NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS

Transcription

1 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS DAVID CRIENS Abstract. We study no arbitrage conditions for financial models in which either the stocks itself or its log returns are continuous Itô processes. More precisely, we derive deterministic conditions for the existence and nonexistence of equivalent local martingale measures and strict martingale densities. For models with a random switching mechanism we also study the set of equivalent local martingale measures which are structure preserving. In particular, for one dimensional Markov switching models we provide sufficient and necessary conditions for the existence of structure preserving equivalent local martingale measures. Mathematically, our proofs are based on local changes of measures and existence and uniqueness conditions.. Introduction It is an important question in mathematical finance whether a financial model contains arbitrage opportunities. In fact, no arbitrage conditions are necessary to solve problems of pricing, hedging and portfolio optimization in a meaningful manner. More recently, there is also a growing interest in the construction of models containing specific arbitrage opportunities, see 59 for a discussion. The classical concepts of no arbitrage are the notion of no free lunch with vanishing risk NFLVR as defined by Delbaen and Schachermayer 5; 6 and the notion of no generalized arbitrage NGA as defined by Cherny 7 and Yan 64. The difference between NFLVR and NGA is captured by the concept of a financial bubble in the sense of Cox and Hobson. More precisely, a financial bubble exists if NFLVR holds while NGA fails. In the Benchmark Approach of Platen and coauthors see 53 a pricing theory relying on a weaker notion than NFLVR has been developed. The absence of arbitrage in the Benchmark Approach is implied by the no unbounded profit with bounded risk NUPBR condition, which can be considered as a minimal no arbitrage condition for portfolio optimization, see 4. The Stochastic Portfolio Theory of Fernholz see 23 relies on the no relative arbitrage NRA condition, which is a weaker condition than NGA. As observed by Mijatović and Urusov 48, for one dimensional diffusion models the NRA condition is not necessarily implied by NFLVR. For further notions of no arbitrage we refer to the article of Fontana 25. For continuous semimartingale models the celebrated fundamental theorem of Delbaen and Schachermayer 5; 6 states that NFLVR is equivalent to the existence of an equivalent local martingale measure ELMM. Similar fundamental theorems are known for NGA and NUPBR. More precisely, NGA is equivalent to the existence of an equivalent martingale measure EMM, see 7, and NUPBR is known to be equivalent to the existence of a strict local martingale density SLMD, see 9 or 63. The NRA condition is implied by the existence of a strict martingale density SMD, see 39. If no SMD exists, complete markets satisfy RA, see 22. Date: March 2, Mathematics Subject Classification. 6G44, 6H, 9B7. Key words and phrases. no arbitrage, continuous financial market, Itô process, financial bubbles, Markov switching model. D. Criens - Technical University of Munich, Department of Mathematics, Germany, david.criens@tum.de. This is a revised version of the preprint on the arxiv with identifier: arxiv: v3.

2 2 D. CRIENS In this article we introduce a general continuous framework and study the no arbitrage conditions NFLVR, NGA and NRA via deterministic conditions for the existence and nonexistence of ELMMs and SMDs. In particular, we obtain deterministic conditions for the existence of a financial bubble. We will assume NUPBR as a minimal condition via the existence of a so-called market price of risk, which in fact is equivalent to NUPBR. Let us explain our financial market in more detail. We assume that our real-world probability space supports a d-dimensional Brownian motion W t t, a d-dimensional continuous process S t t with dynamics. ds t = b t dt + σ t dw t, S = s, where b t t and σ t t are progressively measurable processes, and a càdlàg process ξ t t with values in a Polish space. The process ξ t t represents a possible switching mechanism. In classical Markov switching models, S t t is of the form.2 ds t = bs t, ξ t dt + σs t, ξ t dw t, S = s, and ξ t t is a continuous time Markov chain. We discuss this special case also in more detail. In general, however, ξ t t can be chosen very flexible. We fix a finite time horizon T, and assume that our market contains m d risky assets P k t t [,T ] for k =,..., m. We discuss two of the most popular types of models. First, we assume that S t t [,T ] has values in R d m {x R d : x i > for i =,..., m}. In this case, we assume that S k t t [,T ] = P k t t [,T ] for k =,..., m, where S k t t [,T ] denotes the k-th coordinate of S t t [,T ]. We call this type of model a diffusiontype model. It is for instance studied by Mijatović and Urusov 48 and Ruf 57. Second, we consider a stochastic exponential model, i.e. we assume that dp k t = P k t ds k t, k =,..., m. This type of model is in the spirit of the classical Black-Scholes and Heston model. A diffusion version is for instance studied by Criens 3. Conditions for no arbitrage are systematically studied by Delbaen and Shirakawa 7, Mijatović and Urusov 48, Lyasoff 46, Bernard et al. 3 and Criens 3. The framework in 46 is closely related to the stochastic exponential model as studied in this article. The main result in 46 shows that the existence of an ELMM is determined by the equivalence of a probability measure to the Wiener measure. Our main results are different due to their deterministic nature. The models in 3; 3; 7; 48 are of diffusion-type. In the context of Markov switching models we are only aware of the work of Elliott et al. 9; 2, where Esscher-type EMMs for a Markov switching Black-Scholes and a Markov switching Heston model are constructed. We provide a general systematic study. Fontana et al. 26 study NUPBR and NFLVR for a stochastic exponential model in which S t t [,T ] is of the form ds t = µ t dt + σ t, S t {t τ} + σ 2 t, S t {t>τ} dwt, where τ is a stopping time, which represents a change point of the economical situation caused, for instance, by a sudden adjustment in the interest rates or a default of a major financial institution. Our results extend the conditions in 26 for the existence of NFLVR via a local integrability condition on the market price of risk. Furthermore, we give conditions for NGA and GA. Let us now comment in more detail on our contributions, which are threefold. First, we derive general deterministic conditions for the existence and nonexistence of ELMMs and SMDs. These results can be seen as generalizations of those in 3; 48 to a general Itô process setting. In the one dimensional setting our conditions are integral tests evolving functions a, a and ζ which are assumed to satisfy as t σ 2 t ζtas t.

3 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 3 Second, when the σ-fields σξ t, t [, T ] and σw t, t [, T ] are independent, we show that our conditions imply the existence of structure preserving ELMMs, i.e. ELMMs under which the law of ξ t t [,T ] is not affected and such that the process S t t [,T ] is again an Itô process driven by a Brownian motion B t t [,T ] which is still independent of ξ t t [,T ]. Furthermore, we explain how the law of ξ t t [,T ] can be modified. Using one common type of model for the risk neutral and the real world dynamics is interesting from a practical point of view, because it reduces computation complexity. Therefore, we think that structure preserving ELMMs are particularly suitable for applications. Finally, we give necessary and sufficient conditions for the existence of structure preserving ELMM for one dimensional Markov switching models, where we allow countably many different states. More precisely, if S t t is given by.2 our conditions are integral tests evolving the functions x σx, e for fixed states e. To prove the existence of an ELMM we have to verify the martingale property of a candidate density process. We employ different methods for the general setup and the Markov switching case. In the general setup we use a local change of measure and Lyapunov-type arguments. This approach is related to classical existence proofs for martingale problems, which can be found in the monographs of Stroock and Varadhan 62 and Pinsky 52. In 3 a similar technique is applied for a pure diffusion setting. We derive new Lyapunovtype conditions and a Khasminskii-type integral test for general Itô processes. For the case where ξ t t is a Markov chain we use an argument based on the concept of local uniqueness as introduced by Jacod and Mémin 3. The idea traces back to work of Jacod and Mémin 3, Kabanov et al. 34; 35 and Jacod 29, who studied local absolute continuity of general semimartingales. More recently, the approach was used by Cheridito et al. 6 to study local equivalence of possibly explosive jump-diffusions and by Kallsen and Muhle- Karbe 38 to study the martingale property of exponential affine semimartingales. The main difficulties are to show local uniqueness and to prove suitable existence conditions for Markov switching diffusions. We show local uniqueness with a Yamada-Watanabe-type argument and derive existence conditions by an explicit construction of a solution, which is inspired by an argument in 28. A related idea was also applied in 49; 6 to prove conditions for the existence in multidimensional settings with and without state dependent switching. The existence of an EMM or a SMD is proven along the same lines. Our argument for the nonexistence of an ELMM or a SMD is based on a contradiction argument. Using a comparison result in the spirit of Ikeda and Watanabe 27, we use the result of Bruggeman and Ruf 4 that one dimensional diffusions explode arbitrarily fast with positive probability, to prove a Khasminskii-type integral test for the non existence of Itô processes on finite time intervals. Relating one dimensional Markov switching diffusions to classical diffusions, we obtain necessary and sufficient conditions for the existence and non existence on finite time intervals. To show the existence of a structure preserving ELMM we adapt arguments from Ethier and Kurtz 2. The change from a structure preserving ELMM to an ELMM with modified dynamics of ξ t t [,T ] is based on an adaption of a Girsanov-type theorem for Markov processes as given in 5; 55. Let us also comment on the approaches in 3; 48. The main idea in 48 is to relate the martingale property of a candidate density process to so-called separation times, which were introduced in 8. The setting in 3 is a special canonical setup in which the martingale property of the candidate density process is determined by the finiteness of a linear functional under a test probability measure. This approach is closely related to the work of Föllmer 24, Carr et al. 5, Ruf 58, Kardaras et al. 42 and Criens and Glau 4. Finally, we comment on the structure of the article. In Section 2 we introduce our mathematical foundations and in Section 3 we explain the financial models. In Section 4 we present our main results for the general one dimensional framework and in Section 5 we explain how structure preserving ELMM can be modified. In Section 6 we discuss the one dimensional Markov switching setting and in Section 7 we give conditions for the

4 4 D. CRIENS general multidimensional models. We comment on our proofs in Section 8 and the proofs themselves are given in the Sections 9 and. A last comment concerning our standing assumptions: All standing assumptions of the Sections 4 7 are only assumed for the subsection they are stated in. 2. The Mathematical Setting Throughout the whole article, let Ω, F, F t t, P be the underlying filtered probability space with complete right-continuous filtration F t t. If not stated otherwise, all terms such as martingale, local martingale, stopping time etc. refer to F t t. Standing Assumption. The underlying filtered probability space supports an R d - valued Brownian motion W t t and an adapted continuous R d -valued process S t t such that S t = s + b s ds + σ s dw s, t, where b t t is a progressively measurable R d -valued process, σ t t is a progressively measurable R d R d -valued process and s R d. It is implicit that the integrals are well-defined, i.e. P -a.s. where bs + σ s 2 ds < for all t, σ 2 trace σσ with σ being the adjoint of σ. Moreover, the underlying filtered probability space supports an adapted càdlàg E-valued process ξ t t, where E is a Polish space. It is classical to call S t t an Itô process. Our motivation for including the process ξ t t into our framework comes from Markov switching models, which we will discuss in a more detailed manner below. To be more precise, a so-called Markov switching process is of the form S t = s + bs s, ξ s ds + σs s, ξ s dw s, t, where b and σ are suitable Borel functions and ξ t t is a continuous time Markov chain. For this type of model, our aim is not only to study the existence of an ELMM, but to show the existence of structure preserving ELMMs. These can be valuable for applications, when one wants to work with a Markov switching model for the real-world and the risk-neutral dynamics to reduce the computational complexity. Besides the Markov switching case, the process S t t could also be a usual diffusion or a solution to a stochastic differential equation SDE with path-dependent functionals as coefficients see, e.g., 29, to give only two further examples. Next, we parameterize the law of ξ t t via an abstract martingale problem. Let D be the space of all càdlàg functions R + E and let D be the σ-field generated by the coordinate process X t t, which is defined by X t ω = ωt for ω D and t. Furthermore, let Dt o t be the natural filtration of X t t, i.e. Dt o σx s, s [, t], and set D t s>t Do s for all t. Let B n n N be a sequence of nonempty bounded open sets in E such that clb n B n+ and n N B n = E. We set 2. ρ n ω inf t : ωt B n or ωt B n, ω D, n N. As always, the infimum of the empty set is defined to be. If there is a no reason for confusion, we will sometimes also write ρ n when we mean ρ n ξ: Ω [, ]. Note that ρ n is a D o t t -stopping time and that ρ n as n see 2, Proposition 2..5, Problem Let θ be the shift operator, i.e. for s, t and ω D we set θ t ωs ωt + s,

5 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 5 and let Θ be the stopping operator, i.e. for s, t and ω D we set Θ t ωs ωs t. The following are the parameters of the martingale problem: i R, ]. ii A set Σ D, which is stable under stopping, i.e. if ω Σ, then Θ t ω Σ for all t. Moreover, we call Σ to be R-good if for all n N there exists a Σ n D o R ρ n such that Θ R Σ Σ n and Σ = { ω D : ω, θ R ρnωω Σ n Σ }. iii A family A of continuous functions E R such that for each f A, n N and t we have sup fωs ρ n ω : ω Θ t Σ, s [, t] <. iv A map L: A P, where P is the set of all D o t t -progressively measurable processes, such that ρnω Lfω, s ds < for all n N, ω D, f A and t, and for each n N, f A and t v e E. sup Lfω, s ρ n ω : ω Θ t Σ, s [, t] <. Remark. A natural choice for Σ is D. Another important choice for Σ is the set {ω D : t ωt is continuous}, which is in D due to 2, Exercise In both cases Σ is R-good, because we can take Σ n to be Θ R ρ n Σ. The set Σ and the initial value e will not change in the following, such that we suppress them in our notation. If R =, we use the convention that [, R] R +. Definition. We say that ξ t t is a solution process to the martingale problem A, L, R, if for all f A and n N the process f t ρn ξ t ρn Lfξ, sds, t [, R], is a martingale and P ξ t R t Σ = P ξ = e =. Furthermore, we say that the martingale problem is uniquely solvable if the law of ξ t t [,R] is unique. Let us give two examples for martingale problems. Example. Suppose that E = R and let ξ t t be a solution process to the SDE dξ t = uξ, tdt + vξ, tdb t + gξ, t, x { gξ,t,x } d p q t + gξ, t, x { gξ,t,x >} dp t, where B t t is a one dimensional Brownian motion and p is a Poisson random measure with compensator q = dt F. Here, the coefficients u, v and g are supposed to be pathdependent predictable functionals, see 29, Section 4. for the precise formalism. Under appropriate local boundedness assumption on the coefficients see, e.g., 29, Hypothesis 3.5 we can choose R =, Σ = D, A = Cb 2 R and Lfξ, t f ξ t uξ, t + 2 f ξ t v 2 ξ, t + fξt + gξ, t, x fξ t f ξ t gξ, t, x { gξ,t,x } F dx. The martingale problem is, for instance, uniquely solvable when the coefficients are locally Lipschitz continuous see 29, Chapter 4 or if the coefficients have a Markovian

6 6 D. CRIENS structure, are locally bounded and v is continuous and non-degenerated see 29, Theorem 3.58, Exercise 3.2. Example 2. Suppose that E = {,..., N}, with N, and that ξ t t is a continuous time Markov chain with Q-matrix Q, which is a Feller process in the classical sense, 2 see, e.g., 54 or 2, Section I..5 for conditions on the Q-matrix. It is important to clarify our terminology: When we speak of a Markov chain, we always mean a Markov chain for the filtration F t t. Due to 54, Theorem 5 and Dynkin s formula see, e.g., 55, Proposition VII..6, the process ξ solves the martingale problem for R =, Σ = D, A = { f c : Qf c } and Lfξ, t = Qfξ t. In particular, due to 44, Theorem 4..3, the martingale problem is uniquely solvable. 3. The Financial Market We fix a finite time horizon T,. In the following, we distinguish between a diffusion-type model DM, where components of S t t [,T ] represent the assets, and a stochastic exponential model SEM, where S t t [,T ] represents the log-returns of the assets. We think that discussing these two types will cover most continuous models. Let us formally introduce them: Definition 2. i We say that we consider a DM m, d, if the market consists of m d risky assets Pt k t [,T ] for k =,..., m and P k t = S k t, t [, T ], where S k t t [,T ] denotes the k-th component of S t t [,T ]. In particular, S t t [,T ] is R d m-valued, where R d m { x R d : x i > for all i =,..., m }. ii We say that we consider a SEM m, d, if the market consists of m d risky assets P k t t [,T ] for k =,..., m and dp k t = P k t ds k t, P k = p k >. In the following we will either consider a DM or a SEM. Of course, for both cases the concepts of ELMMs and SLMDs are the same. Definition 3. i We call a probability measure Q on Ω, F an ELMM, if it has the following two properties: a Q P. b For all k =,..., m the process Pt k t [,T ] is a local Q-martingale. ii We call a strictly positive local P -martingale Z t t [,T ] a SLMD, if for all k =,..., m the process Z t Pt k t [,T ] is a local P -martingale. Remark 2. Regardless whether one considers a DM or an SEM, if d = m and an ELMM Q exists, it follows from Girsanov s theorem see 32, Theorem III.3.24 that there exists a Brownian motion B t t [,T ] such that 3. S t = s + σ s db s, t [, T ]. of course, when N = we mean E = N 2 i.e. the corresponding transition semigroup is a self-map on the continuous functions vanishing at infinity

7 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 7 To be more precise, Girsanov s theorem yields that there exists a predictable process h t t [,T ] of finite variation such that B t W t + h t, t [, T ], is a Q-Brownian motion. Using 29, Proposition 7.26, we have Q-a.s. S t = s + b s ds σ s dh s + σ s db s, t [, T ]. Because the stochastic integral σ sdb s t [,T ] is a local Q-martingale and S t t [,T ] is a local Q-martingale by the definition of an ELMM, we conclude that H t b s ds σ s dh s, t [, T ], is a predictable local Q-martingale of finite variation. Thus, Q-a.s. H t for all t [, T ] and 3. follows. We stress that this observation implies that if no Itô process Y t t [,T ] of the type dy t = σ t db t with Y = s exists, then also no ELMM exists. 3 Typically, due to the generality of our setting, there are several ELMMs. A simple example is the following: Example 3. Consider an SEM, with S t = ξ s dw s, t [, T ], and let ξ t t [,T ] be a real-valued Brownian motion independent of W t t [,T ]. Then, P is an ELMM and so is any probability measure Q a defined by the Radon-Nikodym derivative dq a = e aξ T 2 a2t dp, a R\{}. Under Q a the process W t t [,T ] remains a Brownian motion and the process ξ t t [,T ] turns into a Brownian motion with a linear non-trivial drift. In many applications, the drivers ξ t t [,T ] and W t t [,T ] are independent processes. In such cases, we ask whether there exists an ELMM which preserves the Itô process structure together with the joint distribution of the drivers, or at least the independence. In the latter case it is also interesting to study how the law of ξ t t [,T ] can be changed. To keep track of the law of ξ t t [,T ] and to give a precise description how the law is affected by a change of measure we defined the abstract martingale problem. Definition 4. We call an ELMM Q a structure preserving ELMM SPELMM, if there exists a Q-Brownian motion B t t [,T ] such that S t t [,T ] is an Itô process driven by B t 4 t [,T ] and P ξ t, W t t [,T ] = Q ξ t, B t t [,T ]. Finally, we introduce the concept of a market price of risk MPR. Throughout the article, we denote a σσ. 3 This observation can also be seen without doing any computation: By the definition an ELMM, the process S t t [,T ] is a local Q-martingale. Furthermore, because Q P and the fact that equivalent changes of measures do not affect the quadratic variation process since it is a limit in probability, the process S t t [,T ] is a continuous local Q-martingale with quadratic variation process σsσ s ds t [,T ]. A classical representation theorem for continuous local martingales see, e.g., 55, Theorem V.3.9 yields now the existence of an Itô process Y t t [,T ] of the type dy t = σ tdb t with Y = s. 4 i.e. dst = v tdt + u tdb t for progressively measurable processes v t t [,T ] and u t t [,T ] ; of course, when Q is an ELMM we can choose v

8 8 D. CRIENS Definition 5. An R d -valued progressively measurable process M t t [,T ] is called MPR, if P -a.s. and P -a.s. for all k =,..., m where e k is the k-th unit vector. T a s M s, M s ds < e k, b s + a s M s ds =, t [, T ], Remark 3. The existence of a MPR is equivalent to the existence of a SLMD. This can be proven similar to 3, Theorem 3.. Consequently, assuming the existence of a MPR is equivalent to assuming that NUPBR holds. Remark 4. If a is invertible, then a b is a candidate for a MPR. If b = σθ, then σ + θ, where σ + denotes the Moore-Penrose inverse of σ, is also a candidate for a MPR. Next, we will provide conditions for the existence of ELMMs and SMDs in all generality and for SPELMM when ξ t t,t ] and W t t [,T ] are independent. 4. Conditions for the Existence and Absence of Arbitrage for One Dimensional Models 4.. The one dimensional Diffusion-Type Model. Let us start with a DM,. Note that in this case the process S t t [,T ] is, -valued. Furthermore, we impose the following standing assumption: Standing Assumption 2. σ t for all t [, T ]. We start formulating conditions. Our first condition concerns the existence of a good version of a MPR. We define 4. τ n inf t [, T ]: S t n+, n, n N. Condition. There exists a MPR M t t [,T ] such that the process Z t exp M s, dss c 4.2 a s M s, M s ds, t [, T ], 2 where S c t t [,T ] is the continuous local martingale part of S t t [,T ], i.e. S c t σ s dw s, t [, T ], is a local martingale with localizing sequence τ n n N. The assumptions in Condition are the existence of a MPR and that τ n n N is a localization sequence, because Z t t [,T ] is a well-defined local martingale already from the definition of a MPR. Due to Novikov s condition, τ n n N is a localizing sequence when for all n N [ ] T τn E exp a s M s, M s ds <. 2 Condition is only local in the S variable, because τ n only depends on the paths of S t t [,T ]. The global structure in the ξ variable will be present also in the results presented below. This is inflicted by the generality of our setting. In Section 6 below we study the special case where the law of ξ t t is a continuous time Markov chain. For this framework we present existence and nonexistence conditions which have a local structure in both variables.

9 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 9 Condition 2. The martingale problem A, L, T is uniquely solvable and ξ t t is a solution. Furthermore, the σ-fields σw t, t [, T ] and σξ t, t [, T ] are independent. Remark 5. Condition 2 will only be assumed when we study SPELMMs. The independence assumption is satisfied when we consider a Markov switching model. More precisely, when ξ t t is as in Example 2, it follows as in the proof of Lemma 2 below that σξ t, t R + and σw t, t R + are independent. Key of this fact is our assumption that ξ t t and W t t solve martingale problems for the same filtration and that ξ t t has only finitely many jumps in a finite time interval. Condition 3. There exists a Borel function a:,, and a Borel function ζ : [, T ] [, such that ζ L and for all t [, T ] and all ω Ω a t ω ζtas t ω. We define H to be the class of all Borel functions h: [, [, which are starting at zero, are strictly increasing and satisfy for all ɛ >. ɛ dz h 2 z = Condition 4. There exists a continuous function a:,, such that for all n N there is an h n H such that for all x, y [ n+, n] 4.3 a 2 x a 2 y hn x y and for all t [, T ] and all ω Ω a t ω as t ω. The main results of this section are the following. Theorem. i Suppose that the Conditions and 3 hold. Let a be as in Condition 3. If we have a L loc, z dz =, az then an ELMM exists. If, in addition, Condition 2 holds, then a SPELMM exists. ii Suppose that Condition 4 holds and let a be as in the very same. If we have then no ELMM exists. z dz <, az This result illustrates that the existence and nonexistence of an ELMM depends on the behavior on the left boundary of the state space. In Condition 4 we have no time localization, because in time-inhomogeneous cases we cannot expect to get the nonexistence of an ELMM for arbitrary finite time horizons. For usual diffusions, the main conditions boil down to those in 48. Remark 6. When f :,, is Borel, a simple condition for fz const. z 2, z. zdz fz = is We denote by K the set of all Borel functions κ: [, [,, which are starting at zero, are strictly increasing and concave and satisfy for all ɛ >. ɛ dz κz =

10 D. CRIENS Condition 5. There exists a continuous function a:,, such that for all n N there is an h n H and a κ n K such that for all x, y [ n+, n] a 2 x a 2 y h n x y, 4.4 x ax y ay κ n x y, and for all t [, T ] and all ω Ω a t ω as t ω. Theorem 2. Suppose that Condition holds. i Assume that Condition 3 holds and let a as in the very same. Moreover, suppose that a L loc, z az dz = z dz =. az Then, an EMM exists. If, in addition, Condition 2 holds, then a SPEMM exists. ii Suppose that the Conditions 3 and 5 hold and let a and a be as in these. Furthermore, assume that a L loc, z dz =, az z dz <. az Then, there exists an ELMM which is no EMM, i.e. the model contains a financial bubble in the sense of. If, in addition, Condition 2 holds, then there exists a SPELMM which is no EMM. The additional conditions which determine whether an ELMM is an EMM only depend on the right boundary of the state space. Again, for usual diffusions, the main conditions boil down to those in 48. A comment on the proofs of the Theorems and 2 and a mathematical explanation why the additional conditions for the existence of an EMM only depend on the right boundary can be found in Section 8 below. Remark 7. When f :,, is Borel, a simple condition for fz const. z 2, z. zdz fz = is Theorem 3. i Suppose that the Conditions and 3 hold and let a be as in the latter. Moreover, assume that a L loc, z dz =. az Then, a SMD exists. In particular, if M t for all t [, T ], then S t t [,T ] is a P -martingale. ii Suppose that Condition 5 holds and let a be as in the very same. If then no SMD exists. z dz <, az Again, for usual diffusions, the main conditions boil down to those in 48. Example 4. Let β α > and assume that Condition holds. i Let τ be an a priori given finite stopping time and suppose that a t = S α t {t τ} + S β t {t>τ}, t. We consider this example as a version of the CEV model see 2 in the spirit of the work of Fontana et al. 26 on which we comment more detailed in Example 5 below. The stopping time τ can be interpreted as a change point of the economical situation, which could be caused by a sudden adjustment in the interest rates or

11 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS a default of a major financial institution. Set ax = minx α, x β and ax = maxx α, x β. Clearly, we have as t a t as t, t [, T ]. Moreover, a satisfies 4.4 with h n x κ n x x. To see this, note that fx = x α and gx = x β are locally Lipschitz on,. Then, because minfx, gx minfy, gy fx fy + gx gy, which follows from the identity minf, g = 2 f + g f g, also minf, g is locally Lipschitz on,. Because also x and x are locally Lipschitz on, and compositions and products of locally Lipschitz functions are again locally Lipschitz, the claim follows. Now, we note that z az dz = z α dz = α 2, and that z az dz = z β dz = β 2. Consequently, Theorem yields that an ELMM exists if α 2 and no ELMM exists if β < 2. Furthermore, we have z az dz = z β dz = β 2, and z az dz = z α dz = α 2. Thus, the Theorems 2 and 3 imply that an EMM exists if β = α = 2, a SMD exists if β 2 and no SMD hence also no EMM exists if α > 2. Consequently, a financial bubble exists if α > 2. In summary, we obtain the following conditions: β < 2 β = α = 2 α > 2 SMD, no ELMM, no EMM EMM, ELMM, SMD Financial bubble, ELMM, no SMD, no EMM Finally, we comment on the choice of b t t, the existence of S t t and Condition. If α 2, we could take b t = S t, t, which is in the spirit of the CEV model. Due to 4, Proposition 5.2.3, Corollary , Theorems 5.5.5, there exist two, -valued continuous processes X t t and Y t t with dynamics where dy t = Y t dt + Y α 2 t dw t, Y = s, dx t = X t dt + X β 2 t db t, X = Y τ, B t W t+τ W τ, t, is a Brownian motion. Then, it can be shown as in the proof of Lemma 5 below that { Y t, t τ, S t X t τ, t > τ, 5 The conclusion also follows from a transformation of the CEV processes into CIR processes as explained in 33, Section

12 2 D. CRIENS satisfies ds t = S t dt + S α t {t τ} + S β t {t>τ} dwt = b t dt + a t dw t, S = s. Furthermore, we can take M t = St α {t τ} S β t {t>τ}, t [, T ], as MPR and Novikov s condition implies that M t t [,T ] satisfies Condition. In the case β < 2, we can take b t = a t S t, t, and similar arguments as before yield that S t t exists and that Condition is satisfied. ii Suppose that a t = S α t + inf s [,t] S s, t. This is a toy example for a path-dependent coefficient. We choose ax = x α and ax = xα +s and as in part i of this example it follows that an ELMM exists if and only if α 2, and a SMD exists if and only id α 2, and an EMM exists if and only if α = The one dimensional Stochastic Exponential Model. Next, we discuss the SEM,. As we will see below, this model differs substantially from DTM,. In this section, we redefine τ n inf t [, T ]: S t n, n. Standing Assumption 3. σ t for all t [, T ]. Condition 6. There exists a Borel function a: R, and a Borel function ζ : [, T ] [, such that ζ L and for all t [, T ] and all ω Ω a t ω ζtas t ω. Condition 7. There exists a continuous function a: R, such that for all n N there is an h n H and a κ n K such that for all x, y [ n, n] a 2 x a 2 y h n x y, ax ay κ n x y, and for all t [, T ] and all ω Ω a t ω as t ω. The main result of this section is the following: Theorem 4. i Suppose that the Conditions and 6 hold and let a be as in the latter. If a L loc, then an ELMM exists. Moreover, if, in addition, Condition 2 holds, then a SPELMM exists. ii Suppose that the Conditions and 6 hold and let a be as in the latter. If a L loc, dz az =, then an EMM exists. Moreover, if, in addition, Condition 2 holds, then a SPEMM exists.

13 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 3 iii Suppose that the Conditions, 6 and 7 hold. Let a and a be as in the latter. If a L loc, dz az <, then there exists no SMD, but an ELMM which is no EMM. Moreover, if, in addition, Condition 2 holds, then there exists a SPELMM which is no EMM. We have a very mild condition for the existence of an ELMM. A similar condition was given in 3 in a pure diffusion setting. Furthermore, we stress that the existence of an EMM immediately implies the existence of a SMD, see 32, Proposition III.3.8. Thus, the conditions given in Theorem 4 ii also imply the existence of a SMD. The proof of this result is identical to the proofs of the Theorems, 2 and 3 and is therefore omitted. Remark 8. When f : R, is Borel, a simple condition for fz const. z, z. dz fz = is Example 5. Fontana et al. 26 study NUPBR and NFLVR for a model with a change point. The main interest lies in the influence of underlying filtrations, which represent different levels of informations. Under a weak form of the classical H -hypothesis the model can be included into our framework. More precisely, in this case S t t [,T ] is of the form ds t = µ t dt + σ t, S t {t τ} + σ 2 t, S t {t>τ} dwt, where τ is a stopping time. The coefficient σ i is assumed to be positive, continuous and Lipschitz continuous in the second variable uniformly in the first, see 26, Condition I. Theorem 4 provides a local condition for NFLVR, namely Condition. In particular, for the two special cases described in 26, Section 3.3, we have 4.5 µ t = µ t, S t {t τ} + µ 2 t, S t {t>τ}, where µ i is locally bounded, and NFLVR holds due to Theorem 4. This extends the observation in 26 that NUPBR holds in this situation. Furthermore, again in view of Theorem 4, if in addition to 4.5 for i =, 2 σ i t, x 2 const. x, x, t [, T ], then an EMM exists, i.e. NGA holds. 5. Modifying SPELMMs In the previous sections we discussed conditions for the existence of SPELMMs. It is a natural question whether it is possible to find an ELMM which affects also the dynamics of ξ t t [,T ] in a tractable manner. In this section, we explain that once we found an SPELMM, we can also find an ELMM which preserves independence of the drivers, but changes the law of ξ t t [,T ]. The structure of the changed dynamics is classical for martingale problems, see 5 or 55, Section VII.3. Condition 8. The martingale problem A, L, T is uniquely solvable and ξ t t is a solution process. Furthermore, the σ-fields σξ t, t R + and σw t, t R + are independent. Theorem 5. Suppose that Condition 8 holds. Let f A be positive such that T Lfω, s fωs ds < for all ω D and suppose that the process fξ t t fe exp Lfξ, s 5. fξ s ds, t [, T ], is a martingale. Set A { g A: gf A },

14 4 D. CRIENS and L Lfg glf g. f Suppose that L satisfies the same local boundedness assumptions as L. Then, there exists a probability measure Q such that Q P, the process W t t is a Q-Brownian motion, the σ-fields σξ t, t R + and σw t, t R + are Q-independent and under Q the process ξ t t solves the martingale problem A, L, T. This theorem shows that in many cases where ξ t t [,T ] has a non-trivial role the existence of a SPELMM implies the existence of infinitely many different ELMMs, see Example 3 for a simple special case. We discuss a more explicit version of Theorem 5 in the following section, where we give a detailed discussion of the special case where ξ t t is a continuous time Markov chain. Remark 9. When one is interested in EMMs instead of ELMMs, one can also use Theorem 5 to find a suitable EMM with changed dynamics of ξ t t [,T ] by considering an ELMM as real-world measure. Remark. Note that L g = Lg + where Γ is the opérateur carré du champ, i.e. Γf, g, f Γf, g Lfg flg glf. A useful criterion for the martingale property of 5. is given by the following proposition, which can be seen as an extension of observations in 6; 29; 62. To formulate it we require more terminology. In the following X t t denotes the coordinate process on D, i.e. X t ω = ωt for all ω D and t. When we talk about a martingale on D, D we always refer to the right-continuous filtration D t t. Definition 6. A set A A is called a determining set for the martingale problem A, L, if for all e E a probability measure µ on D, D is the law of a solution process to the martingale problem A, L, with initial value e if and only if µσ = µx = e = and f t ρn X t ρn LfX, sds, t, is a µ-martingale for all n N and f A. Example 6. For R d -valued diffusions with locally bounded coefficients, i.e. Σ = {ω D : t ωt is continuous}, A = C 2 R d and Lfξ, t = fξ t, bξ t + 2 trace 2 fξ t aξ t, where b: R d R d and a: R d S d S d is the space of real valued symmetric non negative definite d d matrices are locally bounded Borel functions, the set A { f i, g ij : i, j =,..., d }, where f i x = x i and g ij x = x i x j, is a determining set, see, e.g., 4, Proposition Proposition. Suppose that ξ t t solves the martingale problem A, L, T, that Σ is T - good and that f, A and L are as in Theorem 5. Moreover, assume that either E is discrete or that there exists a countable determining set for the martingale problem A, L, and that L gξ, t = Kgξ t, where K maps A into the space of Borel functions E R. Finally, we assume that the process 5. is a local martingale with localizing sequence ρ n n N and that the martingale problem A, L, has a unique solution for all deterministic initial values. Then, the process 5. is a martingale.

15 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 5 This proposition shows that in Markovian settings we can essentially modify the law of ξ t t [,T ] to a solution to any martingale problem A, L, T for which the global martingale problem A, L, is well-posed. Example 7. Let us discuss a version of Theorem 5 for one dimensional diffusions. We assume that E = R and that dξ t = uξ t dt + vξ t db t, where B t t is a one dimensional Brownian motion and u: R R and v : R R are locally bounded Borel functions. Then, we can take Σ = {ω D : t ωt is continuous}, A = C 2 R and Lgξ, t = uξ t g ξ t + 2 v2 ξ t g ξ t, g A. Let c C R and set x fx exp cydy, which is positive and satisfies f A. A short computation reveals that 5. is a local martingale with localizing sequence ρ n n N. For g A we have L gξ, t = u + f f v2 ξ t g ξ t + 2 v2 ξ t g ξ t = u + v 2 c ξ t g ξ t + 2 v2 ξ t g ξ t. Thus, the process 5. is a martingale whenever the SDE dy t = u + v 2 c Y t dt + vy t db t, where B t t is a one dimensional Brownian motion, has a unique solution for all deterministic initial values. In particular, we see that the structure of the change of measure is the same as in the classical version of Girsanov s theorem for diffusions. 6. Conditions for the existence and absence of arbitrage for One Dimensional Markov Switching Models In this section we discuss Markov switching models as an important special case. Standing Assumption 4. We have E = {,..., N} for N and ξ t t is an E-valued irreducible continuous time Markov chain with Q-matrix Q which is a Feller process and ξ = e. Let us clarify the terminology: A continuous time Markov chain ξ t t is called irreducible if for all t or, equivalently, some t 6 we have P ξ t = e ξ = e 2 > for all e, e 2 E. Remark. Note that Standing Assumption 4 implies Condition 8, see Example 2 and Remark 5. We stress that we allow countably many possible states for ξ t t. Standing Assumption 5. Let I =, when we consider the DM,, and I = R when we consider the SEM,. For all t [, T ] and ω Ω we have b t ω = bs t ω, ξ t ω, σ t ω = σs t ω, ξ t ω, where b: I E R and σ : I E R are Borel functions. Moreover, the map x σx, e is continuous for all e E and σ. 6. We set where we redefine τ n 6 see, e.g., 45, Remark 2.48 γ n τ n ρ n, n N, { inf t [, T ]: S t n+, n, if I =,, inf t [, T ]: S t n, n, if I = R.

16 6 D. CRIENS Condition 9. There exists a MPR M t t [,T ] such that the process 4.2 is a local martingale with localizing sequence γ n n N. Condition. I =, and for all e E and n N there exists an h n,e H such that for all x, y [ n+, n] 6.2 σx, e σy, e h n,e x y. Condition. I = R and for all e E and n N there exists an h n,e H such that for all x, y [ n, n] the inequality 6.2 holds. We have the following results: Theorem 6. We consider the DM,, i.e. I =,. Assume that the Conditions 9 and hold. i An SPELMM exists if for all e E ii An SPEMM exists if for all e E z σ 2 z, e dz = z σ 2 dz =. z, e z σ 2 dz =. z, e iii An SMD exists if for all e E z σ 2 dz =. z, e Theorem 7. We consider the SEM,, i.e. I = R. Assume that the Conditions 9 and hold. i An SPELMM exists. ii An SPEMM exists if for all e E dz σ 2 z, e =. The following theorems are converse to the previous theorems: Theorem 8. We consider the DM,, i.e. I =,. i Suppose that there is an e E such that for all n N there exists an h n,e H such that for all x, y [, n] the inequality 6.2 holds, and n+ z σ 2 dz <. z, e Then, there exists no ELMM such that ξ t t is an irreducible recurrent 7 continuous time Markov chain which is a Feller process. ii Suppose that there is an e E such that for all n N there exists an h n,e H such that for all x, y [, n] the inequality 6.2 holds, and n+ z σ 2 dz <. z, e Then, there exists no EMM such that ξ t t is an irreducible recurrent continuous time Markov chain which is a Feller process. 7 A continuous time Markov chain ξt t is called recurrent if P {t R + : ξ t = e} is unbounded ξ = e = for all e {,..., N}. If the chain is irreducible it is recurrent if and only if the previous property holds for some e {,..., N}. In particular, if ξ t t is irreducible and N <, then ξ t t is recurrent, see 5, Theorems.5.6, 3.4..

17 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 7 Theorem 9. We consider the SEM,, i.e. I = R. Suppose there exists an e E such that for all n N there exists an h n,e H such that for all x, y [ n, n] the inequality 6.2 holds, and dz σ 2 z, e <. Then, there exists no EMM such that ξ t t is an irreducible recurrent continuous time Markov chain which is a Feller process. The proofs of the Theorems 7 and 9 are omitted due to their similarity to the proofs of Theorems 6 and 8. Finally, let us comment on an application of Theorem 5 in the present setting. Proposition 2. Suppose that N < and Q = q ij i,j E. Let f = f,..., f N R N with f i > for all i E and let A and L as in Theorem 5. Then, A = A and L = qi,j i,j E with { f j qij qij f i, i j, 6.3 k i q ik f k fi, i = j. Proof: See 5, Proposition 5.. Remark 2. The assumption N < in the previous proposition can be relaxed, but this requires additional assumptions. Since the proposition has the purpose of illustration, we omit this technical issue. We now outline an explicit application of Theorem 5 and the previous proposition: Suppose that N < and that Q is a SPELMM. Then, Theorem 5 and Proposition 2 yield that for any positive vector f = f,..., f N R N we find another ELMM Q such that the process ξ t t [,T ] has the law of an irreducible Markov chain with the modified Q-matrix q ij i,j E given by 6.3 and is independent of the driving Brownian motion. In this case the existence of a SPELMM implies the existence of infinitely many ELMMs which are structure preserving in a wider sense. 7. Conditions for the Absence and Existence of Arbitrage for Multidimensional Models 7.. The multidimensional Diffusion-Type Model. We now turn to the multidimensional DM m, d. In this section we present a variety of deterministic conditions for the existence and nonexistence of ELMMs and SMDs, which are more technical than in the previous sections. We consider this section as well as the following as a collection of conditions. Let I n n N R d m and redefine Now, we collect several conditions: τ n inf t [, T ]: S t I n. Condition 2. There exists a MPR M t t [,T ] such that the process 4.2 is a local martingale with localizing sequence τ n n N and an R d -valued progressively measurable process c t t [,T ] such that P -a.s. for all t [, T ] and k =,..., d 7. e k, b s + a s M s ds = e k, c s ds. Example 8. Let us discuss Condition 2 for a switching version of the diffusion model studied by Ruf 57. Suppose that for k =,..., d d dst k = St k κθ k S t, t, ξ t dt + κ ki S t, t, ξ t dwt i, i=

18 8 D. CRIENS where κ: R d d [, T ] E Rd R d and θ : R d d [, T ] E Rd are Borel functions. Let κ + the Moore-Penrose inverse of κ. We see that a ik t St i = St k κκ ki S t, t, ξ t. One candidate for a MPR is given by because Mt k κ+ θ k S t, t, ξ t St k, a t M t k = S k t κκ κ + θ k S t, t, ξ t = S k t κθ k S t, t, ξ t, which shows that 7. holds. Moreover, we have a t M t, M t = κθs t, t, ξ t, κ + θs t, t, ξ t = κ + κθs t, t, ξ t, θs t, t, ξ t. In particular, if κ has linearly independent columns, then κ + κ = id and a t M t, M t = θs t, t, ξ t 2. In this case, one can readily formulate local boundedness conditions on θ such that τ n n N is a localizing sequence for 4.2. For r > and two continuous functions f, g : [r, R with g > we set x y y 2 exp u r 7.2 vf, g, rx exp 2fzdz 2fzdz dudy. gu r r Condition 3. Condition 2 holds with I n = {x R d m : x < n} and let c t t [,T ] be as in the very same. Moreover, assume that there exists a Borel function ζ : [, T ] [, such that ζ L and t [, T ] and for all ω Ω trace a t ω + 2 S t ω, c t ω ζt + S t ω 2. Condition 4. Condition 2 holds with I n = {x R d m : n+ < x < n} and let c t t [,T ] be as in the very same. Moreover, there exists a Borel function ζ : [, T ] [, such that ζ L and for all t [, T ] and all ω Ω trace a t ω + c t ω ζt S t ω 2. Condition 5. Condition 2 holds with I n = {x R d m : x < n} and let c t t [,T ] be as in the very same. For all i =,..., m there exists a Borel function ζ i : [, T ] [, such that ζ i L and for all t [, T ] and ω Ω trace a t ω + 2 S t ω, c t ω + S tω,e i a tωe i ζ i t + S t ω 2. Condition 6. Condition 2 holds with I n = {x R d m : n+ < x < n} and let c t t [,T ] be as in the very same. For all i =,..., m there exists a Borel function ζ i : [, T ] [, such that ζ i L and for all t [, T ] and ω Ω trace a t ω + 2 S t ω, c t ω + S tω,e i a tωe i ζ i t S t ω 2. Condition 7. Condition 2 holds with I n = {x R d m : x < n} and let c t t [,T ] be as in the very same. Moreover, there exists an r > and a Borel function ζ : [, T ] [, such that ζ L and for all t [, T ], ω Ω: S t ω 2r trace a t ω + c t ω ζt. Moreover, there are continuous functions A: [r,, and B : [r, R such that for all t [, T ], ω Ω: S t ω 2r ζta S tω 2 2 at ωωs, S t ω, a t ωs t ω, S t ω B S tω 2 trace at ω + 2 S t ω, c t ω, 2 r

19 NO ARBITRAGE IN CONTINUOUS FINANCIAL MARKETS 9 and lim v t 2 B, A, r t =. Condition 8. Condition 2 holds with I n = {x R d m : n+ < x < n} and c t t [,T ] is as in the very same. Moreover, there exists an r > and a Borel function ζ : [, T ] [, such that ζ L and continuous functions A : [r,,, B : [r, R, A 2 :, r], and B 2 :, r] R such that for all t [, T ], ω Ω: S t ω 2r ζta S tω 2 2 at ωωs, S t ω, a t ωs t ω, S t ω B S tω 2 trace at ω + 2 S t ω, c t ω, for all t [, T ], ω Ω: S t ω 2r 2 and ζta 2 S tω 2 2 at ωωs, S t ω, a t ωs t ω, S t ω B 2 S tω 2 trace at ω + 2 S t ω, c t ω, 2 lim v t 2 B, A, r t = lim v t 2 B2, A 2, r t =. Condition 9. Condition 2 holds with I n = {x R d m : x < n} and let c t t [,T ] be as in the very same. Moreover, for i =,..., m there exists an r i > and a Borel function ζ i : [, T ] [, such that ζ i L and for all t [, T ], ω Ω: S t ω 2r i trace a t ω + c t ω + S tω,e i a tωe i ζ i t. Moreover, there are continuous functions A i : [r i,, and B i : [r i, R such that for all t [, T ], ω Ω: S t ω 2r i ζ i ta Stω 2 i 2 at ωωs, S t ω, a t ωs t ω, S t ω B Stω 2 i 2 trace at ω + 2 S t ω, c t ω + S a tω,e i tωe i, and lim v t 2 B i, A i, r i t =. Condition 2. Condition 2 holds with I n = {x R d m : n+ < x < n} and c t t [,T ] is as in the very same. Moreover, for i =,..., m there exists an r i > and a Borel function ζ i : [, T ] [, such that ζ i L and continuous functions A i : [r i,,, Bi : [r i, R, A 2 i :, r i], and B 2 :, r i ] R such that for all t [, T ], ω Ω: S t ω 2r i Stω 2 at ωωs, S t ω, ζ i ta i a t ωs t ω, S t ω B i 2 Stω 2 2 trace at ω + 2 S t ω, c t ω + S tω,e i a tωe i, for all t [, T ], ω Ω: S t ω 2r i ζ i ta 2 Stω 2 i 2 at ωωs, S t ω, a t ωs t ω, S t ω Bi 2 Stω 2 2 trace at ω + 2 S t ω, c t ω + S a tω,e i tωe i, and lim t v 2 B i, A i, r i t = lim t v 2 B2 i, A 2 i, r i t =.

20 2 D. CRIENS Condition 2. There exist continuous functions A:,, and B :, R and for all n N there exist an h n H and a κ n K such that for all x, y [ n+, n] 7.3 A 2 x A 2 y hn x y, AxBx AyBy κ n x y, and for all t [, T ], ω Ω A S tω 2 2 at ωs t ω, S t ω, a t ωs t ω, S t ω B S tω 2 trace at ω, and 2 lim v t 2 B, A, t <. Condition 22. There exist continuous functions A:,, and B :, R and for all n N there exist an h n H and a κ n K such that 7.3 holds for all x, y [ n+, n], and for all t [, T ], ω Ω A S tω 2 2 at ωs t ω, S t ω, a t ωs t ω, S t ω B S tω 2 trace at ω, and 2 lim v t 2 B, A, t <. Condition 23. There exist an i {,..., d}, continuous functions A:,, and B :, R and for all n N there exist an h n H and a κ n K such that 7.3 holds for all x, y [ n+, n], and for all t [, T ], ω Ω A S tω 2 2 at ωs t ω, S t ω, a t ωs t ω, S t ω B S tω 2 2 trace at ω + 2 S t ω, S a tω,e i tωe i, and lim v t 2 B, A, t <. Condition 24. There exist an i {,..., d}, continuous functions A:,, and B :, R and for all n N there exist an h n H and a κ n K such that 7.3 holds for all x, y [ n+, n], and for all t [, T ], ω Ω A S tω 2 2 at ωs t ω, S t ω, a t ωs t ω, S t ω B S tω 2 2 trace at ω + 2 S t ω, S a tω,e i tωe i, and lim v t 2 B, A, t <. The main result of this section are the following two theorems. Theorem. i If one of the Conditions 3, 4, 7 and 8 holds, then an ELMM exists. If, in addition, Condition 2 holds, then a SPELMM exists. ii If one of the Conditions 3, 4, 7 and 8 and one of the Conditions 5, 6, 9 and 2 hold, then an EMM exists. If, in addition, Condition 2 holds, then a SPEMM exists. iii If one of the Conditions 5, 6, 9 and 2 holds, then a SMD exists. In particular, if M t for all t [, T ], then St k t [,T ] is a P -martingale for all k =,..., m. Theorem. Suppose that d = m. i If one of the Conditions 2 and 22 holds, then there exists no ELMM.