aa r X i v : . [ m a t h . P R ] J a n Game-theoretic Brownian motion
Vladimir Vovk vovk @ cs.rhul.ac.ukhttp://vovk.net October 29, 2018
Abstract
This paper suggests a perfect-information game, along the lines ofL´evy’s characterization of Brownian motion, that formalizes the processof Brownian motion in game-theoretic probability. This is perhaps thesimplest situation where probability emerges in a non-stochastic environ-ment.
This paper is part of the recent revival of interest (see, e.g., [4, 14, 15, 10, 6, 9])in game-theoretic probability. It further develops the game-theoretic approachto continuous-time processes along the lines of the papers [16, 18, 19]. Unlikethose papers, which only demonstrate the emergence of randomness-type prop-erties in a continuous trading protocol, this paper gives an example where afull-fledged probability measure emerges (although in a significantly more re-strictive protocol). Only the very simple case of “game-theoretic Brownianmotion” is considered, but we can expect that probabilities will also emerge inless restrictive protocols.The words such as “positive”, “negative”, “before”, and “after” will be un-derstood in the wide sense of ≥ or ≤ , as appropriate; when necessary, we willadd the qualifier “strictly”.The latest version of this working paper can be downloaded from the website http://probabilityandfinance.com . We consider a perfect-information game between two players, Reality and Scep-tic. Reality chooses a continuous function ω : [0 , ∞ ) → R with ω (0) = 0, butbefore she announces her choice Sceptic chooses his strategy of trading in twosecurities, one with the price process ω ( t ), t ∈ [0 , ∞ ), and the other with theprice process ω ( t ) − t , t ∈ [0 , ∞ ). This game, which we call the L´evy game (after [11], Theorem 18.6), is formalized as follows.1et Ω be the set of all continuous functions ω : [0 , ∞ ) → R with ω (0) = 0.For each t ∈ [0 , ∞ ), F t is defined to be the σ -algebra generated by the functions ω ∈ Ω ω ( s ), s ∈ [0 , t ], and F ∞ := ∨ t F t ; we will often write F for F ∞ .A process S is a family of functions S t : Ω → [ −∞ , ∞ ], t ∈ [0 , ∞ ), each S t being F t -measurable; we only consider processes with lower continuous (oftencontinuous) sample paths t S t ( ω ). An event is an element of the σ -algebra F .Stopping times τ : Ω → [0 , ∞ ] w.r. to the filtration ( F t ) and the corresponding σ -algebras F τ are defined as usual. We simplify ω ( τ ( ω )) and S τ ( ω ) ( ω ) to ω ( τ )and S τ ( ω ), respectively; the argument ω will often be omitted in other cases aswell.The class of allowed strategies for Sceptic is defined in two steps. An ele-mentary betting strategy G consists of an increasing sequence of stopping times τ ≤ τ ≤ · · · and, for each n = 1 , , . . . , a pair of bounded F τ n -measurable func-tions, M n and V n . It is required that, for any ω ∈ Ω, lim n →∞ τ n ( ω ) = ∞ . Tosuch G and an initial capital c ∈ R corresponds the elementary capital process K G,ct ( ω ) := c + ∞ X n =1 (cid:18) M n ( ω ) (cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) + V n ( ω ) (cid:16)(cid:0) ω ( τ n +1 ∧ t ) − ( τ n +1 ∧ t ) (cid:1) − (cid:0) ω ( τ n ∧ t ) − ( τ n ∧ t ) (cid:1)(cid:17)(cid:19) ,t ∈ [0 , ∞ ) (1)(with the zero terms in the sum ignored). The numbers M n ( ω ) and V n ( ω ) willbe called Sceptic’s stakes (on ω ( t ) and ω ( t ) − t , respectively) chosen at time τ n , and K G,ct ( ω ) will sometimes be referred to as Sceptic’s capital at time t ; wemay also say that Sceptic bets M n ( ω ) on ω ( t ) and V n ( ω ) on ω ( t ) − t at time τ n . (We are following standard probability textbooks, such as [20], Chapter 10,in using gambling rather than financial terminology.)A positive capital process is any process S that can be represented in theform S t ( ω ) := ∞ X n =1 K G n ,c n t ( ω ) , (2)where the elementary capital processes K G n ,c n t ( ω ) are required to be positive,for all t and ω , and the positive series P ∞ n =1 c n is required to converge. The sum(2) is always positive but allowed to take value ∞ . Since K G n ,c n ( ω ) = c n doesnot depend on ω , S ( ω ) also does not depend on ω and will often be abbreviatedto S .The upper probability of a set E ⊆ Ω is defined as P ( E ) := inf (cid:8) S (cid:12)(cid:12) ∀ ω ∈ Ω : lim inf t →∞ S t ( ω ) ≥ I E ( ω ) (cid:9) , (3)where S ranges over the positive capital processes and I E stands for the indicatorfunction of E . (The lim inf t →∞ can be replaced by sup t ∈ [0 , ∞ ) in this definition:see [18], Lemma 1.) The lower probability of E ⊆ Ω is P ( E ) := 1 − P ( E c ) , E c := Ω \ E stands for the complement of E . Remark.
Our definition of a positive capital process corresponds to the intu-itive picture where Sceptic divides his initial capital into a sequence of inde-pendent accounts, with a prudent (not risking bankruptcy) elementary bettingstrategy applied to each account. On the other hand, we could make the defini-tion of upper probability more similar to the standard definition of expectationfor positive random variables: a positive capital process could be equivalentlydefined as the limit of an increasing sequence of positive elementary capitalprocesses with uniformly bounded initial capitals.
It is obvious that upper probability is countably (in particular, finitely) subad-ditive:
Lemma 1.
For any sequence of subsets E , E , . . . of Ω , P ∞ [ n =1 E n ! ≤ ∞ X n =1 P ( E n ) . Therefore, P is an outer measure in Carath´eodory’s sense. Recall that a set A ⊆ Ω is P -measurable if, for each E ⊆ Ω, P ( E ) = P ( E ∩ A ) + P ( E ∩ A c ) . (4)A standard result (see [3], Sections 1–11, or, e.g., [7], Theorem 2.1) shows thatthe family A of all P -measurable sets forms a σ -algebra and that the restrictionof P to A is a probability measure on (Ω , A ). Theorem 1.
Each event A ∈ F is P -measurable, and the restriction of P to F coincides with the Wiener measure W on (Ω , F ) . In particular, P ( A ) = P ( A ) = W ( A ) for each A ∈ F . The rest of this paper is devoted to proving this result. A capital process is a process of the form S − C , where S is a positive capitalprocess and C is a real constant. The upper expectation of a bounded functional F : Ω → R is E ( F ) := inf (cid:8) S (cid:12)(cid:12) ∀ ω ∈ Ω : lim inf t →∞ S t ( ω ) ≥ F ( ω ) (cid:9) , where S ranges over the capital processes. (This generalizes upper probability: P ( E ) = E ( I E ) for all E ⊆ Ω.) Theorem 1 will immediately follow from thefollowing result: 3 heorem 2. If F is a bounded F -measurable functional on Ω , E ( F ) = Z Ω F ( ω ) W ( dω ) . (5)Indeed, let us deduce Theorem 1 from Theorem 2. Let A ∈ F . Since theinequality ≤ in (4) follows from Lemma 1, to show that A ∈ A we are onlyrequired to show P ( E ∩ A ) + P ( E ∩ A c ) ≤ P ( E ) + ǫ (6)for each ǫ >
0. Fix such an ǫ .Let S be a positive capital process such that S < P ( E ) + ǫ and ∀ ω ∈ Ω :lim inf t →∞ S t ( ω ) ≥ I E ( ω ). Set F ( ω ) := 1 ∧ lim inf t →∞ S t ( ω ), so that F is abounded (taking values in [0 , F -measurable functional satisfying E ( F ) < P ( E ) + ǫ , I E ∩ A ≤ F I A , and I E ∩ A c ≤ F I A c (the last two inequalities follow from I E ≤ F ). Therefore, it suffices to prove E ( F I A ) + E ( F I A c ) ≤ E ( F ) . This immediately follows from F I A + F I A c = F and Theorem 2.The remaining statements of Theorem 1 are obvious corollaries of Theorem 2:for A ∈ F , P ( A ) = E ( I A ) = Z Ω I A dW = W ( A ) , P ( A ) = 1 − P ( A c ) = 1 − W ( A c ) = W ( A ) . In this and the following two sections we will prove some auxiliary results thatwill be needed in the proof of Theorem 2. This section’s results, however, alsohave considerable substantive significance: they show that our definitions are“free of contradiction”. (In fact, these definitions have been chosen to make thiseasy.)The following result says that the L´evy game is coherent , in the sense that P (Ω) = 1 (i.e., no positive capital process increases its value between time 0 and ∞ by more than a positive constant for all ω ∈ Ω).
Proposition 1. P (Ω) = 1 .Proof. If ω is generated as sample path of (measure-theoretic) Brownian mo-tion, any positive elementary capital process will be a positive continuous localmartingale (since, by the optional sampling theorem, every partial sum in (1)will be a continuous martingale), and so it suffices to apply the maximal in-equality for positive supermartingales to the partial sums corresponding to agiven positive capital process. 4he lower expectation of a bounded functional F : Ω → R is defined as E ( F ) := − E ( − F ) . Corollary 1.
For every bounded functional on Ω , E ( F ) ≤ E ( F ) .Proof. Suppose E ( F ) > E ( F ) for some F ; by the definition of E , this wouldmean that E ( F ) + E ( − F ) <
0. Since E is finitely subadditive, this wouldimply E (0) <
0, which is equivalent to P (Ω) < Corollary 2.
For every set A ⊆ Ω , P ( A ) ≤ P ( A ) . In the special case where A is P -measurable (see (4)), Proposition 1 implies P ( A ) = P ( A ).The facts that we have just established allow us to simplify our goal: (5)will follow from E ( F ) ≤ Z Ω F ( ω ) W ( dω ) . (7)Indeed, (7) implies E ( F ) = − E ( − F ) ≥ − Z Ω ( − F ( ω )) W ( dω ) = Z Ω F ( ω ) W ( dω ) , and so Corollary 1 implies E ( F ) = E ( F ) = Z Ω F ( ω ) W ( dω ) . Theorem 2 (as well as Theorem 1) also holds, and will be slightly easier to prove,for the natural modification of the L´evy game in which Sceptic is allowed to beton ω ( t ) − ω ( τ ) and ( ω ( t ) − ω ( τ )) − ( t − τ ) at any time τ . Formally, we obtainthe definition of upper probability in the modified L´evy game when we replace(1) by K G,ct ( ω ) := c + ∞ X n =1 (cid:18) M n ( ω ) (cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) + V n ( ω ) (cid:16)(cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) − (cid:0) ( τ n +1 ∧ t ) − ( τ n ∧ t ) (cid:1)(cid:17)(cid:19) ;we will say that the corresponding elementary betting strategy bets M n ( ω ) (orstakes M n ( ω ) units) on ω ( t ) − ω ( τ n ) and bets V n ( ω ) (or stakes V n ( ω ) units) on( ω ( t ) − ω ( τ n )) − ( t − τ n ) at time τ n . The rest of the definition is as before:positive capital processes are defined by (2) and upper probability is then defined5y (3). The definitions of lower probability and upper and lower expectationalso carry over to the modified L´evy game.The L´evy game and modified L´evy game are very close, as can be seen fromthe identity M n ( ω ) (cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) + V n ( ω ) (cid:16)(cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) − (cid:0) ( τ n +1 ∧ t ) − ( τ n ∧ t ) (cid:1)(cid:17) = (cid:0) M n ( ω ) − ω ( τ n ∧ t ) V n ( ω ) (cid:1)(cid:0) ω ( τ n +1 ∧ t ) − ω ( τ n ∧ t ) (cid:1) + V n ( ω ) (cid:16)(cid:0) ω ( τ n +1 ∧ t ) − ( τ n +1 ∧ t ) (cid:1) − (cid:0) ω ( τ n ∧ t ) − ( τ n ∧ t ) (cid:1)(cid:17) . (8)However, the absence of an upper bound on ω prevents us from asserting thatthe two games lead to the same notion of upper probability. (Remember thatthe stakes are required to be bounded, which is implicitly used in the proof ofProposition 1.) If there is a risk of confusion, we will write P ′ , P ′ , E ′ , and E ′ instead of P , P , E , and E , respectively, for the modified L´evy game.It is obvious that Lemma 1 continues to hold for the modified L´evy game.This is also true about Proposition 1 and Corollaries 1–2 (although this fact isnot used in this paper outside this section). As already mentioned, Theorem 1and Theorem 2 also hold for the modified L´evy game: we will prove (7) directlyfor this game, and the reduction to (7) depended on arguments of general nature,not involving the specifics of the L´evy game. The set Ω is equipped with the standard metric ρ ( ω , ω ) := ∞ X n =1 − n sup t ∈ [0 ,n ] ( | ω ( t ) − ω ( t ) | ∧ , (9)which makes Ω a complete separable metric space with Borel σ -algebra F . Inthis topology, P ′ is tight: Lemma 2.
For each α > there exists a compact set K ⊆ Ω such that P ′ ( K ) ≥ − α . For ω ∈ Ω and T ∈ (0 , ∞ ), the modulus of continuity of ω on [0 , T ] is definedas m Tδ ( ω ) := sup s,t ∈ [0 ,T ]: | s − t |≤ δ | ω ( s ) − ω ( t ) | , δ > . The proof of Lemma 2 will be based on the following result.
Lemma 3.
For each α > and T > , P ′ n ∀ δ > m Tδ ≤ α − / T / δ / o ≥ − α. (10)6f course, Theorem 1 and its counterpart for the modified L´evy game will implymuch subtler results than Lemma 3 and its counterpart for the L´evy game,such as L´evy’s modulus of continuity formula. It is interesting, however, thatthis section’s results do not require that Sceptic should be allowed to bet on ω ( t ) − ω ( τ ) (i.e., he can achieve his goal even if he is required to always choose M n ( ω ) := 0). Proof of Lemma 3.
We will use the method of [17], pp. 213–216. For each n =1 , , . . . , divide the time interval [0 , T ] into 2 n equal subintervals of length 2 − n T .Fix, for a moment, an n , and set β = β n := (cid:0) / − (cid:1) − n/ α (where 2 / − β n sum to α ) and ω i := ω ( i − n T ) , i = 0 , , . . . , n . With lower probability at least 1 − β/ n X i =1 ( ω i − ω i − ) ≤ T /β (11)(since there is a positive elementary capital process taking value T + P ji =1 ( ω i − ω i − ) − j − n T at time j − n T , j = 0 , , . . . , n , and this elementary capitalprocess will make 2 T /β at time T out of initial capital T if (11) fails to happen).For each ω ∈ Ω, define J ( ω ) := { i = 1 , . . . , n : | ω i − ω i − | ≥ ǫ } , where ǫ = ǫ ( δ, n ) will be chosen later. It is clear that | J ( ω ) | ≤ T /βǫ on the set(11). Consider the elementary betting strategy that bets 1 on ( ω ( t ) − ω ( τ )) − ( t − τ ) at each time τ ∈ [( i − − n T, i − n T ] with i ∈ J ( ω ) when | ω ( τ ) − ω i − | = ǫ for the first time during [( i − − n T, i − n T ] and gets rid of the stake at time i − n T . This strategy will make at least ǫ out of (2 T /βǫ )2 − n T provided bothevent (11) and event ∃ i ∈ { , . . . , n } : | ω i − ω i − | ≥ ǫ happen. (And we can make the corresponding elementary capital process posi-tive by allowing Sceptic to bet at most 2 T /βǫ times.) This corresponds to mak-ing at least 1 out of (2 T /βǫ )2 − n T . Solving the equation (2 T /βǫ )2 − n T = β/ ǫ = (4 T − n /β ) / . Therefore,max i =1 ,..., n | ω i − ω i − | ≤ ǫ = 2(4 T − n /β ) / = 2 / (cid:16) / − (cid:17) − / α − / T / − n/ (12)with lower probability at least 1 − β . By the countable subadditivity of upperprobability (Lemma 1), (12) holds for all n = 1 , , . . . with lower probability atleast 1 − P n β n = 1 − α . 7ntervals of the form [( i − − n T, i − n T ], for n ∈ { , , . . . } and i ∈{ , , , . . . , n } , will be called dyadic . Given an interval [ s, t ] of length at most δ > , T ], we can cover its interior (without covering any points in itscomplement) by adjacent dyadic intervals with disjoint interiors such that, forsome m ∈ { , , . . . } : there are between one and two dyadic intervals of length2 − m T ; for i = m + 1 , m + 2 , . . . , there are at most two dyadic intervals of length2 − i T (start from finding the point in [ s, t ] of the form 2 − k T with the smallestpossible k and cover ( s, − k T ] and [2 − k T, t ) by dyadic intervals in the greedymanner). Combining (12) and 2 − m T ≤ δ , we obtain: m Tδ ( ω ) ≤ × / (cid:16) / − (cid:17) − / α − / T / × (cid:16) − m/ + 2 − ( m +1) / + 2 − ( m +2) / + · · · (cid:17) = 2 / (cid:16) / − (cid:17) − / (cid:16) − − / (cid:17) − α − / T / − m/ ≤ / (cid:16) / − (cid:17) − / (cid:16) − − / (cid:17) − α − / T / ( δ/T ) / = 2 / (cid:16) / − (cid:17) − / (cid:16) − − / (cid:17) − α − / T / δ / , which is stronger than (10).Now we can prove the following elaboration of Lemma 2, which will also beused in the next section. Lemma 4.
For each α > , P ′ n ∀ T ≥ ∀ δ > m Tδ ≤ α − / T / δ / o ≥ − α. (13) Proof.
Replacing α in (10) by α T := (1 − − / ) T − / α for T = 1 , , , , . . . (where 1 − − / is the normalizing constant ensuring that the α T sum to α over T ), we obtain P ′ n ∀ δ > m Tδ ≤
157 (1 − − / ) − / α − / T / δ / o ≥ − (1 − − / ) T − / α. The countable subadditivity of upper probability now gives P ′ n ∀ T ∈ { , , , . . . } ∀ δ > m Tδ ≤
157 (1 − − / ) − / α − / T / δ / o ≥ − α, which in turn gives P ′ n ∀ T ≥ ∀ δ > m Tδ ≤
157 (1 − − / ) − / α − / (2 T ) / δ / o ≥ − α, which is stronger than (13). 8emma 2 immediately follows from Lemma 4 and the Arzel`a–Ascoli theorem(as stated in [8], Theorem 2.4.9).Inequality (11) will also be useful in the next section; the following lemmapackages it in a convenient form. Lemma 5.
For each α > , P ′ ( ∀ T ∈ { , , , . . . } ∀ n ∈ { , , . . . } : n X i =1 (cid:16) ω ( i − n T ) − ω (( i − − n T ) (cid:17) ≤ α − T n/ ) ≥ − α. (14) Proof.
Replacing β/ − (2 / − T − − n/ α , where T rangesover { , , , . . . } and n over { , , . . . } , we obtain P ′ ( n X i =1 (cid:16) ω ( i − n T ) − ω (( i − − n T ) (cid:17) ≤ / − − α − T n/ ) ≥ − − (2 / − T − − n/ α ;by the countable subadditivity of upper probability this implies P ′ ( ∀ T ∈ { , , , . . . } ∀ n ∈ { , , . . . } : n X i =1 (cid:16) ω ( i − n T ) − ω (( i − − n T ) (cid:17) ≤ / − − α − T n/ ) ≥ − α, which is stronger than (14). To establish (7) (with E replaced by E ′ ) we only need to establish E ′ ( F ) < R F dW + ǫ for a positive constant ǫ . We start from a series of reductions:1. We can assume that F is lower semicontinuous on Ω. Indeed, if it is not,by the Vitali–Carath´eodory theorem (see, e.g., [13], Theorem 2.24) forany compact K ⊆ Ω there exists a lower semicontinuous function G on K such that G ≥ F on K and R K GdW ≤ R K F dW + ǫ . Without loss ofgenerality we assume sup G ≤ sup F , and we extend G to all of Ω by setting G := sup F outside K . Choosing K with large enough W ( K ) (which canbe done since the probability measure W is tight: see, e.g., [2], Theorem1.4), we will have G ≥ F and R GdW ≤ R F dW + 2 ǫ . Achieving S ≤ R GdW + ǫ and lim inf t →∞ S t ( ω ) ≥ G ( ω ), where S is a capital process, willautomatically achieve S ≤ R F dW + 3 ǫ and lim inf t →∞ S t ( ω ) ≥ F ( ω ).9. We can further assume that F is continuous on Ω. Indeed, since eachlower semicontinuous function on a metric space is a limit of an increasingsequence of continuous functions (see, e.g., [5], Problem 1.7.15(c)), givena lower semicontinuous function F on Ω we can find a series of positivecontinuous functions G n on Ω, n = 1 , , . . . , such that inf F + P ∞ n =1 G n = F . The sum S of inf F and positive capital processes S , S , . . . achieving S n ≤ R G n dW + 2 − n ǫ and lim inf t →∞ S nt ( ω ) ≥ G n ( ω ), n = 1 , , . . . , willachieve S ≤ R F dW + ǫ and lim inf t →∞ S t ( ω ) ≥ F ( ω ).3. We can further assume that F depends on ω ∈ Ω only via ω | [0 ,T ] for some T ∈ (0 , ∞ ). Indeed, let us fix ǫ > E ′ ( F ) ≤ R F dW + Cǫ for some positive constant C assuming E ′ ( G ) ≤ R GdW for all G thatdepend on ω only via ω | [0 ,T ] for some T ∈ (0 , ∞ ). Choose a compactset K ⊆ Ω with W ( K ) > − ǫ and P ′ ( K ) > − ǫ (cf. Lemma 2). Set F T ( ω ) := F ( ω T ), where ω T is defined by ω T ( t ) := ω ( t ∧ T ) and T issufficiently large in the following sense. Since F is uniformly continuouson K and the metric is defined by (9), F and F T can be made arbitrarilyclose in C ( K ) (spaces C ( . . . ) are always equipped with the sup norm in thispaper); in particular, let k F − F T k C ( K ) < ǫ . Choose capital processes S and S such that S ≤ R F T dW + ǫ , lim inf t →∞ S t ( ω ) ≥ F T ( ω ), S ≤ ǫ ,lim inf t →∞ S t ( ω ) ≥ I K c ( ω ). The sum S := S + 2 sup | F | S + ǫ will satisfy S ≤ Z F T dW + (2 sup | F | + 2) ǫ ≤ Z K F T dW + (3 sup | F | + 2) ǫ ≤ Z K F dW + (3 sup | F | + 3) ǫ ≤ Z F dW + (4 sup | F | + 3) ǫ and lim inf t →∞ S t ( ω ) ≥ F T ( ω ) + 2 sup | F | I K c ( ω ) + ǫ ≥ F ( ω ) . Without loss of generality, we assume T ∈ { , , , . . . } .4. We can further assume that F depends on ω only via the values ω ( iT /N ), i = 1 , . . . , N (remember that ω (0) = 0), for some N ∈ { , , . . . } . Indeed,let us fix ǫ > E ′ ( F ) ≤ R F dW + Cǫ for some positive constant C assuming E ′ ( G ) ≤ R GdW for all G that depend on ω only via ω ( iT /N ), i = 1 , . . . , N , for some N . Let K ⊆ Ω be the compact set in Ω definedas K := (cid:8) ω | ∀ δ > m Tδ ≤ f ( δ ) (cid:9) for some f : (0 , ∞ ) → (0 , ∞ ) satisfyinglim δ → f ( δ ) = 0 (cf. the Arzel`a–Ascoli theorem) and chosen in such a waythat W ( K ) > − ǫ and P ′ ( K ) > − ǫ . Let g be a modulus of continuityof F on K ; we know that lim δ → g ( δ ) = 0. Set F N ( ω ) := F ( ω N ), where ω N is the piecewise linear function whose graph is obtained by joining thepoints ( iT /N, ω ( iT /N )), i = 0 , , . . . , N , and ( ∞ , ω ( T )), and N is so largethat g ( f ( T /N )) ≤ ǫ . Since ω ∈ K = ⇒ k ω − ω N k C ([0 ,T ]) ≤ f ( T /N ) = ⇒ ρ ( ω, ω N ) ≤ f ( T /N )(we assume, without loss of generality, that the graph of ω is horizontalover [ T, ∞ )), we have k F − F N k C ( K ) ≤ ǫ . Choose capital processes S S such that S ≤ R F N dW + ǫ , lim inf t →∞ S t ( ω ) ≥ F N ( ω ), S ≤ ǫ ,lim inf t →∞ S t ( ω ) ≥ I K c ( ω ). The sum S := S + 2 sup | F | S + ǫ will satisfy S ≤ Z F N dW + (2 sup | F | + 2) ǫ ≤ Z K F N dW + (3 sup | F | + 2) ǫ ≤ Z K F dW + (3 sup | F | + 3) ǫ ≤ Z F dW + (4 sup | F | + 3) ǫ and lim inf t →∞ S t ( ω ) ≥ F N ( ω ) + 2 sup | F | I K c ( ω ) + ǫ ≥ F ( ω ) .
5. We can further assume that F ( ω ) = U ( ω ( T /N ) , ω (2 T /N ) , . . . , ω ( T )) (15)where the function U : R N → R is not only continuous but also hascompact support. (We will sometimes say that U is the generator of F .) Indeed, let us fix ǫ > E ′ ( F ) ≤ R F dW + Cǫ for somepositive constant C assuming E ′ ( G ) ≤ R GdW for all G whose generatorhas compact support. Let B R be the open ball of radius R and centredat the origin in the space R N with the ℓ ∞ norm. We can rewrite (15)as F ( ω ) = U ( s ( ω )) where s : Ω → R N reduces each ω ∈ Ω to s ( ω ) :=( ω ( T /N ) , ω (2 T /N ) , . . . , ω ( T )). Choose R so large that W ( s − ( B R )) > − ǫ and P ′ ( s − ( B R )) > − ǫ (the existence of such R follows from theArzel`a–Ascoli theorem and Lemma 2). Alongside F , whose generator isdenoted U , we will also consider F ∗ with generator U ∗ ( σ ) := ( U ( σ ) if σ ∈ B R σ ∈ B c R ;in the remaining region B R \ B R , U ∗ is defined arbitrarily (but makingsure that U ∗ is continuous and takes values in [inf U, sup U ]; this can bedone by the Tietze–Urysohn theorem, [5], Theorem 2.1.8). Choose capitalprocesses S and S such that S ≤ R F ∗ dW + ǫ , lim inf t →∞ S t ( ω ) ≥ F ∗ ( ω ), S ≤ ǫ , lim inf t →∞ S t ( ω ) ≥ I s − ( B cR ) ( ω ). The sum S := S +2 sup | F | S will satisfy S ≤ Z F ∗ dW + (2 sup | F | + 1) ǫ ≤ Z s − ( B R ) F ∗ dW + (3 sup | F | + 1) ǫ = Z s − ( B R ) F dW + (3 sup | F | + 1) ǫ ≤ Z F dW + (4 sup | F | + 1) ǫ and lim inf t →∞ S t ( ω ) ≥ F ∗ ( ω ) + 2 sup | F | I s − ( B cR ) ( ω ) ≥ F ( ω ) .
11. Since every continuous U : R N → R with compact support can be arbi-trarily well approximated in C ( R N ) by an infinitely differentiable functionwith compact support (see, e.g., [1], Theorem 2.29(d)), we can further as-sume that the generator U of F is an infinitely differentiable function withcompact support.7. By Lemma 2, it suffices to prove that, given ǫ > K in Ω, some capital process S ≥ inf F − S ≤ R F dW + ǫ achieveslim inf t →∞ S t ( ω ) ≥ F ( ω ) for all ω ∈ K . Indeed, we can choose K with P ′ ( K ) so close to 1 that the sum of S and a positive capital process even-tually attaining 2 sup | F | +2 on K c will give a capital process starting from R F dW + 2 ǫ and exceeding F ( ω ) in the limit.From now on we fix a compact K ⊆ Ω, assuming, without loss of generality,that the statements inside the curly braces in (13) and (14) are satisfied forsome α > i = N −
1, define a function U i : R × [0 , ∞ ) × R i → R by U i ( s, D ; s , . . . , s i ) := Z ∞−∞ U i +1 ( s , . . . , s i , s + z ) N ,D ( dz ) , (16)where U N stands for U and N ,D is the Gaussian probability measure on R withmean 0 and variance D ≥
0. Next define, for i = N − U i ( s , . . . , s i ) := U i ( s i , T /N ; s , . . . , s i ) . (17)Finally, we can alternately use (16) and (17) for i = N − , . . . , , U i and U i (with (17) interpreted as U := U (0 , T /N ) when i = 0). Notice that U = R F dW .Informally, the functions (16) and (17) constitute Sceptic’s goal: assum-ing ω ∈ K , he will keep his capital at time iT /N , i = 0 , , . . . , N , close to U i ( ω ( T /N ) , ω (2 T /N ) , . . . , ω ( iT /N )) and his capital at any other time t ∈ [0 , T ]close to U i ( ω ( t ) , D ; ω ( T /N ) , ω (2 T /N ) , . . . , ω ( iT /N )) where i := ⌊ N t/T ⌋ and D := ( i + 1) T /N − t . This will ensure that his capital at time T is close to F ( ω )when his initial capital is U = R F dW .It is easy to check that each function U i ( s, D ; s , . . . , s i ) satisfies the heatequation in the variables s and D : ∂U i ∂D ( s, D ; s , . . . , s i ) = 12 ∂ U i ∂s ( s, D ; s , . . . , s i ) (18)for all s ∈ R , all D >
0, and all s , . . . , s i ∈ R . This is the key element of theproof.Sceptic will only bet at the times that are multiples of T /LN , where L ∈{ , , . . . } will later be chosen large. For i = 0 , . . . , N and j = 0 , . . . , L let us set t i,j := iT /N + jT /LN, S i,j := ω ( t i,j ) , D i,j := T /N − jT /LN. A i,j , we set dA i,j := A i,j +1 − A i,j .Using Taylor’s formula and omitting the arguments ω ( T /N ) , . . . , ω ( iT /N ),we obtain, for i = 0 , . . . , N − j = 0 , . . . , L − dU i ( S i,j , D i,j ) = ∂U i ∂s ( S i,j , D i,j ) dS i,j + ∂U i ∂D ( S i,j , D i,j ) dD i,j + 12 ∂ U i ∂s ( S ′ i,j , D ′ i,j )( dS i,j ) + ∂ U i ∂s∂D ( S ′ i,j , D ′ i,j ) dS i,j dD i,j + 12 ∂ U i ∂D ( S ′ i,j , D ′ i,j )( dD i,j ) , (19)where ( S ′ i,j , D ′ i,j ) is a point strictly between ( S i,j , D i,j ) and ( S i,j +1 , D i,j +1 ).Applying Taylor’s formula to ∂ U i /∂s , we find ∂ U i ∂s ( S ′ i,j , D ′ i,j ) = ∂ U i ∂s ( S i,j , D i,j )+ ∂ U i ∂s ( S ′′ i,j , D ′′ i,j )∆ S i,j + ∂ U i ∂D∂s ( S ′′ i,j , D ′′ i,j )∆ D i,j , where ( S ′′ i,j , D ′′ i,j ) is a point strictly between ( S i,j , D i,j ) and ( S ′ i,j , D ′ i,j ), and∆ S i,j and ∆ D i,j satisfy | ∆ S i,j | ≤ | dS i,j | , | ∆ D i,j | ≤ | dD i,j | . Plugging thisequation and the heat equation (18) into (19), we obtain dU i ( S i,j , D i,j ) = ∂U i ∂s ( S i,j , D i,j ) dS i,j + 12 ∂ U i ∂s ( S i,j , D i,j ) (cid:0) ( dS i,j ) + dD i,j (cid:1) + 12 ∂ U i ∂s ( S ′′ i,j , D ′′ i,j )∆ S i,j ( dS i,j ) + 12 ∂ U i ∂D∂s ( S ′′ i,j , D ′′ i,j )∆ D i,j ( dS i,j ) + ∂ U∂s∂D ( S ′ i,j , D ′ i,j ) dS i,j dD i,j + 12 ∂ U∂D ( S ′ i,j , D ′ i,j )( dD i,j ) . (20)At this point we are at last ready to define Sceptic’s elementary betting strategy:namely, he plays in such a way that the increment of his capital between times t i,j and t i,j +1 is equal to the sum of the first two terms on the right-hand sideof (20). Both ∂U i /∂s and ∂ U i /∂s are bounded as averages of ∂U i +1 /∂s and ∂ U i +1 /∂s , and so, eventually, averages of ∂U/∂s and ∂ U/∂s , respectively.Let us show that the last four terms on the right-hand side of (20) are negli-gible when L is sufficiently large (assuming T , N , and U fixed). All the partialderivatives involved in those terms are bounded: the heat equation implies ∂ U i ∂D∂s = ∂ U i ∂s ∂D = 12 ∂ U i ∂s ,∂ U i ∂s∂D = 12 ∂ U i ∂s ,∂ U i ∂D = 12 ∂ U i ∂D∂s = 14 ∂ U i ∂s , ∂ U i /∂s and ∂ U i /∂s , being averages of ∂ U i +1 /∂s and ∂ U i +1 /∂s ,and eventually averages of ∂ U/∂s and ∂ U/∂s , are bounded. We can assumethat | dS i,j | ≤ C L − / , N − X i =0 L − X j =0 ( dS i,j ) ≤ C L / (cf. (13) and (14), respectively) for ω ∈ K and some constants C and C (re-member that T , N , U , and, of course, α are fixed; without loss of generality weassume that N and L are powers of 2). This makes the cumulative contributionof the four terms have at most the order of magnitude O ( L − / ); therefore,Sceptic can achieve his goal for ω ∈ K by making L sufficiently large.To ensure that his capital never drops strictly below inf F −
1, Sceptic stopsplaying as soon as his capital hits inf F −
1. This will never happen when ω ∈ K (for L sufficiently large). The following simple lemma will allow us to deduce the results for the L´evygame.
Lemma 6.
For each α > and T > , we have P ( sup t ∈ [0 ,T ] | ω ( t ) | ≤ α − / T / ) ≥ − α (21) in the L´evy game.Proof. Starting from initial capital α , bet α/T on ω ( t ) − t at time 0 and stopplaying (set the stake to 0) at time T ∧ inf { t | | ω ( t ) | = α − / T / } ; the initialcapital α will grow to at least α + αT ( α − T − T ) = 1 if the inner inequality in(21) is violated.It is instructive to compare the bounds given by the inner inequalities in(10) with δ := T and in (21): the only difference is the constant factor 157 inthe former. Lemma 3 itself will also continue to hold in the L´evy game; we willstate it in a slightly weakened form: Lemma 7.
For each α > and T > , P n ∀ δ > m Tδ ≤ α − / T / δ / o ≥ − α. Proof.
Let τ := inf (cid:8) t ≥ | | ω ( t ) | = α − / T / + 1 (cid:9) and let ω τ : [0 , ∞ ) → R be the stopped ω , ω τ ( t ) := ω ( τ ∧ t ). Consider the same elementary bettingstrategies as in the proof of Lemma 3 except that they now stop playing at time τ . Identity (8) with ω τ in place of ω shows that the corresponding elementarycapital processes will also be elementary capital processes in the L´evy game.14heir combination (analogous to the one in the proof of Lemma 3) witnessesthat sup t ∈ [0 ,T ] | ω ( t ) | ≤ α − / T / = ⇒ ∀ δ > m Tδ ≤ α − / T / δ / with lower probability at least 1 − α in the L´evy game; it remains to combinethis with Lemma 6.In the same way as we obtained Lemma 4 from Lemma 3, we can now obtainthe following corollary from Lemma 7: Lemma 8.
For each α > , P n ∀ T ≥ ∀ δ > m Tδ ≤ α − / T / δ / o ≥ − α. Lemma 8 shows that Lemma 2 will also hold for the L´evy game. In a similarway we can get rid of the prime in P ′ in Lemma 5.The proof of (7) for the L´evy game proceeds as in the case of the modifiedL´evy game: indeed, the 7 reductions in Section 8 do not depend on the gamebeing played, and the elementary betting strategy constructed afterwards alwaysmakes bounded stakes (see its description after (20)).
10 Conclusion
In this short section we will state two open problems. First, what is the class A of all P -measurable subsets of Ω? It is easy to see that the statement ofTheorem 1 that the sets in F are P -measurable can be strengthened: Proposition 2.
Each set A ∈ F W in the completion of F w.r. to W is P -measurable.Proof. To establish (6) for A ∈ F W we choose A , A ∈ F such that A ⊆ A ⊆ A and W ( A ) = W ( A ), and define F as in the proof of Theorem 1(see Section 4). Since now E ( F ) < P ( E ) + ǫ , I E ∩ A ≤ F I A ≤ F I A , and I E ∩ A c ≤ F I A c ≤ F I A c , it suffices to notice that E ( F I A ) + E ( F I A c ) ≤ E ( F )immediately follows from Theorem 2.In particular, it would be interesting to know whether A coincides with F W .The second problem is: will the (modified) L´evy game remain coherent if themeasurability restrictions on stopping times and stakes are dropped? (In otherwords, if each σ -algebra considered is extended to become closed under arbitrary,and not just countable, unions and intersections.) A positive answer would leadto simpler and more intuitive definitions. A negative answer would also be ofgreat interest, providing a counter-intuitive phenomenon akin to the Banach–Tarski paradox. A related question is whether dropping the requirement that M and V should be bounded will lead to loss of coherence.15 cknowledgments This work was partially supported by EPSRC (grant EP/F002998/1), MRC(grant G0301107), VLA, EU FP7 (grant 201381), and the Cyprus ResearchPromotion Foundation.
References [1] Robert A. Adams and John J. F. Fournier.
Sobolev Spaces . AcademicPress, Amsterdam, second edition, 2003.[2] Patrick Billingsley.
Convergence of Probability Measures . Wiley, New York,1968.[3] Constantin Carath´eodory. ¨Uber das lineare Mass von Punktmengen—eineVerallgemeinerung des L¨angenbegriffs.
Nachrichten der Akademie der Wis-senschaften zu G¨ottingen. II. Mathematisch-Physikalische Klasse , 4:404–426, 1914.[4] A. Philip Dawid and Vladimir Vovk. Prequential probability: principlesand properties.
Bernoulli , 5:125–162, 1999.[5] Ryszard Engelking.
General Topology . Heldermann, Berlin, second edition,1989.[6] Yasunori Horikoshi and Akimichi Takemura. Implications of contrarian andone-sided strategies for the fair-coin game.
Stochastic Processes and theirApplications , to appear, doi:10.1016/j.spa.2007.11.007.[7] Olav Kallenberg.
Foundations of Modern Probability . Springer, New York,second edition, 2002.[8] Ioannis Karatzas and Steven E. Shreve.
Brownian Motion and StochasticCalculus . Springer, New York, second edition, 1991.[9] Masayuki Kumon and Akimichi Takemura. On a simple strategy weaklyforcing the strong law of large numbers in the bounded forecasting game.
Annals of the Institute of Statistical Mathematics , to appear.[10] Masayuki Kumon, Akimichi Takemura, and Kei Takeuchi. Game-theoreticversions of strong law of large numbers for unbounded variables.
Stochas-tics , 79:449–468, 2007.[11] Paul L´evy.
Processus stochastiques et mouvement brownien . Gauthier-Villars, Paris, second edition, 1965.[12] Jarl Waldemar Lindeberg. Eine neue Herleitung des Exponential-gesetzesin der Wahrsheinlichkeitsrechnung.
Mathematische Zeitschrift , 15:211–225,1922. 1613] Walter Rudin.
Real and Complex Analysis . McGraw-Hill, New York, thirdedition, 1987.[14] Glenn Shafer and Vladimir Vovk.
Probability and Finance: It’s Only aGame!
Wiley, New York, 2001.[15] Kei Takeuchi.
Kake no suuri to kinyu kogaku (Mathematics of Betting andFinancial Engineering, in Japanese) . Saiensusha, Tokyo, 2004.[16] Kei Takeuchi, Masayuki Kumon, and Akimichi Takemura. A new for-mulation of asset trading games in continuous time with essential forcingof variation exponent. Technical Report arXiv:0708.0275v1 [math.PR], arXiv.org e-Print archive, August 2007.[17] Vladimir Vovk. Forecasting point and continuous processes: prequentialanalysis.
Test , 2:189–217, 1993.[18] Vladimir Vovk. Continuous-time trading and emergence of randomness.Technical Report arXiv:0712.1275 [math.PR], arXiv.org e-Print archive,December 2007.[19] Vladimir Vovk. Continuous-time trading and emergence of volatility. Tech-nical Report arXiv:0712.1483 [math.PR], arXiv.org e-Print archive, De-cember 2007.[20] David Williams.