 8).

4.3 Third Self-Synchronization Method

At last, we introduce another type of stochastic element in the application of the rule Φ Given

an integer number L, the surrounding of site i at each time step is redefined A site i l is

randomly chosen among the L neighbors of site i to the left,(i−L, , i−1) Analogously,

a site i r is randomly chosen among the L neighbors of site i to the right, (i+1, , i+L)

The rule Φ is now applied on the site i using the triplet(i l , i, i r)instead of the usual nearest

neighbors of the site This new version of the rule is called ΦL, being ΦL=1 =Φ Later the

operator Γpacts in identical way as in the first method Therefore, the dynamical evolution

law is:

σ(t+1) =Γp[(σ1(t), σ2(t))] =Γp[(σ(t), ΦL[σ(t)])] (13)

The DA density as a function of p is plotted in Fig 9 for the rule 18 and in Fig 10 for other

rules It can be observed again that the rule 18 is a singular case that, even for different L,

maintains the memory and continues to self-synchronize It means that the influence of the

rule is even more important than the randomness in the election of the surrounding sites The

system self-synchronizes and decays to the corresponding stable structure Contrary, for the

rest of the rules, the DA density decreases linearly with p even for L=1 as shown in Fig 10

Fig 11 Space-time configurations of automata with N = 100 sites iterated during T= 400

time steps evolving under rules 18 and 150 for p  p c Left panels show the automatonevolution in time (increasing from top to bottom) and the right panels display the evolution

of the corresponding DA

The systems oscillate randomly among their different marginally stable structures as in theprevious methods (Sanchez & Lopez-Ruiz, 2006)

5 Symmetry Pattern Transition in Cellular Automata with Complex Behavior

In this section, the stochastic synchronization method introduced in the former sections(Morelli & Zanette, 1998) for two CA is specifically used to find symmetrical patterns in theevolution of a single automaton To achieve this goal the stochastic operator, below described,

is applied to sites symmetrically located from the center of the lattice It is shown that a metry transition take place in the spatio-temporal pattern The transition forces the automaton

sym-to evolve sym-toward complex patterns that have mirror symmetry respect sym-to the central axe ofthe pattern In consequence, this synchronization method can also be interpreted as a controltechnique for stabilizing complex symmetrical patterns

Cellular automata are extended systems, in our case one-dimensional strings composed of N sites or cells Each site is labeled by an index i =1, , N, with a local variable s icarrying a

binary value, either 0 or 1 The set of sites values at time t represents a configuration (state

or pattern) σ t of the automaton During the automaton evolution, a new configuration σ t+1at

time t+1 is obtained by the application of a rule or operator Φ to the present configuration(see former section):

Rule 18 Rule 150

Fig 12 Time configurations of automata with N = 100 sites iterated during T = 400 time

steps evolving under rules 18 and 150 using p > p c The space symmetry of the evolving

patterns is clearly visible

5.1 Self-Synchronization Method by Symmetry

Our present interest (Sanchez & Lopez-Ruiz, 2008) resides in those CA evolving under rules

capable to show asymptotic complex behavior (rules of class III and IV) The technique applied

here is similar to the synchronization scheme introduced by Morelli and Zanette (Morelli &

Zanette, 1998) for two CA evolving under the same rule Φ The strategy supposes that the two

systems have a partial knowledge one about each the other At each time step and after the

application of the rule Φ, both systems compare their present configurations Φ[σ t1]and Φ[σ t2]

along all their extension and they synchronize a percentage p of the total of their different sites.

The location of the percentage p of sites that are going to be put equal is decided at random

and, for this reason, it is said to be an stochastic synchronization If we call this stochastic

operator Γp, its action over the couple(Φ[σ t1], Φ[σ t2])can be represented by the expression:

(σ t+11 , σ t+12 ) =Γp(Φ[σ t1], Φ[σ t2]) = (Γp◦Φ)(σ t1, σ t2) (15)

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

p

Rule 30 Rule 122 Rule 150

Fig 13 Asymptotic density of the DA for different rules is plotted as a function of the coupling

probability p Different values of p c for each rule appear clearly at the points where ρ →0

The automata with N=4000 sites were iterated during T=500 time steps The mean values

of the last 100 steps were used for density calculations

Rule 18 22 30 54 60 90 105 110 122 126 146 150 182

p c 0.25 0.27 1.00 0.20 1.00 0.25 0.37 1.00 0.27 0.30 0.25 0.37 0.25

Table 1 Numerically obtained values of the critical probability p cfor different rules displayingcomplex behavior Rules that can not sustain symmetric patterns need fully coupling of the

symmetric sites, i.e (p c=1)

The same strategy can be applied to a single automaton with a even number of sites (Sanchez

& Lopez-Ruiz, 2008) Now the evolution equation, σ t+1= (Γp◦Φ)[σ t], given by the successiveaction of the two operators Φ and Γp , can be applied to the configuration σ tas follows:

1 the deterministic operator Φ for the evolution of the automaton produces Φ[σ t], and,

2 the stochastic operator Γp, produces the result Γp(Φ[σ t]), in such way that, if sites

sym-metrically located from the center are different, i.e s i=s N−i+1, then Γp equals s N−i+1

to s i with probability p Γ pleaves the sites unchanged with probability 1−p.

A simple way to visualize the transition to a symmetric pattern can be done by splitting the

automaton in two subsystems (σ t1, σ t2),

• σ1

t , composed by the set of sites s(i)with i=1, , N/2 and

• σ t2, composed the set of symmetrically located sites s(N−i+1)with i=1, , N/2,

and displaying the evolution of the difference automaton (DA), defined as

Rule 18 Rule 150

Fig 12 Time configurations of automata with N = 100 sites iterated during T = 400 time

steps evolving under rules 18 and 150 using p > p c The space symmetry of the evolving

patterns is clearly visible

5.1 Self-Synchronization Method by Symmetry

Our present interest (Sanchez & Lopez-Ruiz, 2008) resides in those CA evolving under rules

capable to show asymptotic complex behavior (rules of class III and IV) The technique applied

here is similar to the synchronization scheme introduced by Morelli and Zanette (Morelli &

Zanette, 1998) for two CA evolving under the same rule Φ The strategy supposes that the two

systems have a partial knowledge one about each the other At each time step and after the

application of the rule Φ, both systems compare their present configurations Φ[σ t1]and Φ[σ t2]

along all their extension and they synchronize a percentage p of the total of their different sites.

The location of the percentage p of sites that are going to be put equal is decided at random

and, for this reason, it is said to be an stochastic synchronization If we call this stochastic

operator Γp, its action over the couple(Φ[σ t1], Φ[σ t2])can be represented by the expression:

(σ1t+1 , σ2t+1) =Γp(Φ[σ t1], Φ[σ t2]) = (Γp◦Φ)(σ t1, σ t2) (15)

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

p

Rule 30 Rule 122 Rule 150

Fig 13 Asymptotic density of the DA for different rules is plotted as a function of the coupling

probability p Different values of p c for each rule appear clearly at the points where ρ→ 0

The automata with N=4000 sites were iterated during T=500 time steps The mean values

of the last 100 steps were used for density calculations

Rule 18 22 30 54 60 90 105 110 122 126 146 150 182

p c 0.25 0.27 1.00 0.20 1.00 0.25 0.37 1.00 0.27 0.30 0.25 0.37 0.25

Table 1 Numerically obtained values of the critical probability p cfor different rules displayingcomplex behavior Rules that can not sustain symmetric patterns need fully coupling of the

symmetric sites, i.e (p c=1)

The same strategy can be applied to a single automaton with a even number of sites (Sanchez

& Lopez-Ruiz, 2008) Now the evolution equation, σ t+1= (Γp◦Φ)[σ t], given by the successiveaction of the two operators Φ and Γp , can be applied to the configuration σ tas follows:

1 the deterministic operator Φ for the evolution of the automaton produces Φ[σ t], and,

2 the stochastic operator Γp, produces the result Γp(Φ[σ t]), in such way that, if sites

sym-metrically located from the center are different, i.e s i=s N−i+1, then Γp equals s N−i+1

to s i with probability p Γ pleaves the sites unchanged with probability 1−p.

A simple way to visualize the transition to a symmetric pattern can be done by splitting the

automaton in two subsystems (σ t1, σ2t),

• σ1

t , composed by the set of sites s(i)with i=1, , N/2 and

• σ t2, composed the set of symmetrically located sites s(N−i+1)with i=1, , N/2,

and displaying the evolution of the difference automaton (DA), defined as

The mean density of active sites for the difference automaton, defined as

represents the Hamming distance between the sets σ1and σ2 It is clear that the automaton

will display a symmetric pattern when limt→∞ρ t =0 For class III and IV rules, a symmetry

transition controlled by the parameter p is found The transition is characterized by the DA

behavior:

when p<p c → limt→∞ρ t=0 (complex non-symmetric patterns),

when p>p c → limt→∞ρ t=0 (complex symmetric patterns)

The critical value of the parameter p csignals the transition point

In Fig 11 the space-time configurations of automata evolving under rules 18 and 150 are

shown for p  p c The automata are composed by N =100 sites and were iterated during

T =400 time steps Left panels show the automaton evolution in time (increasing from top

to bottom) and the right panels display the evolution of the corresponding DA For p p c,

complex structures can be observed in the evolution of the DA As p approaches its critical

value p c, the evolution of the DA become more stumped and reminds the problem of

struc-tures trying to percolate the plane (Pomeau, 1986; Sanchez & Lopez-Ruiz, 2005-a) In Fig 12

the space-time configurations of the same automata are displayed for p>p c Now, the space

symmetry of the evolving patterns is clearly visible

Table 1 shows the numerically obtained values of p cfor different rules displaying complex

be-havior It can be seen that some rules can not sustain symmetric patterns unless those patterns

are forced to it by fully coupling the totality of the symmetric sites (p c =1) The rules whose

local dynamics verify φ( 1, s0, s2) =φ( 2, s0, s1)can evidently sustain symmetric patterns, and

these structures are induced for p c<1 by the method here explained

Finally, in Fig 13 the asymptotic density of the DA, ρ t for t→∞, for different rules is plotted

as a function of the coupling probability p The values of p c for the different rules appear

clearly at the points where ρ→0

6 Conclusion

A method to measure statistical complexity in extended systems has been implemented It

has been applied to a transition to spatio-temporal complexity in a coupled map lattice and

to a transition to synchronization in two stochastically coupled cellular automata (CA) The

statistical indicator shows a peak just in the transition region, marking clearly the change of

dynamical behavior in the extended system

Inspired in stochastic synchronization methods for CA, different schemes for

self-synchronization of a single automaton have also been proposed and analyzed

Self-synchronization of a single automaton can be interpreted as a strategy for searching and

con-trolling the structures of the system that are constant in time In general, it has been found

that a competition among all such structures is established, and the system ends up

oscillat-ing randomly among them However, rule 18 is a unique position among all rules because,

even with random election of the neighbors sites, the automaton is able to reach the

configu-ration constant in time

Also a transition from asymmetric to symmetric patterns in time-dependent extended systems

has been described It has been shown that one dimensional cellular automata, started from

fully random initial conditions, can be forced to evolve into complex symmetrical patterns by stochastically coupling a proportion p of pairs of sites located at equal distance from the center

of the lattice A nontrivial critical value of p must be surpassed in order to obtain symmetrical

patterns during the evolution This strategy could be used as an alternative to classify thecellular automata rules -with complex behavior- between those that support time-dependentsymmetric patterns and those which do not support such kind of patterns

7 References

Anteneodo, C & Plastino, A.R (1996) Some features of the statistical LMC complexity Phys.

Lett A, Vol 223, No 5, 348-354.

Argentina, M & Coullet, P (1997) Chaotic nucleation of metastable domains Phys Rev E, Vol.

56, No 3, R2359-R2362

Bennett, C.H (1985) Information, dissipation, and the definition of organization In: Emerging

Syntheses in Science, David Pines, (Ed.), 297-313, Santa Fe Institute, Santa Fe.

Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D.L & Zhou, C.S (2002) The synchronization

of chaotic systems Phys Rep., Vol 366, No 1-2, 1-101.

Calbet, X & López-Ruiz, R (2001) Tendency toward maximum complexity in a

non-equilibrium isolated system Phys Rev E, Vol 63, No.6, 066116(9).

Chaitin, G (1966) On the length of programs for computing finite binary sequences J Assoc.

Comput Mach., Vol 13, No.4, 547-569.

Chaté, H & Manneville, P (1987) Transition to turbulence via spatio-temporal intermittency

Phys Rev Lett., Vol 58, No 2, 112-115.

Crutchfield, J.P & Young, K (1989) Inferring statistical complexity Phys Rev Lett., Vol 63, No.

2, 105-108

Feldman D.P & Crutchfield, J.P (1998) Measures of statistical complexity: Why? Phys Lett.

A, Vol 238, No 4-5, 244-252.

Grassberger, P (1986) Toward a quantitative theory of self-generated complexity Int J Theor.

Phys., Vol 25, No 9, 907-915.

Hawking, S (2000) “I think the next century will be the century of complexity", In San José

Mercury News, Morning Final Edition, January 23.

Houlrik, J.M.; Webman, I & Jensen, M.H (1990) Mean-field theory and critical behavior of

coupled map lattices Phys Rev A, Vol 41, No 8, 4210-4222.

Ilachinski, A (2001) Cellular Automata: A Discrete Universe, World Scientific, Inc River Edge,

NJ

Kaneko, K (1989) Chaotic but regular posi-nega switch among coded attractors by

cluster-size variation Phys Rev Lett., Vol 63, No 3, 219-223.

Kolmogorov, A.N (1965) Three approaches to the definition of quantity of information Probl.

Inform Theory, Vol 1, No 1, 3-11.

Lamberti, W.; Martin, M.T.; Plastino, A & Rosso, O.A (2004) Intensive entropic non-triviality

measure Physica A, Vol 334, No 1-2, 119-131.

Lempel, A & Ziv, J (1976) On the complexity of finite sequences IEEE Trans Inform Theory,

The mean density of active sites for the difference automaton, defined as

represents the Hamming distance between the sets σ1and σ2 It is clear that the automaton

will display a symmetric pattern when limt→∞ρ t =0 For class III and IV rules, a symmetry

transition controlled by the parameter p is found The transition is characterized by the DA

behavior:

when p<p c → limt→∞ρ t=0 (complex non-symmetric patterns),

when p>p c → limt→∞ρ t=0 (complex symmetric patterns)

The critical value of the parameter p csignals the transition point

In Fig 11 the space-time configurations of automata evolving under rules 18 and 150 are

shown for p  p c The automata are composed by N = 100 sites and were iterated during

T =400 time steps Left panels show the automaton evolution in time (increasing from top

to bottom) and the right panels display the evolution of the corresponding DA For p p c,

complex structures can be observed in the evolution of the DA As p approaches its critical

value p c, the evolution of the DA become more stumped and reminds the problem of

struc-tures trying to percolate the plane (Pomeau, 1986; Sanchez & Lopez-Ruiz, 2005-a) In Fig 12

the space-time configurations of the same automata are displayed for p>p c Now, the space

symmetry of the evolving patterns is clearly visible

Table 1 shows the numerically obtained values of p cfor different rules displaying complex

be-havior It can be seen that some rules can not sustain symmetric patterns unless those patterns

are forced to it by fully coupling the totality of the symmetric sites (p c =1) The rules whose

local dynamics verify φ( 1, s0, s2) =φ( 2, s0, s1)can evidently sustain symmetric patterns, and

these structures are induced for p c<1 by the method here explained

Finally, in Fig 13 the asymptotic density of the DA, ρ t for t→∞, for different rules is plotted

as a function of the coupling probability p The values of p c for the different rules appear

clearly at the points where ρ→0

6 Conclusion

A method to measure statistical complexity in extended systems has been implemented It

has been applied to a transition to spatio-temporal complexity in a coupled map lattice and

to a transition to synchronization in two stochastically coupled cellular automata (CA) The

statistical indicator shows a peak just in the transition region, marking clearly the change of

dynamical behavior in the extended system

Inspired in stochastic synchronization methods for CA, different schemes for

self-synchronization of a single automaton have also been proposed and analyzed

Self-synchronization of a single automaton can be interpreted as a strategy for searching and

con-trolling the structures of the system that are constant in time In general, it has been found

that a competition among all such structures is established, and the system ends up

oscillat-ing randomly among them However, rule 18 is a unique position among all rules because,

even with random election of the neighbors sites, the automaton is able to reach the

configu-ration constant in time

Also a transition from asymmetric to symmetric patterns in time-dependent extended systems

has been described It has been shown that one dimensional cellular automata, started from

fully random initial conditions, can be forced to evolve into complex symmetrical patterns by stochastically coupling a proportion p of pairs of sites located at equal distance from the center

of the lattice A nontrivial critical value of p must be surpassed in order to obtain symmetrical

patterns during the evolution This strategy could be used as an alternative to classify thecellular automata rules -with complex behavior- between those that support time-dependentsymmetric patterns and those which do not support such kind of patterns

7 References

Anteneodo, C & Plastino, A.R (1996) Some features of the statistical LMC complexity Phys.

Lett A, Vol 223, No 5, 348-354.

Argentina, M & Coullet, P (1997) Chaotic nucleation of metastable domains Phys Rev E, Vol.

56, No 3, R2359-R2362

Bennett, C.H (1985) Information, dissipation, and the definition of organization In: Emerging

Syntheses in Science, David Pines, (Ed.), 297-313, Santa Fe Institute, Santa Fe.

Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D.L & Zhou, C.S (2002) The synchronization

of chaotic systems Phys Rep., Vol 366, No 1-2, 1-101.

Calbet, X & López-Ruiz, R (2001) Tendency toward maximum complexity in a

non-equilibrium isolated system Phys Rev E, Vol 63, No.6, 066116(9).

Chaitin, G (1966) On the length of programs for computing finite binary sequences J Assoc.

Comput Mach., Vol 13, No.4, 547-569.

Chaté, H & Manneville, P (1987) Transition to turbulence via spatio-temporal intermittency

Phys Rev Lett., Vol 58, No 2, 112-115.

Crutchfield, J.P & Young, K (1989) Inferring statistical complexity Phys Rev Lett., Vol 63, No.

2, 105-108

Feldman D.P & Crutchfield, J.P (1998) Measures of statistical complexity: Why? Phys Lett.

A, Vol 238, No 4-5, 244-252.

Grassberger, P (1986) Toward a quantitative theory of self-generated complexity Int J Theor.

Phys., Vol 25, No 9, 907-915.

Hawking, S (2000) “I think the next century will be the century of complexity", In San José

Mercury News, Morning Final Edition, January 23.

Houlrik, J.M.; Webman, I & Jensen, M.H (1990) Mean-field theory and critical behavior of

coupled map lattices Phys Rev A, Vol 41, No 8, 4210-4222.

Ilachinski, A (2001) Cellular Automata: A Discrete Universe, World Scientific, Inc River Edge,

NJ

Kaneko, K (1989) Chaotic but regular posi-nega switch among coded attractors by

cluster-size variation Phys Rev Lett., Vol 63, No 3, 219-223.

Kolmogorov, A.N (1965) Three approaches to the definition of quantity of information Probl.

Inform Theory, Vol 1, No 1, 3-11.

Lamberti, W.; Martin, M.T.; Plastino, A & Rosso, O.A (2004) Intensive entropic non-triviality

measure Physica A, Vol 334, No 1-2, 119-131.

Lempel, A & Ziv, J (1976) On the complexity of finite sequences IEEE Trans Inform Theory,

López-Ruiz, R & Pérez-Garcia, C (1991) Dynamics of maps with a global multiplicative

cou-pling Chaos, Solitons and Fractals, Vol 1, No 6, 511-528.

López-Ruiz, R (1994) On Instabilities and Complexity, Ph D Thesis, Universidad de Navarra,

Pamplona, Spain

López-Ruiz, R.; Mancini, H.L & Calbet, X (1995) A statistical measure of complexity Phys.

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López-Ruiz, R & Fournier-Prunaret, D (2004) Complex behaviour in a discrete logistic model

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Zero-sum stopping game associated with threshold probability

Yoshio Ohtsubo

0

Zero-sum stopping game associated

with threshold probability

Yoshio Ohtsubo

Kochi University

Japan

Abstract

We consider a zero-sum stopping game (Dynkin’s game) with a threshold probability criterion

in discrete time stochastic processes We first obtain fundamental characterization of value

function of the game and optimal stopping times for both players as the result of the classical

Dynkin’s game, but the value function of the game and the optimal stopping time for each

player depend upon a threshold value We also give properties of the value function of the

game with respect to threshold value These are applied to an independent model and we

explicitly find a value function of the game and optimal stopping times for both players in a

special example

1 Introduction

In the classical Dynkin’s game, a standard criterion function is the expected reward (e.g

DynkinDynkin (1969) and NeveuNeveu (1975)) It is, however, known that the criterion is

quite insufficient to characterize the decision problem from the point of view of the decision

maker and it is necessary to select other criteria to reflect the variability of risk features for

the problem (e.g WhiteWhite (1988)) In a optimal stopping problem, Denardo and

Roth-blumDenardo & Rothblum (1979) consider an optimal stopping problem with an exponential

utility function as a criterion function in finite Markov decision chain and use a linear

pro-gramming to compute an optimal policy In Kadota et al.Kadota et al (1996), they investigate

an optimal stopping problem with a general utility function in a denumerable Markov chain

They give a sufficient condition for an one-step look ahead (OLA) stopping time to be optimal

and characterize a property of an OLA stopping time for risk-averse and risk-seeking utilities

BojdeckiBojdecki (1979) formulates an optimal stopping problem which is concerned with

maximizing the probability of a certain event and give necessary and sufficient conditions for

existence of an optimal stopping time He also applies the results to a version of the

discrete-time disorder problem OhtsuboOhtsubo (2003) considers optimal stopping problems with

a threshold probability criterion in a Markov process, characterizes optimal values and finds

optimal stopping times for finite and infinite horizon cases, and he in Ohtsubo (2003) also

investigates optimal stopping problem with analogous objective for discrete time stochastic

process and these are applied to a secretary problem, a parking problem and job search

prob-lems

5

On the other hand, many authors propose a variety of criteria and investigate Markov

deci-sion processes for their criteria, instead of standard criteria, that is, the expected discounted

total reward and the average expected reward per unit (see WhiteWhite (1988) for survey)

Especially, WhiteWhite (1993), Wu and LinWu & Lin (1999), Ohtsubo and ToyonagaOhtsubo

& Toyonaga (2002) and OhtsuboOhtsubo (2004) consider a problem in which we minimize a

threshold probability Such a problem is called risk minimizing problem and is available for

applications to the percentile of the losses or Value-at-Risk (VaR) in finance (e.g FilarFilar et

al (1995) and UryasevUryasev (2000))

In this paper we consider Dynkin’s game with a threshold probability in a random sequence

In Section 3 we characterize a value function of game and optimal stopping times for both

players and show that the value function of game has properties of a distribution function

with respect to a threshold value except a right continuity In Section 4 we investigate an

independent model, as applications of our game, and we explicitly find a value function which

is right continuous and optimal stopping times for both players

2 Formulation of problem

Let(Ω,F , P)be a probability space and(F n)n∈N an increasing family of sub-σ-fields of F,

where N = {0, 1, 2,· · · } is a discrete time space Let X = (X n)n∈N , Y = (Y n)n∈N , W =

(W n)n∈Nbe sequences of random variables defined on(Ω,F , P)and adapted to(F n)such

that X n ≤ W n ≤ Y n almost surely (a.s.) for all n ∈ N and P(supn X+

n +supn Y n − <∞) = 1,

where x+ =max(0, x) and x − = (−x)+ The second assumption holds if random

vari-ables supn X+

n and supn Y n −are integrable, which are standard conditions given in the

classi-cal Dynkin’s game Also let Z be an arbitrary integrable random variable on(Ω,F , P) For

each n ∈ N, we denote by Γ nthe class of(F n)–stopping times τ such that τ ≥ n a s

We consider the following zero-sum stopping game There are two players and the first and

the second players choose stopping times τ and σ in Γ0, respectively Then the reward paid to

the first player from the second is equal to

g(τ , σ) =X τ I (τ<σ)+Y σ I (σ<τ)+W τ I (τ=σ<∞)+ZI (τ=σ=∞),

where I A is the indicator function of a set A in F In the classical Dynkin’s game the aim of the

first player is to maximize the expected gain E[g(τ , σ)]with respect to τ ∈Γ0and that of the

second is to minimize this expectation with respect to σ ∈Γ0 In our problem the objective of

the first player is to minimize the threshold probability P[g(τ , σ)≤ r]with respect to τ ∈Γ0

and the second maximizes the probability with respect to σ ∈ Γ0for a given threshold value

r.

We can define processes of minimax and maxmin values corresponding to our problem by

V n(r) = ess inf ess sup

τ∈Γ n σ∈Γ n P[g(τ , σ)≤ r|F n],

V n(r) = ess sup ess inf

σ∈Γ n τ∈Γ n P[g(τ , σ)≤ r|F n],

respectively, where P[g(τ , σ) ≤ r|F n]is a conditional probability of an event{g(τ , σ) ≤ r}

givenF n See NeveuNeveu (1975) for the definition of ess sup and ess inf We also define

sequences of minimax and maxmin values by

X n(r) =I(X n ≤r), Y n(r) =I(Y n ≤r), W n(r) =I(W n ≤r), Z(r) =I(Z≤r)

Since X n ≤ W n ≤ Y n, we see that Y n(r)≤ Wn(r)≤ Xn(r)for all r Thus our problem is just a special version of the classical Dynkin’s game for a fixed threshold value r.

We first have three propositions below for a fixed r from the result of Dynkin’s game (e.g see

NeveuNeveu (1975) and OhtsuboOhtsubo (2000)) In the following proposition, the notationmid(a, b, c)denotes the middle value among constants a, b and c For example, when a < b < c

then mid(a, b, c) =b If a < b, mid(a, b, c) =max(a, min(b, c)) =min(b, max(a, c))

Proposition 3.1. Let r be arbitrary.

(a) For each n ∈ N, V n(r) =V n(r), say V n(r), and v n(r) =v n(r) =E[V n(r)], say v n(r).

(b)(V n(r))is the unique sequence of random variables satisfying the equalities

V n=mid(Xn(r), Y n(r), E[V n+1 |F n]), n ∈ N and the inequalities

X n(r)≤ V n ≤ Yn(r), n ∈ N, where(Xn(r))is the largest submartingale dominated by min(Xn(r), E[Z(r)|F n])and(Yn(r))is the smallest supermartingale dominating max(Yn(r), E[Z(r)|F n]), that is,

On the other hand, many authors propose a variety of criteria and investigate Markov

deci-sion processes for their criteria, instead of standard criteria, that is, the expected discounted

total reward and the average expected reward per unit (see WhiteWhite (1988) for survey)

Especially, WhiteWhite (1993), Wu and LinWu & Lin (1999), Ohtsubo and ToyonagaOhtsubo

& Toyonaga (2002) and OhtsuboOhtsubo (2004) consider a problem in which we minimize a

threshold probability Such a problem is called risk minimizing problem and is available for

applications to the percentile of the losses or Value-at-Risk (VaR) in finance (e.g FilarFilar et

al (1995) and UryasevUryasev (2000))

In this paper we consider Dynkin’s game with a threshold probability in a random sequence

In Section 3 we characterize a value function of game and optimal stopping times for both

players and show that the value function of game has properties of a distribution function

with respect to a threshold value except a right continuity In Section 4 we investigate an

independent model, as applications of our game, and we explicitly find a value function which

is right continuous and optimal stopping times for both players

2 Formulation of problem

Let(Ω,F , P)be a probability space and(F n)n∈N an increasing family of sub-σ-fields of F,

where N = {0, 1, 2,· · · } is a discrete time space Let X = (X n)n∈N , Y = (Y n)n∈N , W =

(W n)n∈Nbe sequences of random variables defined on(Ω,F , P)and adapted to(F n)such

that X n ≤ W n ≤ Y n almost surely (a.s.) for all n ∈ N and P(supn X+

n +supn Y n − <∞) = 1,

where x+ = max(0, x) and x − = (−x)+ The second assumption holds if random

vari-ables supn X+

n and supn Y n −are integrable, which are standard conditions given in the

classi-cal Dynkin’s game Also let Z be an arbitrary integrable random variable on(Ω,F , P) For

each n ∈ N, we denote by Γ nthe class of(F n)–stopping times τ such that τ ≥ n a s

We consider the following zero-sum stopping game There are two players and the first and

the second players choose stopping times τ and σ in Γ0, respectively Then the reward paid to

the first player from the second is equal to

g(τ , σ) =X τ I (τ<σ)+Y σ I (σ<τ)+W τ I (τ=σ<∞)+ZI (τ=σ=∞),

where I A is the indicator function of a set A in F In the classical Dynkin’s game the aim of the

first player is to maximize the expected gain E[g(τ , σ)]with respect to τ ∈Γ0and that of the

second is to minimize this expectation with respect to σ ∈Γ0 In our problem the objective of

the first player is to minimize the threshold probability P[g(τ , σ)≤ r]with respect to τ ∈Γ0

and the second maximizes the probability with respect to σ ∈Γ0for a given threshold value

r.

We can define processes of minimax and maxmin values corresponding to our problem by

V n(r) = ess inf ess sup

τ∈Γ n σ∈Γ n P[g(τ , σ)≤ r|F n],

V n(r) = ess sup ess inf

σ∈Γ n τ∈Γ n P[g(τ , σ)≤ r|F n],

respectively, where P[g(τ , σ) ≤ r|F n]is a conditional probability of an event{g(τ , σ) ≤ r}

givenF n See NeveuNeveu (1975) for the definition of ess sup and ess inf We also define

sequences of minimax and maxmin values by

X n(r) =I(X n ≤r), Y n(r) = I(Y n ≤r), W n(r) =I(W n ≤r), Z(r) =I(Z≤r)

Since X n ≤ W n ≤ Y n, we see that Y n(r)≤ Wn(r)≤ Xn(r)for all r Thus our problem is just a special version of the classical Dynkin’s game for a fixed threshold value r.

We first have three propositions below for a fixed r from the result of Dynkin’s game (e.g see

NeveuNeveu (1975) and OhtsuboOhtsubo (2000)) In the following proposition, the notationmid(a, b, c)denotes the middle value among constants a, b and c For example, when a < b < c

then mid(a, b, c) =b If a < b, mid(a, b, c) =max(a, min(b, c)) =min(b, max(a, c))

Proposition 3.1. Let r be arbitrary.

(a) For each n ∈ N, V n(r) =V n(r), say V n(r), and v n(r) =v n(r) =E[V n(r)], say v n(r).

(b)(V n(r))is the unique sequence of random variables satisfying the equalities

V n=mid(Xn(r), Y n(r), E[V n+1 |F n]), n ∈ N and the inequalities

X n(r)≤ V n ≤ Yn(r), n ∈ N, where(Xn(r))is the largest submartingale dominated by min(Xn(r), E[Z(r)|F n])and(Yn(r))is the smallest supermartingale dominating max(Yn(r), E[Z(r)|F n]), that is,

Proposition 3.2. Let r be arbitrary For each k, n : k ≥ n, γ k

n(r)≥ γ k+1 n (r)and for each n ∈ N,

limk→∞ γ k n(r) =Xn(r).

For k ≥ n, let

β k k(r) =Xk(r),

β k n(r) =mid(Xn(r), Y n(r), E[β k n+1(r)|F n]), n < k,

Proposition 3.3. Let r be arbitrary For each k ≥ n, β k

n(r)≤ β k+1 n and for each n, lim k→∞ β k n(r) =

V n(r).

Theorem 3.1 For eachn, V n(·) has properties of a distribution function on R except for the

right continuity

Proof We first notice that  Z(r) =I(Z≤r) is a nondecreasing function in r From the definition

of a conditional expectation and the dominated convergence theorem, E[Z(r)|F k]for each k

is also nondecreasing at r Since  X k(r) =I(X k ≤r) is nondecreasing at r for each k ∈N, we see

that γ k

k(r) =min(Xk(r), E[Z(r)|F k])is a nondecreasing function in r By induction, γ k

n(r)is

nondecreasing in r for each k ≥ n Since a sequence {γ k n(r)}∞k=nof functions is nonincreasing

and X n(r) = limk→∞ γ k n(r), it follows that β n(r) = Xn(r)is nondecreasing for each n

Sim-ilarly, it follows by induction that β k

n(r)is nondecreasing at r for each n ≤ k, since  Y n(r)is

nondecreasing at r From Proposition 2.3, the monotonicity of a sequence {β k n(r)}∞k=nimplies

that V n(r) =limk→∞ β k n(r)is a nondecreasing function in r.

Next, since we have V n(r) ≤ Xn(r)and we see that X n(r) = I(X n ≤r) = 0 for a sufficiently

small r, it follows that lim r→−∞ V n(r) =0 Similarly, we see that limr→∞ V n(r) =1, since we

have V n(r) ≥ Yn(r)and we see that Y n(r) =1 for a sufficiently large r Thus this theorem is

completely proved

We give an example below in which the value function V n(r)is not right continuous at some

r.

Example 3.1 LetX n=W n=− 1, Y n=1/n for each n and let Z=1 We shall obtain the value

function V n(r) by Propositions 3.2 and 3.3 Since X k(r) = I[−1,∞)(r)and Z(r) = I[1,∞)(r),

we have γ k

k(r) = I[1,∞)(r) By induction, we easily see that γ k

n(r) = I[1,∞)(r)for each k ≥ n and hence β n(r) = Xn(r) = limk→∞ γ k n = I[1,∞)(r) Next, since Y k−1(r) = I[1/(k−1),∞)(r),

We shall consider an independent sequences as a special model Let

(W n)n∈N be a sequence of independent distributed random variables with

P(supn |W n | < ∞) = 1, and let Z be a random variable which is independent of(W n)n∈N

For each n ∈ N let F n be the σ-field generated by {W k ; k ≤ n} Also, for each n ∈ N, let

X n=W n − c and Y n=W n+d, where c and d are positive constants.

SinceF nis independent of{W k ; k > n}, the relation in Proposition 3.1 (b) is represented as

Example 4.1 LetW be a uniformly distributed random variable on an interval[0, 1]and

assume that W n has the same distribution as W for all n ∈ N and that 0 < c, d <1/2 Thensince(W n)n∈N is a sequence of independently and identically distributed random variables,

V n(r)does not depend on n Hence, letting V(r) =V n(r), n ∈ N and v(r) =E[V(r)], we have

V(r) =mid(I(W≤r+c) , I(W≤r−d) , v(r))

When W < r − d, we have I(W≤r+c)=I(W≤r−d)=1, so V(r) =1 When W ≥ r+c, we have

V(r) =0, since I(W≤r+c)=I(W≤r−d)=0 Thus we obtain

V(r) = I(W≤r−d)+v(r)I(r−d≤W<r+c).Taking the expectation on the both sides, we see that

v(r) =P(W ≤ r − d) +v(r)P(r − d ≤ W < r+c)

If r < d then we have v(r) =v(r)P(0≤ W < r+c) Since r < d <1/2<1− c, P(0≤ W <

r+c ) < 1 and hence v(r) = 0 If d ≤ r <1− c, then we obtain v(r) = (r − d)/(1− c − d),

since P(W ≤ r − d) =r − d and P(r − d ≤ W < r+c) =c+d Similarly, if r ≥1− c then we have v(r) =1 Thus it follows that

E[Y(r)] =Y(r) =Yn(r) =P[Z ≤ r]I(−∞,d)(r) +I[d,∞)(r)

Now v(r)is a distribution function in r Let U is a random variable corresponding to v(r)

Then we see that E[U] = (1− c+d))/2

We shall next compare our model with the classical Dynkin’s game in this example Let

J n= ess inf ess sup

τ ∈Γ n σ ∈Γ n

E[g(τ , σ)|F n],

J n= ess sup ess inf

σ∈Γ n τ∈Γ n E[g(τ , σ)|F n],

Proposition 3.2. Let r be arbitrary For each k, n : k ≥ n, γ k

n(r)≥ γ k+1 n (r)and for each n ∈ N,

limk→∞ γ k n(r) =Xn(r).

For k ≥ n, let

β k k(r) =Xk(r),

β k n(r) =mid(Xn(r), Y n(r), E[β k n+1(r)|F n]), n < k,

Proposition 3.3. Let r be arbitrary For each k ≥ n, β k

n(r)≤ β k+1 n and for each n, lim k→∞ β k n(r) =

V n(r).

Theorem 3.1 For eachn, V n(·) has properties of a distribution function on R except for the

right continuity

Proof We first notice that  Z(r) =I(Z≤r) is a nondecreasing function in r From the definition

of a conditional expectation and the dominated convergence theorem, E[Z(r)|F k]for each k

is also nondecreasing at r Since  X k(r) =I(X k ≤r) is nondecreasing at r for each k ∈N, we see

that γ k

k(r) =min(Xk(r), E[Z(r)|F k])is a nondecreasing function in r By induction, γ k

n(r)is

nondecreasing in r for each k ≥ n Since a sequence {γ n k(r)}∞k=nof functions is nonincreasing

and X n(r) =limk→∞ γ k n(r), it follows that β n(r) = Xn(r)is nondecreasing for each n

Sim-ilarly, it follows by induction that β k

n(r)is nondecreasing at r for each n ≤ k, since  Y n(r)is

nondecreasing at r From Proposition 2.3, the monotonicity of a sequence {β k n(r)}∞k=nimplies

that V n(r) =limk→∞ β k n(r)is a nondecreasing function in r.

Next, since we have V n(r) ≤ Xn(r)and we see that X n(r) = I(X n ≤r) = 0 for a sufficiently

small r, it follows that lim r→−∞ V n(r) =0 Similarly, we see that limr→∞ V n(r) =1, since we

have V n(r) ≥ Yn(r)and we see that Y n(r) =1 for a sufficiently large r Thus this theorem is

completely proved

We give an example below in which the value function V n(r)is not right continuous at some

r.

Example 3.1 LetX n=W n=− 1, Y n=1/n for each n and let Z=1 We shall obtain the value

function V n(r)by Propositions 3.2 and 3.3 Since X k(r) = I[−1,∞)(r)and Z(r) = I[1,∞)(r),

we have γ k

k(r) = I[1,∞)(r) By induction, we easily see that γ k

n(r) = I[1,∞)(r)for each k ≥ n and hence β n(r) = Xn(r) = limk→∞ γ k n = I[1,∞)(r) Next, since Y k−1(r) = I[1/(k−1),∞)(r),

We shall consider an independent sequences as a special model Let

(W n)n∈N be a sequence of independent distributed random variables with

P(supn |W n | < ∞) = 1, and let Z be a random variable which is independent of(W n)n∈N

For each n ∈ N let F n be the σ-field generated by {W k ; k ≤ n} Also, for each n ∈ N, let

X n=W n − c and Y n=W n+d, where c and d are positive constants.

SinceF nis independent of{W k ; k > n}, the relation in Proposition 3.1 (b) is represented as

Example 4.1 LetW be a uniformly distributed random variable on an interval[0, 1]and

assume that W n has the same distribution as W for all n ∈ N and that 0 < c, d <1/2 Thensince(W n)n∈N is a sequence of independently and identically distributed random variables,

V n(r)does not depend on n Hence, letting V(r) =V n(r), n ∈ N and v(r) =E[V(r)], we have

V(r) =mid(I(W≤r+c) , I(W≤r−d) , v(r))

When W < r − d, we have I(W≤r+c)=I(W≤r−d)=1, so V(r) =1 When W ≥ r+c, we have

V(r) =0, since I(W≤r+c)=I(W≤r−d)=0 Thus we obtain

V(r) =I(W≤r−d)+v(r)I(r−d≤W<r+c).Taking the expectation on the both sides, we see that

v(r) =P(W ≤ r − d) +v(r)P(r − d ≤ W < r+c)

If r < d then we have v(r) =v(r)P(0≤ W < r+c) Since r < d <1/2<1− c, P(0≤ W <

r+c ) < 1 and hence v(r) = 0 If d ≤ r <1− c, then we obtain v(r) = (r − d)/(1− c − d),

since P(W ≤ r − d) =r − d and P(r − d ≤ W < r+c) =c+d Similarly, if r ≥1− c then we have v(r) =1 Thus it follows that

E[Y(r)] =Y(r) =Yn(r) =P[Z ≤ r]I(−∞,d)(r) +I[d,∞)(r)

Now v(r)is a distribution function in r Let U is a random variable corresponding to v(r)

Then we see that E[U] = (1− c+d))/2

We shall next compare our model with the classical Dynkin’s game in this example Let

J n= ess inf ess sup

τ ∈Γ n σ ∈Γ n

E[g(τ , σ)|F n],

J n= ess sup ess inf

σ∈Γ n τ∈Γ n E[g(τ , σ)|F n],

be minimax and maxmin value processes, respectively Then we have J n= J n=J, say, since

J n=J n does not depend upon n in this example Also, by solving the relation

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to a disorder problem Stochastics, Vol.3, 61–71.

Chow, Y S.; Robbins, H & Siegmund, D (1971) Great Expectations: The Theory of Optimal

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conditional probabilities defined by Hausdorff outer and inner measures 87

Stochastic independence with respect to upper and lower conditional probabilities defined by Hausdorff outer and inner measures

Serena Doria

0

Stochastic independence with respect to upper

and lower conditional probabilities defined

by Hausdorff outer and inner measures

Serena Doria

University G.d’Annunzio

Italy

1 Introduction

A new model of coherent upper conditional prevision is proposed in a metric space It is

defined by the Choquet integral with respect to the s-dimensional Hausdorff outer measure

if the conditioning event has positive and finite Hausdorff outer measure in its dimension s.

Otherwise if the conditioning event has Hausdorff outer measure in its dimension equal to

zero or infinity it is defined by a 0-1 valued finitely, but not countably, additive probability

If the conditioning event has positive and finite Hausdorff outer measure in its dimension the

coherent upper conditional prevision is proven to be monotone, comonotonically additive,

submodular and continuous from below

Given a coherent upper conditional prevision the coherent lower conditional prevision is

de-fined as its conjugate

In Doria (2007) coherent upper and lower conditional probablities are obtained when only 0-1

valued random variables are considered

The aim of this chapter is to introduce a new definition of stochastic independence with

re-spect to coherent upper and lower conditional probabilities defined by Hausdorff outer and

inner measures

A concept related to the definition of conditional probability is stochastic independence In

a continuous probability space where probability is usually assumed equal to the Lebesgue

measure, we have that finite, countable and fractal sets (i.e the sets with non-integer

Haus-dorff dimension) have probability equal to zero For these sets the standard definition of

independence given by the factorization property is always satisfied since both members of

the equality are zero

The notion of s-independence with respect to Hausdorff outer and inner measures is

intro-duced to check probabilistic dependence for sets with probability equal to zero, which are

always independent according to the standard definition given by the factorization property

Moreover s-independence is compared with the notion of epistemic independence with

re-spect to upper and lower conditional probabilities (Walley, 1991)

The outline of the chapter is the following

In Section 2 it is proven that a conditional prevision defined by the Radon-Nikodym derivative

may be not coherent and examples are given

6

In Section 3 coherent upper conditional previsions are defined in a metric space by the

Cho-quet integral with respect to Hausdorff outer measure if the conditioning event has positive

and finite Hausdorff outer measure in its dimension Otherwise they are defined by a 0-1

valued finitely, but not countably, additive probability Their properties are proven

In Section 4 the notion of s-irrelevance and s-independence with respect to coherent upper

and lower conditional probabilities defined by Hausdorff outer and inner measures are

intro-duced It is proven that the notions of epistemic irrelevance and s-irrelevance are not always

related In particular we give conditions for which an event B is epistemically irrelevant to an

event A, but it is not s-irrelevant In the Euclidean metric space it is proven that a necessary

condition for s-irrelevance between events is that the Hausdorff dimension of the two events

and their intersection is equal to the Hausdorff dimension of Ω Finally sufficient conditions

for s-irrelevance between Souslin subsets of nare given

In Section 5 some fractal sets are proven to be s-dependent since they do not satisfy the

neces-sary condition for s-independence In particular the attractor of a finite family of similitudes

and its boundary are proven to be s-dependent if the open set condition holds Moreover a

condition for which two middle Cantor sets are s-dependent is given

It is important to note that all these sets are stochastically independent according the axiomatic

definition given by the factorization property if probability is defined by the Lebesgue

mea-sure

In Section 6 curves filling the space, such as Peano curve and Hilbert curve are proven to be

s-independent

2 Conditional expectation and coherent conditional prevision

Partial knowledge is a natural interpretation of conditional probability This interpretation

can be formalized in a different way in the axiomatic approach and in the subjective approach

where conditional probability is respectively defined by the Radon-Nikodym derivative or by

the axioms of coherence In both cases conditional probability is obtained as the restriction of

conditional expectation or conditional prevision to the class of indicator functions of events

Some critical situations, which highlight as the axiomatic definition of conditional probability

is not always a useful tool to represent partial knoweledge, are proposed in literature and

ana-lyzed in this section In particular the role of the Radon-Nikodym derivative in the assessment

of a coherent conditional prevision is investigated

It is proven that, every time that the σ-field of the conditioning events is properly contained in

the σ-field of the probability space and it contains all singletons, the Radon-Nikodym

deriva-tive cannot be used as a tool to define coherent conditional previsions This is due to the fact

that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable

with respect to the σ-field of the conditioning events, contradicts a necessary condition for the

coherence

Analysis done points out the necessity to introduce a different tool to define coherent

condi-tional previsions

2.1 Conditional expectation and Radon-Nikodym derivative

In the axiomatic approach Billingsley (1986) conditional expectation is defined with respect

to a σ-field G of conditioning events by the Radon-Nikodym derivative Let(Ω, F, P) be a

probability space and let F and G be two σ-fields of subsets of Ω with G contained in F and let

X be an integrable random variable on(Ω, F, P) Let P be a probability measure on F; define

a measure ν on G by ν(G)=G XdP This measure is finite and absolutely continuous with

respect to P So there exists a function, the Radon-Nikodym derivative denoted by E[X|G],

defined on Ω, G-measurable, integrable and satisfying the functional equation

If X is the indicator function of any event A belonging to F then E[X|G] =E[A|G] =P[A|G]

is a version of the conditional probability

Conditional probability can be used to represent partial information (Billingsley, 1986, Section33)

A probability space(Ω, F, P)can be use to represent a random phenomenon or an experiment

whose outcome is drawn from according to the probability given by P Partial information

about the experiment can be represented by a sub σ-field G of F in the following way: an

observer does not know which ω has been drawn but he knows for each H ∈ G, if ω belongs

to H or if ω belongs to H c A sub σ-field G of F can be identified as partial information about the random experiment, and, fixed A in F, conditional probability can be used to represent partial knowledge about A given the information on G If conditional probability is defined

by the Radon-Nykodim derivative, denoted by P[A|G], by the standard definition

(Billings-ley, 1986, p.52) we have that an event A is independent from the σ-field G if it is independent

from each H ∈ G, that isP[A|G] =P(A)with probability 1 In (Billingsley, 1986, Example33.11) it is shown that the interpretation of conditional probability in terms of partial knowl-

edge breaks down in certain cases Let Ω = [0,1], let F be the Borel σ-field of [0,1] and let

P be the Lebesgue measure on F Let G be the sub σ-field of sets that are either countable

or co-countable Then P(A)is a version of the conditional probability P[A|G]define by the

Radon-Nikodym derivative because P(G)is either 0 or 1 for evey G ∈ G So an eventA is

independent from the information represented by G and this is a contradiction according to the fact that the information represented by G is complete since G contains all the singletons

of Ω

2.2 Coherent upper conditional previsions

In the subjective probabilistic approach (de Finetti 1970, Dubins 1975 and Walley 1991)

coher-ent upper conditional previsions P(·|B)are functionals, defined on a linear space of boundedrandom variables, satisfying the axioms of coherence

In Walley (1991) coherent upper conditional previsions are defined when the conditioningevents are sets of a partition

Definition 1. Let Ω be a non-empty set let B be a partition of Ω For every B ∈ B let K(B)be a linear space of bounded random variables defined on B Then separately coherent upper conditional previsions are functionals P(·|B)defined on K(B), such that the following conditions hold for every X and Y in

K(B)and every strictly positive constant λ:

• 1) P (X|B) ≤ sup(X|B);

• 2) P(λ X|B) = λ P(X|B)(positive homogeneity);

• 3) P(X+Y)|B)≤ P(X|B) +P(Y|B);

• 4) P(B|B) =1.

conditional probabilities defined by Hausdorff outer and inner measures 89

In Section 3 coherent upper conditional previsions are defined in a metric space by the

Cho-quet integral with respect to Hausdorff outer measure if the conditioning event has positive

and finite Hausdorff outer measure in its dimension Otherwise they are defined by a 0-1

valued finitely, but not countably, additive probability Their properties are proven

In Section 4 the notion of s-irrelevance and s-independence with respect to coherent upper

and lower conditional probabilities defined by Hausdorff outer and inner measures are

intro-duced It is proven that the notions of epistemic irrelevance and s-irrelevance are not always

related In particular we give conditions for which an event B is epistemically irrelevant to an

event A, but it is not s-irrelevant In the Euclidean metric space it is proven that a necessary

condition for s-irrelevance between events is that the Hausdorff dimension of the two events

and their intersection is equal to the Hausdorff dimension of Ω Finally sufficient conditions

for s-irrelevance between Souslin subsets of nare given

In Section 5 some fractal sets are proven to be s-dependent since they do not satisfy the

neces-sary condition for s-independence In particular the attractor of a finite family of similitudes

and its boundary are proven to be s-dependent if the open set condition holds Moreover a

condition for which two middle Cantor sets are s-dependent is given

It is important to note that all these sets are stochastically independent according the axiomatic

definition given by the factorization property if probability is defined by the Lebesgue

mea-sure

In Section 6 curves filling the space, such as Peano curve and Hilbert curve are proven to be

s-independent

2 Conditional expectation and coherent conditional prevision

Partial knowledge is a natural interpretation of conditional probability This interpretation

can be formalized in a different way in the axiomatic approach and in the subjective approach

where conditional probability is respectively defined by the Radon-Nikodym derivative or by

the axioms of coherence In both cases conditional probability is obtained as the restriction of

conditional expectation or conditional prevision to the class of indicator functions of events

Some critical situations, which highlight as the axiomatic definition of conditional probability

is not always a useful tool to represent partial knoweledge, are proposed in literature and

ana-lyzed in this section In particular the role of the Radon-Nikodym derivative in the assessment

of a coherent conditional prevision is investigated

It is proven that, every time that the σ-field of the conditioning events is properly contained in

the σ-field of the probability space and it contains all singletons, the Radon-Nikodym

deriva-tive cannot be used as a tool to define coherent conditional previsions This is due to the fact

that one of the defining properties of the Radon-Nikodym derivative, that is to be measurable

with respect to the σ-field of the conditioning events, contradicts a necessary condition for the

coherence

Analysis done points out the necessity to introduce a different tool to define coherent

condi-tional previsions

2.1 Conditional expectation and Radon-Nikodym derivative

In the axiomatic approach Billingsley (1986) conditional expectation is defined with respect

to a σ-field G of conditioning events by the Radon-Nikodym derivative Let(Ω, F, P) be a

probability space and let F and G be two σ-fields of subsets of Ω with G contained in F and let

X be an integrable random variable on(Ω, F, P) Let P be a probability measure on F; define

a measure ν on G by ν(G)=G XdP This measure is finite and absolutely continuous with

respect to P So there exists a function, the Radon-Nikodym derivative denoted by E[X|G],

defined on Ω, G-measurable, integrable and satisfying the functional equation

If X is the indicator function of any event A belonging to F then E[X|G] =E[A|G] =P[A|G]

is a version of the conditional probability

Conditional probability can be used to represent partial information (Billingsley, 1986, Section33)

A probability space(Ω, F, P)can be use to represent a random phenomenon or an experiment

whose outcome is drawn from according to the probability given by P Partial information

about the experiment can be represented by a sub σ-field G of F in the following way: an

observer does not know which ω has been drawn but he knows for each H ∈ G, if ω belongs

to H or if ω belongs to H c A sub σ-field G of F can be identified as partial information about the random experiment, and, fixed A in F, conditional probability can be used to represent partial knowledge about A given the information on G If conditional probability is defined

by the Radon-Nykodim derivative, denoted by P[A|G], by the standard definition

(Billings-ley, 1986, p.52) we have that an event A is independent from the σ-field G if it is independent

from each H ∈ G, that isP[A|G] =P(A)with probability 1 In (Billingsley, 1986, Example33.11) it is shown that the interpretation of conditional probability in terms of partial knowl-

edge breaks down in certain cases Let Ω = [0,1], let F be the Borel σ-field of [0,1] and let

P be the Lebesgue measure on F Let G be the sub σ-field of sets that are either countable

or co-countable Then P(A)is a version of the conditional probability P[A|G]define by the

Radon-Nikodym derivative because P(G)is either 0 or 1 for evey G ∈ G So an eventA is

independent from the information represented by G and this is a contradiction according to the fact that the information represented by G is complete since G contains all the singletons

of Ω

2.2 Coherent upper conditional previsions

In the subjective probabilistic approach (de Finetti 1970, Dubins 1975 and Walley 1991)

coher-ent upper conditional previsions P(·|B)are functionals, defined on a linear space of boundedrandom variables, satisfying the axioms of coherence

In Walley (1991) coherent upper conditional previsions are defined when the conditioningevents are sets of a partition

Definition 1. Let Ω be a non-empty set let B be a partition of Ω For every B ∈ B let K(B)be a linear space of bounded random variables defined on B Then separately coherent upper conditional previsions are functionals P(·|B)defined on K(B), such that the following conditions hold for every X and Y in

K(B)and every strictly positive constant λ:

• 1) P (X|B) ≤ sup(X|B);

• 2) P(λ X|B) = λ P(X|B)(positive homogeneity);

• 3) P(X+Y)|B)≤ P(X|B) +P(Y|B);

• 4) P(B|B) =1.

Coherent conditional upper previsions can always be extended to coherent upper previsions

on the class L(B)of all bounded random variables defined on B.

Suppose that P(X|B) is a coherent upper conditional prevision on K then its conjugate

coher-ent lower conditional prevision is defined by P(−X|B) =−P(X|B) If for every X belonging

to K we have P(X|B) =P(X|B) =P(X|B) then P(X|B)is called a coherent linear conditional

prevision de Finetti (1970) and it is a linear positive functional on K.

Definition 2. Let Ω be a non-empty set let B be a partition of Ω For every B ∈ B let K(B)be a

linear space of bounded random variables defined on B Then linear coherent conditional previsions are

functionals P(·|B)defined on K(B), such that the following conditions hold for every X and Y in K(B)

and every strictly positive constant λ:

In Dubins (1975) coherent conditional probabilities are defined when the family of the

condi-tioning events is a field of subsets of Ω

Definition 3. Let Ω be a non-empty set and let F and G be two fields of subsets of Ω , with G ⊆ F P

is a finitely additive conditional probability on(F, G)if it is a real function defined on F × G0, where

G0=G −  such that the following conditions hold:

• I) given any H ∈ G0 and A1, , An ∈ F and A i ∩ A j = for i = j, the function P(·|H)

defined on F is such that P(A|H)≥ 0, P(n k=1 A k |H) =∑n k=1 P(A k |H), P(Ω|H) =1

• II) P(H |H) =1 if H ∈ G0

• III) given E ∈ F, H ∈ F with A ∈ G0and EA ∈ G0then P(EH|A) =P(E|A)P(H |EA).

From conditions I) and II) we have

II’) P(A|H) =1 if A ∈F,H ∈G0and H ⊂ A.

These conditional probabilities are coherent in the sense of de Finetti, since conditions I), II),

III) are sufficient for the coherence of P on C=F×G0when F and G are fields of subsets of Ω

with G⊆F or G is an additive subclass of F; otherwise if F and G are two arbitrary families

of subsets of Ω, such that Ω∈ F the previous conditions are necessary for the coherence but

not sufficient

2.3 Coherent conditional previsions and the Radon-Nikodym derivative

In this subsection the role of the Radon-Nikodym derivative in the assessment of a coherent

conditional prevision is analyzed

The definitions of conditional expectation and coherent linear conditional prevision can be

compared when the σ-field G is generated by the partition B Let G be equal or contained in

the σ-field generated by a countable class C of subsets of F and let B be the partition generated

by the class C Denote Ω’ = B and ϕ B the function from Ω to Ω’ that associates to every ω ∈Ω

the atom B of the partition B that contains ω; then we have that P(A|G) =P(A|B)◦ ϕ B for

every A ∈F (Koch, 1997, 262).

The next theorem shows that every time that the σ-field G of the conditioning events is

prop-erly contained in F and it contains all singletons of[0, 1]then the conditional prevision, fined by the Radon-Nikodym derivative is not coherent It occurs because one of the defining

de-properties of conditional expectation that is to be measurable with respect to the σ-field of

conditioning events contradicts a necessary condition for coherence of a linear conditional

prevision A bounded random variable is called B-measurable or measurable with respect to

the partition B (Walley, 1991, p.291) if it is constant on the atoms B of the partition If for every

B belonging to B P(X|B) are coherent linear conditional previsions and X is B-measurable

then P(X|B) =X (Walley, 1991, p.292) This necessary condition for coherence is not always satisfied if P(X|B)is defined by the Radon-Nikodym derivative

Theorem 1. Let Ω = [0,1], let F be the Borel σ-field of [0,1] and let P be the Lebesgue measure on F Let

G be a sub σ-field properly contained in F and containing all singletons of [0,1] Let B be the partition

of all singletons of [0,1] and let X be the indicator function of an event A belonging to F - G If we

define the conditional prevision P(X| { ω }) equal to the Radon-Nikodym derivative with probability 1, that is

P(X | { ω}) = E[X|G]

except on a subset N of [0,1] of P-measure zero, then the conditional prevision P(X | { ω}) is not coherent.

Proof If the equality P(X | { ω}) = E[X|G]holds with probability 1, then we have that, with

probability 1, the linear conditional prevision P(X| { ω }) is different from X, the indicator

function of A; in fact having fixed A in F −G the indicator functionX is not G-measurable,

it does not verify a property of the Radon-Nikodym derivative and therefore it cannot beassumed as conditional expectation according to axiomatic definition So the linear con-

ditional prevision P(X | { ω}) does not satisfy the necessary condition for being coherent,

P(X | { ω}) = X for every singleton {ω}of G.

Example 1 (Billingsley, 1986, Example 33.11) Let Ω = [0,1], let F be the Borel σ-field of Ω, let P

be the Lebesgue measure on F and let G be the sub σ-field of F of sets that are either countable

or co-countable Let B be the partition of all singletons of Ω; if the linear conditional prevision

is defined equal, with probability 1, to conditional expectation defined by the Radon-Nikodymderivative, we have that

P(X|B) =E[X|G] =P(X)

So when X is the indicator function of an event A= [a, b]with 0< a < b < 1 then P(X|B) =

P(A)and it does not satisfy the necessary condition for coherence that is P(X | { ω}) = X for

every singleton{ ω }of G.

Evident from Theorem 1 and Example 1 is the necessity to introduce a new tool to definecoherent linear conditional previsions

3 Coherent upper conditional previsions defined by Hausdorff outer measures

In this section coherent upper conditional previsions are defined by the Choquet integral with

respect to Hausdorff outer measures if the conditioning event B has positive and finite dorff outer measure in its dimension Otherwise if the conditioning event B has Hausdorff

Haus-outer measure in its dimension equal to zero or infinity they are defined by a 0-1 valuedfinitely, but not countably, additive probability