W. Schachermayer. Martingale measures for discrete-time processes with infinite horizon. Mathematical Finance, 4-1:25--55, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....satisfying. In a development completely separate from stochastic programming, the mathematical finance literature exploits convex duality to obtain the equivalence of no arbitrage conditions on a market with the existence of an equivalent martingale measure for the market price process, cf. [14, 2], which can then be used to price financial derivatives. These problems generally possess unbounded constraints, and do not satisfy the property of relatively complete recourse, yet miraculously (to the stochastic programmer) do yield dual variables in (these turn out to be the Radon Nikodym ....

.... esoteric characteristics associated with the singular components of elements of that space, cf. 11] The theorems presented here are the culmination of an analysis of the theory of mathematical finance, in particular the state of the art papers by Delbaen and Schachermayer [2] and Schachermayer [14]. In those papers, variants of a classical theorem referred to as the fundamental theorem of asset pricing equate no arbitrage type conditions with the existence of what is known in the literature as an equivalent martingale measure. The no arbitrage type conditions amount to special types of ....

W. Schachermayer. Martingale measures for discrete-time processes with infinite horizon. Mathematical Finance, 4-1:25--55, 1994.

....trading strategy H, where, in general, some additional care is needed, to rule out, e.g. doubling strategies (see [HP 81] One therefore has to restrict to a convenietly chosen sub class of the predictable trading strategies. We don t elaborate on the details here (see, e.g. HP 81] or [DS 94] and simply note that one is naturally led to a subspace (or, more generally, subcone) M of X, consisting of marketed claims , and a linear pricing functional : M R. What does it mean that a market modeled by (X; K; M; does not allow arbitrage Letting M 0 = fx 2 M : x) 0g and C ....

.... possibility of free disposal is in fact the key feature to make the extension of the above theorem work. For a counterexample, showing that the condition, that the closure M 0 of M 0 satis es M 0 K = does not allow, in general, for an extension of the above theorem, we refer to [S 94] The central result ( K 81] Theorem 3) states that, under some mild separability assumption, the condition of no free lunch is equivalent to the extension property of (X; K; M; We now formulate Theorem 3 of [K 81] in a version specializing to the case of L ; F ; P) where it turns ....

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W. Schachermayer, (1994), Martingale Measures for discrete time processes with in nite horizon. Math. Finance, Vol. 4, No. 1, pp. 25-55.

....notation L 0 (resp. L 1 , L 1 ) for the space of all (resp. P integrable, P essentially bounded) real valued random variables on( Omega ; F ; P ) If C is a subset in a linear space, then conv C will denote the minimal convex set containing C. The work of McBeth [16] Schachermayer [19] and Delbaen and Schachermayer [5] has shown the usefulness of the following concept. Definition 5.1 The sequence (f n ) n1 is Fatou convergent to f if (f n ) n1 is uniformly bounded from below and f n f almost surely. A subset C in L 0 which is closed with respect to Fatou convergence ....

....5.1 The sequence (f n ) n1 is Fatou convergent to f if (f n ) n1 is uniformly bounded from below and f n f almost surely. A subset C in L 0 which is closed with respect to Fatou convergence will be called Fatou closed. The following lemma on Fatou convergence is taken from [5] see also [19]. Lemma 5.1 Let (f n ) n1 be a sequence of nonnegative measurable functions. 1) There is a sequence g n 2 conv(f n ; f n 1 ; n 1, which converges almost surely to a function g with values in [0; 1] 2) If conv(f 1 ; f 2 ; is bounded in L 0 then g is finite almost ....

W. Schachermayer. Martingale measures for discrete-time processes with infinite horizon. Mathematical Finance, 4(1):25--55, January 1994.

....Theorem of Asset Pricing ( K 81] th. 3) as it was later referred to by Dybvig and Ross [DR 87] For a general version of the Fundamental Theorem of Asset Pricing as well as for an account on the history of this topic and in particular on the role played by Kreps seminal paper, we refer to [DS 94] and [DS 97] In the present paper we shall make some remarks on Kreps paper: firstly we give in section 2 below a counterexample to the enticing conjecture raised in ( K 81] p. 29) Briefly, the conclusion of our counterexample is that we only can expect nice results in the context of the ....

.... Theorem of Asset Pricing for the case when (X; K) L p ; k Delta k p ; L p nf0g) where 1 p 1, assuming the separability of the underlying probability space( Omega ; F ; P) it was noticed by Stricker [S 90] using a theorem of Yan [Y 80] that this assumption is superfluous (see [S 94] for an exposition of the proof and additional references) loosely speaking, the countability argument needed in the proof of theorem 3 of [K 81] can be obtained by an exhaustion argument from the finiteness (in fact, sigma finiteness would be sufficient) of the underlying measure space( Omega ....

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. Schachermayer, W. (1994), Martingale measures for discrete time processes with infinite horizon, Math. Finance 4, 25--55.

....martingale measures. In this setting it turns out that L ; F ; P) often is the natural space to work in (as opposed to L ; F ; P) for some p 0) as it remains unchanged under the passage from P to an equivalent measure Q (while L ; F ; P) does change, for 0 p 1) We refer, e.g. DS 94] for a general exposition of the above described duality relations and to [KS 97] for an applications of the bipolar theorem 1.3. 4.2 Note: Lemma 2.3 may be viewed as a variation of theorem 1 in [Y 80] which is a result based on previous work of Mokobodzki (as an essential step in Dellacherie s ....

.... is a result based on previous work of Mokobodzki (as an essential step in Dellacherie s proof of the semimartingale characterization theorem due to Bichteler and Dellacherie; see [Me 79] and [Y 80] The proof of Yan s theorem is a blend of a Hahn Banach and an exhaustion argument (see, e.g. S 94] for a presentation of this proof and [Str 90] S 94] for applications of Yan s theorem to Mathematical Finance) In fact, these arguments have their roots in the proof of the Halmos Savage theorem [HS 49] and the theorems of Nikishin and Maurey [N 70] M 74] 4.3 Note: In the course of the ....

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.W. Schachermayer, Martingale measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25 --- 55.

....integrands was used in the paper [DS94] where a short history of this concept is given. The above definition generalizes the admissibility as used in [DS94] in the sense that it replaces a constant function by a fixed nonnegative integrable function w. The concept was also used by W. Schachermayer [S94], Proposition 4.5. Exactly as in [DS94] we introduce the cone C 1;w = ff j there is a w Gamma admissible integrand H such that f (H Delta M)1 g : 21 4.11 Theorem D. Let M be a real valued local martingale and w 1 an integrable function. n1 of M integrable real valued predictable processes ....

W. Schachermayer, Martingale Measures for Discrete time Processes with Infinite Hori-zon, Mathematical Finance 4 (1994), 25--56.

....cone in L Under the assumption that S satisfies (NA) i.e. K L = f0g, we want to find an equivalent martingale measure Q for the process S. The first argument is wellknown in the present context (compare [S92] and theorem 4. 1 below for a general version of this result; we refer to [S94] for an account on the history of this result, in particular on the work of J.A. Yan [Y80] and D. Kreps [K81] 3.1 Lemma. If S satisfies (NA) the convex cone C is weak star closed in ; F ; P) and C L ; F ; P) f0g. Therefore there is a probability measure Q 1 on F ; Q 1 P such that E ....

....Below we will show that also in this case the above equality remains valid, at least for positive random variables g. This result does not immediately follow from the results in [DS95] Another approach to the problem is to enlarge the concept of admissible integrand in a way as was also done in [S94] and [DS96] For a random variable w 1 we call an integrand H to be w admissible if the process H Delta S can be controlled from below by the function w (see 5.3 and 5.4 below for a formal definition of this concept and of the definition of feasible weight function) The aim of this section is ....

W. Schachermayer, Martingale Measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25--55.

....points that relate the theorem to functional analysis. The first and most delicate step consists in proving that under the assumption of (NFLVR) the set C is already weak closed. Yan s theorem [Yan80] whose proof consists of a combination of a Hahn Banach and an exhaustion argument) see, e.g. [S94]) then yields the existence of an equivalent probability measure Q such that, for all f 2 C, E Q [f ] 0. Interpreting Q as a linear functional on L1 , it separates C from the positive cone L 1 . In the case where S is locally bounded this already implies that S is a local martingale for the ....

Schachermayer, W. Martingale Measures for discrete time processes with infinite horizon. Math. Finance, Vol. 4, No. 1 (1994), pp. 25--55.

.... COUNTER EXAMPLE TO SEVERAL PROBLEMS IN THE THEORY OF ASSET PRICING W. Schachermayer Institut fur Mathematik der Universitat Wien, Strudlhofgasse 4, A 1090 Wien, Austria. September 92 Abstract. We construct a continuous bounded stochastic process (S t ) t2[0;1] which admits an equivalent martingale measure but such that the minimal martingale measure in the sense of Follmer and ....

....M) t Key words and phrases. Equivalent Martingale Measure, Follmer Schweizer Decomposition, Girsanov Transformation. Acknowledgement: We are indebted to F. Delbaen for much help and advice on this paper; we also thank C. Stricker for a joyful discussion in Oberwolfach Typeset by A M S T E X 1 2 W. SCHACHERMAYER and d P = G 1 dP as the measure that should be used to minimize the quadratic risk of hedging. This characterisation holds true provided the above defined P exists and is equivalent to P ) Foellmer and Schweizer also gave a characterisation of P (if it exists) as the equivalent ....

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W. Schachermayer (1992), Martingale Measures for discrete time processes with infinite horizon, preprint (submitted to Mathematical Finance), 30 p.

.... Hilbert Space Proof of the Fundamental Theorem of Asset Pricing in Finite Discrete Time W. Schachermayer Institut fur Mathematik der Universitat Wien, Strudlhofgasse 4, A 1090 Wien, Austria. October 92 Abstract. R. Dalang, A. Morton and W. Willinger have proved a beautiful version of the Fundamental Theorem of Asset Pricing which pertains to the case of finite discrete time: In this case the ....

W. Schachermayer (1992), Martingale Measures for discrete time processes with infinite horizon, preprint, 30 p.

....one ECU on a set of probability nearly zero. In the classical case where we do not have a sequence of financial markets but only one fixed market there has been done a lot of work relating the absence of arbitrage or similar concepts (such as no free lunch [11] no free lunch with bounded risk [2,12], no free lunch with vanishing risk [4] to the existence of an equivalent local martingale measure for the price process of the available securities, e.g. 2,4,6,8,9,11,12,13] In a large financial market there is a similar situation. We will specify conditions on the local martingale measures ....

.... a lot of work relating the absence of arbitrage or similar concepts (such as no free lunch [11] no free lunch with bounded risk [2,12] no free lunch with vanishing risk [4] to the existence of an equivalent local martingale measure for the price process of the available securities, e.g. [2,4,6,8,9,11,12,13]. In a large financial market there is a similar situation. We will specify conditions on the local martingale measures of the small spaces, respectively on sequences of such measures, that are necessary and sufficient for the absence of asymptotic arbitrage of first or second kind. Kabanov and ....

W. Schachermayer, Martingale Measures for Discrete Time Processes with Infinite Horizon, Mathematical Finance 4 (1994).

....g 2 K; f gg: 3.2) Under the assumption that S satisfies (NA) i.e. K L 0 = f0g, we want to find an equivalent martingale measure Q for the process S. The first argument is wellknown in the present context (compare [S92] and theorem 4. 1 below for a general version of this result; we refer to [S94] for an account on the history of this result, in particular on the work of J.A. Yan [Y80] and D. Kreps [K81] 3.1 Lemma. If S satisfies (NA) the convex cone C is weak star closed in L 1( Omega ; F ; P) and C L 1 ( Omega ; F ; P) f0g. Therefore there is a probability measure Q 1 on ....

....Below we will show that also in this case the above equality remains valid, at least for positive random variables g. This result does not immediately follow from the results in [DS95] Another approach to the problem is to enlarge the concept of admissible integrand in a way as was also done in [S94] and [DS96] For a random variable w 1 we call an integrand H to be w admissible if the process H Delta S can be controlled from below by the function w (see 5.3 and 5.4 below for a formal definition of this concept and of the definition of feasible weight function) The aim of this section is ....

W. Schachermayer, Martingale Measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25--55.

.... this setting it turns out that L 0( Omega ; F ; P) often is the natural space to work in (as opposed to L p( Omega ; F ; P) for some p 0) as it remains unchanged under the passage from P to an equivalent measure Q (while L p( Omega ; F ; P) does change, for 0 p 1) We refer, e.g. DS 94] for a general exposition of the above described duality relations and to [KS 97] for an applications of the bipolar theorem 1.3. 4.2 Note: Lemma 2.3 may be viewed as a variation of theorem 1 in [Y 80] which is a result based on previous work of Mokobodzki (as an essential step in Dellacherie s ....

.... is a result based on previous work of Mokobodzki (as an essential step in Dellacherie s proof of the semimartingale characterization theorem due to Bichteler and Dellacherie; see [Me 79] and [Y 80] The proof of Yan s theorem is a blend of a Hahn Banach and an exhaustion argument (see, e.g. S 94] for a presentation of this proof and [Str 90] S 94] for applications of Yan s theorem to Mathematical 6 W. BRANNATH AND W. SCHACHERMAYER Finance) In fact, these arguments have their roots in the proof of the Halmos Savage theorem [HS 49] and the theorems of Nikishin and Maurey [N 70] M ....

[Article contains additional citation context not shown here]

.W. Schachermayer, Martingale measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25 --- 55.

....integrands was used in the paper [DS94] where a short history of this concept is given. The above definition generalizes the admissibility as used in [DS94] in the sense that it replaces a constant function by a fixed nonnegative integrable function w. The concept was also used by W. Schachermayer [S94], Proposition 4.5. Exactly as in [DS94] we introduce the cone C 1;w = ff j there is a w Gamma admissible integrand H such that f (H Delta M)1 g : 4.11 Theorem D. Let M be a real valued local martingale and w 1 an integrable function. Given a sequence (H n ) n1 of M integrable real valued ....

W. Schachermayer, Martingale Measures for Discrete time Processes with Infinite Hori-zon, Mathematical Finance 4 (1994), 25--56.

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