The role of measurability in gametheoretic probability
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Abstract
This paper argues that the requirement of measurability (imposed on trading strategies) is indispensable in continuoustime gametheoretic probability. The necessity of the requirement of measurability in measure theory is demonstrated by results such as the Banach–Tarski paradox and is inherited by measuretheoretic probability. The situation in gametheoretic probability turns out to be somewhat similar in that dropping the requirement of measurability allows a trader in a financial security with a nontrivial price path to become infinitely rich while risking only one monetary unit.
Keywords
Axiom of choice Continuous time Gametheoretic probability Incomplete markets MeasurabilityMathematics Subject Classification (2010)
91G99 60G05 60G25 60G40 60G48JEL Classification
C53 G13 G141 Introduction
This paper belongs to the area of gametheoretic probability (see e.g. [9]). The advantage of gametheoretic probability for mathematical finance over the dominant approach of measuretheoretic probability is that it allows us to state and prove results free of any statistical assumptions even in situations where such assumptions are often regarded as essential (see e.g. the probabilityfree Dubins–Schwarz theorem in [11] and the probabilityfree theory of stochastic integration in [7]). In this paper, we consider the framework of an idealized financial market with one tradable security and assume for simplicity a zero interest rate.
The necessity of a nontrivial requirement of measurability (such as Borel, Lebesgue, or universal) is well known in measure theory (without measurability, we have counterintuitive results such as the Banach–Tarski paradox [12]), and it is inherited by measuretheoretic probability. In gametheoretic probability, measurability is usually not needed in discrete time (and is never assumed in e.g. [9]); the present paper, however, shows that in continuous time, imposing some regularity conditions (such as Borel or universal measurability) is essential even in the foundations of gametheoretic probability. If such conditions are not imposed, the basic definitions of gametheoretic probability become uninteresting, or even degenerate; e.g. in the case of continuous price paths, the upper probability of sets can take only two values, namely 1 (if the set contains a constant price path) or 0 (if not).
This paper constructs explicit trading strategies for enriching the trader given a wellorder of the space of all possible price paths; however, such a wellorder exists only under the axiom of choice (which is, despite some anomalous corollaries, universally accepted). Since we cannot construct such wellorders, our strategies cannot be regarded as genuinely constructive. Therefore, they cannot be regarded as practical getrichquick schemes. Moreover, they do not affect the existing results of continuoustime gametheoretic probability, which always explicitly assume measurability (to the best of my knowledge).
Our trading strategies will be very simple and based on Hardin and Taylor’s work on hat puzzles. These authors show in [4] (see also [5, Sect. 7.4]) that the axiom of choice provides us with an Occamtype strategy able to predict the shortterm future (usually albeit not always), which makes it easy to get rich when allowed to trade in a security whose price changes in a nontrivial manner.
Section 2 is devoted to continuous price paths. The assumption of continuity allows us to use leverage and stoploss strategies, and the trader can profit greatly and quickly whenever the price path is not constant. (We only consider trading strategies that never risk bankruptcy. It is clear that profiting from a constant price path is impossible.)
In Sect. 3, we assume instead of continuity that the price path is càdlàg. To make trading possible, we further assume that the price path is positive. It is impossible for the trader to become infinitely rich if the logprice path has finite variation. If the variation is infinite, there will be either points in time such that the variation of the logprice is infinite in any of their left neighbourhoods or points in time such that the variation of the logprice is infinite in any of their right neighbourhoods. Becoming infinitely rich is possible after points of the latter type. Standard stochastic models of financial markets postulate price paths that have such points almost surely.
In the short Sect. 4, we only assume that the price path is positive; the theory in this case is almost identical to the theory for positive càdlàg price paths.
The proofs of all our main results are collected in a separate section, Sect. 5; they are based on Hardin and Taylor’s results. Appendix A provides a more general picture of predicting the shortterm future using the axiom of choice. It answers several very natural questions (and at the end asks 256 more questions answering just one of them).
Our definitions of the basic notions of continuoustime probability (such as stopping times) will be Galmarinotype (see e.g. [2, Theorems 1.2 and 1.4]) and modelled on the ones in the technical report [11] (the journal version uses slightly different definitions), except that the requirements of measurability will be dropped. By “positive” I mean “nonnegative”, adding “strictly” when necessary. The restriction \(f_{C}\) of a function \(f:A\to B\) to a set \(C\) is defined as \(f_{A\cap C}\); this notation will be used even when \(C\not\subseteq A\).
2 Continuous price paths
Remark 2.1
The intuition behind a simple trading strategy is that the trader is allowed to take positions \(h_{k}\), either long or short, in a security whose price at time \(t\in[0,1]\) is denoted \(\omega(t)\). The positions can change only at a discrete sequence of times \(\tau_{1},\tau_{2},\ldots\), which makes the definition (2.3) of the trader’s capital at time \(t\) uncontroversial. To obtain more useful trading strategies, we allow the trader to split his initial capital into a countable number of accounts and to run a separate simple trading strategy for each account; none of the component simple trading strategies is allowed to go into debt. The resulting total capital at time \(t\) is given by (2.4). The upper probability \(\overline {\mathbb {P}}(E)\) defined by (2.5) is the smallest initial capital sufficient for superhedging the binary option on \(E\).
The following theorem is proved in Sect. 5.
Theorem 2.2
Remark 2.3
A more popular version of our definition (2.5) was given by Perkowski and Prömel [7]. Perkowski and Prömel’s definition is more permissive [7, Sect. 2.3], and so the first statement of Theorem 2.2 continues to hold for it as well if we allow nonmeasurable (but still nonanticipative) trading strategies. Like all papers (that I am aware of) on continuoustime gametheoretic probability, the definitions given in [7] assume the measurability of all strategies.
Remark 2.4
The definitions of this section assume that the trader is permitted to short the security (which allows \(h_{k}(\omega)<0\)) and borrow money (which allows leverage, \(H_{k}(\omega)>1\) in the notation of Remark 3.4 below). If shorting and borrowing are not permitted (in the notation of Remark 3.4, if the \(H_{k}\) are only permitted to take values in \([0,1]\)), Theorem 2.2 ceases to be true, but Theorem 3.3 is still applicable.
3 Positive càdlàg price paths
In this section, we prove an analogue of Theorem 2.2 for positive càdlàg price paths \(\omega\); the picture now becomes more complicated. Intuitively, \(\omega:[0,1]\to[0,\infty)\) is a price path of a financial security whose price is known always to stay positive (such as a stock, and from now on it will be referred to as stock). For simplicity, in the bulk of this section, we consider the price paths \(\omega\) satisfying \(\inf\omega>0\); the case of general positive \(\omega\) is considered in Remark 3.8 at the end of the section. Therefore, we redefine \(\Omega\) as the set of all \(\omega:[0,1]\to[0,\infty)\) such that \(\inf\omega>0\). The definitions of adapted processes, stopping times, etc., stay literally as before (but with the new definition of \(\Omega\)). Our goal is to determine the sign of \(\overline {\mathbb {P}}(E)\) (i.e., to determine whether \(\overline {\mathbb {P}}(E)>0\)) for a wide family of sets \(E\subseteq\Omega\).
Theorem 3.1
We can see that \(\overline {\mathbb {P}}(E)>0\) whenever \(E\) contains \(\omega\) with \(\operatorname {var}(\log\omega)<\infty\). Therefore, in the rest of this section, we concentrate on \(\omega\in\Omega\) with \(\operatorname {var}(\log\omega)=\infty\). We start from a classification of such \(\omega\).
Lemma 3.2
Proof
The implication “⇐” is obvious, and so we only check “⇒”. Suppose that \(\operatorname {var}(f,t+)<\infty\) and \(\operatorname {var}(f,t)<\infty\) for all \(t\in[0,1]\). Fix a neighbourhood \(O_{t}=(a_{t},b_{t})\) of each \(t\) such that \(\operatorname {var}(f_{O_{t}})<\infty\). These neighbourhoods form a cover of \([0,1]\). The existence of a finite subcover immediately implies that \(\operatorname {var}(f)<\infty\). □
The following theorem (proved in Sect. 5) tackles the second term of the union in (3.7).
Theorem 3.3
Remark 3.4
Remark 3.5
In the context of the previous remark, imposing the requirement of being determined before \(\tau_{k}\) on the bets \(h_{k}\) rather than relative bets \(H_{k}\) would lead to a useless notion of a predictable positive capital process: all such processes would be constant. Indeed, suppose there is a predictable positive capital process Open image in new window that is not constant (where “predictable” is defined in terms of \(h_{k}\) rather than \(H_{k}\)). Let \(\omega\in\Omega\) be a path on which Open image in new window is not constant. Fix such an \(\omega\) and let \(k\) be the smallest integer such that \(\tau_{k}(\omega)<\tau_{k+1}(\omega)\) and \(h_{k}(\omega)\ne 0\) (such a \(k\) exists since the process Open image in new window is not constant on \(\omega\)). Let us consider the more difficult (since it is safer for the trader) case \(h_{k}(\omega)>0\). Define \(\omega'\) as \(\omega\) over the time interval \([0,\tau_{k}(\omega))\), as a sufficiently large positive number at the time \(\tau_{k}(\omega)\) (namely, it should be such that the product of \(h_{k}(\omega)=h_{k}(\omega')\) and \(\omega'(\tau_{k}(\omega))1\) exceeds the current capital Open image in new window ), and as a linear function over the time interval \([\tau_{k}(\omega),1]\) decreasing from \(\omega'(\tau_{k}(\omega))\) at the left endpoint to 1 at the right endpoint. If \(\tau_{k+1}(\omega')=1\), we have Open image in new window , a contradiction. Since \(\tau_{k+1}(\omega')=\tau_{k}(\omega')\) is impossible by our assumption \(\tau_{k+1}(\omega)>\tau_{k}(\omega)\) and the predictability of \(\tau_{k}\) and \(\tau_{k+1}\), we can assume \(\tau_{k}(\omega')<\tau_{k+1}(\omega')<1\). Defining \(\omega''\) as \(\omega'\) over the time interval \([0,\tau_{k+1}(\omega'))\) and as 1 over the complementary time interval \([\tau_{k+1}(\omega'),1]\), we obtain Open image in new window , again a contradiction.
In the continuous case of Sect. 2 (where \(\omega\) is not required to be positive), the notion of a predictable positive capital process is equivalent to that of a positive capital process, but the notion of a strongly predictable positive capital process, even as given in the previous remark, is useless: again, any such process is a constant.
Theorem 3.6
Remark 3.7
Remark 3.8
4 Positive price paths
Let us now redefine \(\Omega\) to be the set of all positive functions \(\omega:[0,1]\to[0,\infty)\) satisfying \(\inf\omega>0\) (without any continuity requirements). The definitions of adapted processes, stopping times, etc., again stay as in Sect. 2. Theorems 3.3 and 3.6 still hold, as shown by the same arguments in the next section. Remark 3.4 still holds with the same definitions of predictable and strongly predictable positive capital processes. Remark 3.8 still holds for \(\Omega\) being the set of all positive functions \(\omega:[0,1]\to[0,\infty)\).
5 Proofs of the theorems
The next result (Lemma 5.1) is applicable to all \(\Omega\) considered in Sects. 2–4. Fix a wellorder ⪯ of \(\Omega\), which exists by the Zermelo theorem (one of the alternative forms of the axiom of choice; see e.g. [6, Theorem 5.1]). Let \(\omega^{a}\), where \(\omega\in\Omega\) and \(a\in[0,1]\), be the ⪯smallest element of \(\Omega\) such that \(\omega^{a}_{[0,a]}=\omega_{[0,a]}\). Intuitively, using \(\omega^{a}\) as the prediction at time \(a\) for \(\omega\) is an instance of Occam’s razor: out of all hypotheses compatible with the available data \(\omega_{[0,a]}\), we choose the simplest one, where simplicity is measured by the chosen wellorder.
Lemma 5.1
 1.
The set \(W_{\omega}\) is wellordered by ≤. (Therefore, each of its points is isolated on the right, which implies that \(W_{\omega}\) is countable and nowhere dense.)
 2.
If \(t\in[0,1]\setminus W_{\omega}\), there exists \(t'>t\) such that \(\omega^{t}_{[t,t']}=\omega_{[t,t']}\).
 3.
If \(t\in[0,1)\), there exists \(t'>t\) such that \(\omega^{s}_{[t,t']}=\omega_{[t,t']}\) for all \(s\in(t,t')\).
Part 1 of Lemma 5.1 says, informally, that the set \(W_{\omega}\) is small. Part 2 says that at each time point \(t\) outside the small set \(W_{\omega}\), the Occam prediction system that outputs \(\omega^{t}\) as its prediction is correct (over some nontrivial time interval). Part 3 says that even at time points \(t\) in \(W_{\omega}\), the Occam prediction system becomes correct (in the same weak sense) immediately after time \(t\).
Proof of Lemma 5.1
Let us first check that \(W_{\omega}\) is wellordered by ≤. Suppose there is an infinite strictly decreasing chain \(t_{1}>t_{2}>\cdots\) of elements of \(W_{\omega}\). Then we have \(\omega^{t_{1}}\succ\omega^{t_{2}}\succ\cdots\), which contradicts ⪯ being a wellorder.
Each point \(t\in W_{\omega}\setminus\{1\}\) is isolated on the right since \(W_{\omega}\cap(t,t')=\emptyset\), where \(t'\) is the successor in \(W_{\omega}\) of \(t\). Therefore, \(W_{\omega}\) is nowhere dense. To check that \(W_{\omega}\) is countable, map each \(t\in W_{\omega}\setminus\{1\}\) to a rational number in the interval \((t,t')\), where \(t'\) is the successor in \(W_{\omega}\) of \(t\); this mapping is an injection.
As part 2 is obvious (and essentially asserted in (5.1)), let us check part 3. Suppose \(t\in[0,1)\). The set of all \(\omega^{s}\), \(s\in(t,1]\), has a smallest element \(\omega^{t'}\), where \(t'\in(t,1]\). It remains to notice that \(\omega^{s}=\omega^{t'}\) for all \(s\in(t,t')\). □
Remark 5.2
It might be tempting to conjecture that for any \(t\in W_{\omega}\setminus\{1\}\), the function \(s\mapsto\omega^{s}\) does not depend on \(s\in(t,t')\), where \(t'\) is the successor in \(W_{\omega}\) of \(t\). While this statement is true for \(\Omega=C[0,1]\), simple examples show that it is wrong in general; see Lemma A.3 in Appendix A.
5.1 Proof of Theorem 2.2
Let \(\omega\in\Omega\) and let \(c\) be the largest \(t\in[0,1]\) such that \(\omega_{[0,t]}\) is constant (the supremum is attained by the continuity of \(\omega\)). Assuming that \(\omega\) is not constant, we have \(c<1\). Set \(\omega^{c+}:=\omega^{t}\) for \(t\in(c,c+\epsilon)\) for a sufficiently small \(\epsilon>0\) (namely, such that \(t\mapsto\omega^{t}\) is constant over the interval \((c,c+\epsilon)\); such an \(\epsilon\) exists by Lemma 5.1). Choose \(d\in(c,1)\) such that \(\omega^{d}=\omega^{c+}\) (and therefore \(\omega_{(c,d]}=\omega^{c+}_{(c,d]}\) and \(\omega^{t}=\omega^{c+}\) for all \(t\in(c,d]\)). Take rational \(a,b\in(c,d)\) such that \(a< b\) and \(\omega_{[a,b]}\) is not constant; since Open image in new window , (5.2) gives \(\mathfrak{S}_{b}(\omega)=\infty\); and since \(b\) can be arbitrarily close to \(c\), we obtain (2.6).
5.2 Proof of Theorem 3.1
For further (obvious) details, see the proof of Proposition 2 in [10]. □
5.3 Proof of Theorem 3.3
The proof uses the fact that \(\inf I^{+}_{\omega}\in I^{+}_{\omega}\) when \(I^{+}_{\omega}\ne\emptyset\). Notice that \(I^{+}_{\omega}=J^{+}_{\omega}\), where \(J^{+}_{\omega}\) is defined in Remark 3.7.

\(c_{n}=1/n^{2}\) (which ensures that the total initial capital \(\sum_{n} 1/n^{2}\) is finite); \(G_{n}\) will consist of stopping times denoted as \(\tau^{n}_{1},\tau^{n}_{2},\ldots\) and bets denoted as \(h^{n}_{1},h^{n}_{2},\ldots\);

if \(I^{+}_{\omega}=\emptyset\), set \(\tau^{n}_{1}(\omega)=\tau^{n}_{2}(\omega)=\cdots=1\) and \(h^{n}_{1}(\omega)=h^{n}_{2}(\omega)=\cdots=0\) (intuitively, \(G_{n}\) never bets, which makes this part of the definition nonanticipatory); in the rest of this definition we will assume that \(I^{+}_{\omega}\ne\emptyset\) and, therefore, \(\inf I^{+}_{\omega}<1\);

set \(a:=\inf I^{+}_{\omega}\); we know that \(a\in I^{+}_{\omega}\) and \(a<1\);

in view of Lemma 5.1, set \(\omega^{a+}:=\omega^{t}\) for \(t\in(a,a+\epsilon)\) for a sufficiently small \(\epsilon\) (such that \(t\mapsto\omega^{t}\) does not depend on \(t\in(a,a+\epsilon)\));
 define(with \(\inf\emptyset:=(a+2^{n})\wedge 1\));$$ c := \inf \big\{ t \in \big(a,(a+2^{n})\wedge 1\big] :\operatorname {var}^{+}\big(\log\omega^{a+}_{[t,t+2^{n}]}\big) \le n \big\} $$
 set \(d:=(a+c)/2\) and defineand \(h^{n}_{k}(\omega^{a+})\), \(k=1,2,\ldots\), in such a way that$$ \tau^{n}_{k}(\omega^{a+})\in[d,(d+2^{n})\wedge1]\cup\{1\} $$(5.3)(5.4)(5.5)and the fact that (3.11) remains true if only positive simple capital processes are used as \(\mathfrak{S}\) in the definition (2.5) of \(\overline {\mathbb {P}}\) (as can be seen from the proof in Sect. 5.2);$$ \operatorname {var}^{+} \big( \log\omega^{a+}_{[d,d+2^{n}]} \big) > n $$
 setand$$ \tau^{n}_{k}(\omega) := \textstyle\begin{cases} \tau^{n}_{k}(\omega^{a+}) &\quad \text{if $\omega_{[0,\tau^{n}_{k}(\omega^{a+})]}=\omega^{a+}_{[0,\tau^{n}_{k}(\omega^{a+})]}$,}\\ 1 &\quad \text{otherwise,} \end{cases} $$(5.6)$$ h^{n}_{k}(\omega) := \textstyle\begin{cases} h^{n}_{k}(\omega^{a+}) & \quad \text{if $\omega_{[0,\tau^{n}_{k}(\omega^{a+})]}=\omega^{a+}_{[0,\tau^{n}_{k}(\omega^{a+})]}$,}\\ 0 &\quad \text{otherwise}. \end{cases} $$
Let us check (3.8). Suppose the premise part of (3.8) holds for given \(t\in[0,1]\) and \(\omega\in\Omega\). Using the notation introduced in the previous paragraph (and suppressing the dependence on \(\omega\) and \(n\), as before), we can see that \(t>a\). From some \(n\) on, we have \(d+2^{n}< t\) and \(\omega^{s}=\omega^{a+}\) for all \(s\in(a,d+2^{n})\), and so the divergence of the series \(\sum_{n} e^{n}/n^{2}\) implies that \(\mathfrak{S}_{t}(\omega)=\infty\). □
Remark 5.3

\(\sigma^{n}_{k}(\omega^{a+})\in[\tau^{n}_{k}(\omega^{a+}),\tau^{n}_{k+1}(\omega^{a+})]\) and \(\sigma^{n}_{k}(\omega^{a+})\in(\tau^{n}_{k}(\omega^{a+}),\tau^{n}_{k+1}(\omega^{a+}))\) when \(\tau^{n}_{k}(\omega^{a+})<\tau^{n}_{k+1}(\omega^{a+})\);

\(\sigma^{n}_{k}(\omega^{a+})\) are so close to \(\tau^{n}_{k}(\omega^{a+})\) that (5.5) still holds when we replace the stopping times \(\tau^{n}_{k}\) by \(\sigma^{n}_{k}\) in the definition of \(G_{n}\) (with the relative bets corresponding to (5.4) unchanged);
 for an arbitrary \(\omega\in\Omega\),(cf. (5.6)).$$ \sigma^{n}_{k}(\omega) := \textstyle\begin{cases} \sigma^{n}_{k}(\omega^{a+}) & \quad \text{if $\omega_{[0,\tau^{n}_{k}(\omega^{a+})]}=\omega^{a+}_{[0,\tau^{n}_{k}(\omega^{a+})]}$,}\\ 1 & \quad \text{otherwise} \end{cases} $$
5.4 Proof of Theorem 3.6
This proof uses methods of measuretheoretic probability; our probability space is \(\Xi\) equipped with the canonical filtration \((\mathcal{F}_{i})\) and the power of the uniform probability measure on \(\{1,1\}\); \(\mathcal{F}_{i}\) consists of all subsets of \(\Xi\) that are unions of cylinders \(\{(\xi_{1},\xi_{2},\ldots)\in\Xi :\xi_{1}=c_{1},\ldots,\xi_{i}=c_{i}\}\), and the measure of each such cylinder is \(2^{i}\). This is a discrete probability space without any measurability issues (the simple idea of using such a “poor” probability space was used earlier in e.g. [9, Sect. 4.3]).
6 Conclusion
This paper shows that some assumptions of regularity (apart from being nonanticipative, such as universal measurability) should be imposed on continuoustime trading strategies even in gametheoretic probability. This is not a serious problem in applications since only computable trading strategies can be of practical interest, and computable trading strategies will be measurable under any reasonable computational model.

Is it possible to extend Theorems 2.2, 3.3 and 3.6 to the case where only the most recent past is known to the trader, as in [5, p. vii and Sect. 7.3] and [4, Sect. 5]?

Is it possible to extend Theorems 2.2, 3.3 and 3.6 to the case of the trader without a synchronized watch (see [5, Sect. 7.7] or [1])?

The construction in Sect. 5.4 produces a set \(E\subseteq\Omega\) (consisting of all \(\omega_{\xi\to}\) for \(\xi\in\Xi\) with extendable \(\omega_{\xi}\)) that satisfies both \(\overline {\mathbb {P}}(E)=1\) and \(\operatorname {var}(\log\omega)=\infty\) for all \(\omega\in E\). However, \(\operatorname {vi}(\log\omega)=1\) for all \(\omega\in E\) (see e.g. [?, Sect. 4.2] for the definition of the variation index \(\operatorname {vi}\)). Do \(E\subseteq\Omega\) with \(\overline {\mathbb {P}}(E)=1\) and \(\inf_{\omega\in E}\operatorname {vi}(\log\omega)>1\) exist?
Notes
Acknowledgements
I am grateful to the participants in the workshop “Pathwise methods, functional calculus and applications in mathematical finance” (Vienna, 4–6 April 2016) for their comments. Thanks to Yuri Gurevich for numerous discussions on a wide range of topics (as a result of which I discovered Hardin and Taylor’s work on hat puzzles) and to Yuri Kalnishkan for a useful discussion on the topic of this paper. The anonymous reviewers’ thoughtful comments made me change radically the presentation and emphasis; they also prompted me to include Appendix A.
References
 1.Bajpai, D., Velleman, D.J.: Anonymity in predicting the future. Am. Math. Mon. 123, 777–788 (2016) MathSciNetCrossRefGoogle Scholar
 2.Courrège, P., Priouret, P.: Temps d’arrêt d’une fonction aléatoire: relations d’équivalence associées et propriétés de décomposition. Publ. Inst. Stat. Univ. Paris 14, 245–274 (1965) zbMATHGoogle Scholar
 3.Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel. Hermann, Paris (1975). Chapters I–IV zbMATHGoogle Scholar
 4.Hardin, C.S., Taylor, A.D.: A peculiar connection between the Axiom of Choice and predicting the future. Am. Math. Mon. 115, 91–96 (2008) MathSciNetzbMATHGoogle Scholar
 5.Hardin, C.S., Taylor, A.D.: The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems. Springer, Cham (2013) CrossRefzbMATHGoogle Scholar
 6.Jech, T.: Set Theory: The Third Millennium Edition, Revised and Expanded. Springer, Berlin (2003) Google Scholar
 7.Perkowski, N., Prömel, D.J.: Pathwise stochastic integrals for model free finance. Bernoulli 22, 2486–2520 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
 8.Rosenstein, J.G.: Linear Orderings. Academic Press, New York (1982) zbMATHGoogle Scholar
 9.Shafer, G., Vovk, V.: Probability and Finance: It’s Only a Game! Wiley, New York (2001) CrossRefzbMATHGoogle Scholar
 10.Vovk, V.: Rough paths in idealized financial markets. Lith. Math. J. 51, 274–285 (2011). Updated version: arXiv:1005.0279 [qfin.GN] (November 2016) MathSciNetCrossRefzbMATHGoogle Scholar
 11.Vovk, V.: Continuoustime trading and the emergence of probability. Finance Stoch. 16, 561–609 (2012). Updated version: arXiv:0904.4364 [math.PR] (May 2015) MathSciNetCrossRefzbMATHGoogle Scholar
 12.Wagon, S.: The Banach–Tarski Paradox. Cambridge University Press, Cambridge (1985) CrossRefzbMATHGoogle Scholar
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