aa r X i v : . [ m a t h . P R ] M a y Strict Local Martingales: Examples
Xue-Mei Li
Mathematics Institute, The University of Warwick, Coventry, U.K.
Abstract
We show that a continuous local martingale is a strict local martingale if its supremumprocess is not in L α for a positive number α smaller than . Using this we construct afamily of strict local martingales which are not Bessel processes.AMS Mathematics Subject Classification : 60G44, 60G05.Key words: strict local martingales, examples, oscillations, small moments Gauged by the small moments of the supremum of a local martingale, we determinewhether a stochastic integral is a strict local martingale. We construct a family ofstrict local martingales, using neither Bessel processes nor the speed measures of onedimensional diffusions. This can be considered as an addendum to [ELY99, ELY97], awork began around the period Marc Yor used to visit Coventry with his football teamand I just completed my thesis and was obsessed with martingales. I have many fondmemories of visiting Marc at Jussieu and in St Cheron, discussing mathematics, life,and the universe.The initial motivation for studying local martingales was to check the effectivenessof the criterion given in [Li94] for strong p -completeness of a stochastic differentialequation (SDE) on a d -dimensional space (assuming suitable regularity, e.g. C drivingvector fields). Roughly speaking, an SDE is strongly p -complete if its solution flowmoves a p -dimensional sub-manifold into another one, without breaking it. Strong p -completeness is weaker than strong completeness by the latter we mean that thesolution is continuous with respect to the initial data and time for all time. If the SDEand its adjoint SDE (the one with the drift given the negative sign) is also stronglycomplete then the solutions flow are diffeomorphisms for almost all ω . Strong (d-1)completeness is equivalent to strong completeness and strong 1-completeness allowsdifferentiating the solution flow and their corresponding semi-group with respect to theinitial data. The criterion for the strong p -completeness, provided that the solution tothe SDE from a particular initial value does not explode, is [Li94, Thm. 4.1]: sup x ∈ K E (cid:18) sup s ≤ t | Dφ s | p + δ χ t<ζ ( x ) (cid:19) < ∞ , where φ s ( x ) is the solution, Dφ s its derivative with respect to the initial data (solutionto the linearised SDE), ζ ( x ) its life time, K any compact subset and δ > . NTRODUCTION For an explicit SDE in [Li94], which is strongly ( d − ) complete but not strongly( d − ) complete (the only example known to the author), this finally comes down tofor which values of α , E (cid:0) sup s ≤ t | x + B s | α (cid:1) is finite where ( B s ) is a d -dimensionalBrownian motion. It turns out that for a local martingale which is not martingale (nick-named strict(ly) local martingales) there is a dichotomy for a function F ∈ W , : E F ( sup s ≤ t M s ) < ∞ if and only if R ∞ ǫ F ′ ( y ) y dy < ∞ for some ǫ ≥ , see [ELY99,Prop. 2.3, also pp 332], the article took a long while to complete (our second study wascompleted first), but it does include a generalisation of this integrability criterion to aclass of semi-martingales, [ELY99, Prop. 3.5], proved by a perturbative method.The popularity of strict local martingales is largely due to Freddy Delbaen andWalter Schachermayer’s paper [DS95] on incomplete markets, and due to AlexanderCox and David Hobson [CH05], Robert Jarrow, Philip Protter and K. Shimbo [JPS10]for the use of strict local martingales in bubble modelling. See also M. Loewenstainand G. Willard [LW00] and Philip Protter’s excellent survey [Pro13]. The incompletemarket problem concerns equivalent martingale measures and free lunch may followfrom a strict local martingale measure. A classification in terms of the bubble timeof a bubble asset is given by Robert Jarrow, Philip Protter and K. Shimbo [JPS10].A type three financial bubble models a non-zero asset price with bounded life timebursting at on or before the bubble time and is a strict local martingale. The two arelinked as following: birth of bubbles are impossible in complete market, and possiblein incomplete markets, see Philip Protter [Pro15, section3].Given a function m ( t ) of finite variation there exists a strict local martingale ( M t )with E M t = m ( t ) which is a time changed 3-dimensional Bessel processes with timechange r − ( m ( t )) where r ( t ) is the mean process of the Bessel process, see [ELY99,Corollary 3.9] where constructions are also given using the speed measure. However ifthe negative part or the positive part of ( M t ) behaves well, then E ( M t ) cannot vanishfor all t , see [ELY99, Lemma 2.1], the case of a positive local martingale is classic. Apositive strict local martingale has even nicer properties, Soumik Pal and Philip Protter[PP10], following [DS95], showed that they can be obtained as the reciprocal of amartingale under an h -transform. The fact that ( M t ) where M = 0 is strictly local if,and only if, E M t vanishes has implications in probabilistic representations of solutionsof PDE’s and useful for the identification of the domain of the generator of a Markovprocess.Beautiful and complete answers are given to the question whether the solution toa one dimensional SDE without drift is a local martingale, which we discuss below.Let ( M t ) be a solution to the equation M t = x + R t a ( M s ) dW s (where a > , a − ∈ L loc and ( W t ) a one dimensional Brownian motion), then it is a strict localmartingale if and only if R ∞ ǫ xa ( x ) dx = ∞ for some ǫ > . This clean solution is givenin the articles by Freddy Delbaen and H. Sgirakawa [DS02, Thm1.6], and ShinichiKotani [Kot06, Thm 1], see also Hardy Hulley and Eckhard Platen [HP11, Thm1.2]for a description by the first passage times of ( M t ). Also, in [MU12], AlexsanderMijatovi´c and Mikhail Urusov solved the problem whether the exponential martingaleof a stochastic integral is a strict local martingale. These approaches explore the factthat the non-explosion problem for one dimensional elliptic SDE is determined by theFeller test and is equivalent to that the exponential local martingale in the Girsanovtransform removing the drift is a martingale.Despite of the success in classifying one dimensional diffusions which are alsolocal martingales, there is wanting in concrete examples of strict local martingales,especially in variety. The strict local martingales we will construct later are based on XAMPLES an entirely different approach. Acknowledgement.
I met Freddy in Anterwerp in February 1994, who was inconversation with Marc and came to talk to me in ‘The Physics and Stochastic AnalysisConference’ organised by Jan van Casteren. I would like to take this opportunity tothank all of them. The example itself was constructed quite many years ago. I hadthe opportunity to tell Marc about it, who was enthusiastic and thought it should bepublished. It did take me many years to finally write it up. This paper is dedicated tothe friendship and fond memory of Marc, who always loved the beauty of simplicity.
The observation below, presumably known although I know no reference for it, is thatthe sample paths of a strictly local martingale oscillate faster than that of a martin-gale, the wild oscillation might explain why strict local martingales are useful for bub-ble modelling. Recall that a stochastic process ( X t ) is of class DL if { X S } where S ranges through bounded stopping times is uniformly integrable and that a localmartingale ( M s ) satisfying that sup s ≤ t | M s | is integrable is a martingale. Further-more, a local martingale with M integrable and with its negative part of class DL is a super-martingale, and it is a martingale if and only if its mean value is constantin time [ELY99, Prop. 2.2]. Throughout the paper the underlying probability space( Ω , F , F t , P ) satisfies the standard assumptions: right continuity and completeness. Proposition 1.
Let ( M t ) be a right continuous local martingale, T a finite stoppingtime and M T the stopped process.1. Suppose that E (cid:0) sup s ≤ t | M s | α (cid:1) = ∞ for some number α ∈ ( , ) . Then ( M s , s ≤ t ) is a strict local martingale.2. Suppose that E (cid:0) sup s ≤ T | M s | α (cid:1) = ∞ for some number α ∈ ( , ) and T ≤ t .Then ( M Ts , s ≤ t ) is a strict local martingale (also lim sup λ →∞ λ P (cid:0) sup s ≤ T | M s | ≥ λ (cid:1) = ∞ ). If lim t →∞ M t exists we may take T = ∞ .Proof. Let T be a stopping time. Suppose that E (cid:0) sup s ≤ T | M s | α (cid:1) = ∞ for some α ∈ ( , ), then lim sup λ →∞ λ P (cid:18) sup s ≤ T | M s | ≥ λ (cid:19) = ∞ . (*)For otherwise, there exists K such that P (cid:0) sup s ≤ T | M s | ≥ λ (cid:1) ≤ Kλ and E (cid:18) sup s ≤ T | M s | α (cid:19) = Z ∞ P (cid:18) sup s ≤ T | M s | α ≥ λ (cid:19) dλ ≤ Z ∞ Kλ α dλ < ∞ . If ( M s , s ≤ t ) were a martingale, we apply the maximal inequality to M and obtain,for λ > , E | M t | ≥ λ P (cid:18) sup s ≤ t | M s | ≥ λ (cid:19) , contradicting with (*), completing the proof for part (1). If ( M Tt , t ≤ t ) were amartingale, we apply the maximal inequality to M T and obtain, for λ > , E | M Tt | ≥ λ P (cid:18) sup s ≤ t | M Ts | ≥ λ (cid:19) = λ P (cid:18) sup s ≤ T | M s | ≥ λ (cid:19) , this contradicts with (*), concluding part (2). XAMPLES From the proof we see also the following statement.
Remark 2.
A right continuous local martingale ( M s , s ≤ t ) satisfying the condition lim sup λ →∞ λ P (cid:0) sup s ≤ t | M s | ≥ λ (cid:1) = ∞ is a strict local martingale. In Proposition 1 we stated a criterion for the strict localness of a local martingalein terms of small moments of the supremum process. Before using the proposition toconstruct an explicit strict local martingale, we observe that there is a class of localmartingales for which such small moments are always finite. In this class, which wedescribe more carefully in the remark below, is the strict local martingales ( R t ) − δ where δ > and R t is a- δ dimensional Bessel process. See Corollary 2.4 in [ELY99]. Remark 3.
Suppose ( M s , s ≤ t ) is a continuous positive local martingale with M ∈ L . Then ( M s , s ≤ t ) is L bounded, E (cid:0) sup s ≤ t | M s | α (cid:1) < ∞ for any α ∈ ( , ) , and lim λ →∞ λ P (cid:0) sup s ≤ t | M s | ≥ λ (cid:1) = E ( M − M t ) . So ( M s , s ≤ t ) is strictly local if andonly if the latter limit does not vanish for some t .Proof. To see this, we apply Lemma 2.1 in [ELY99] to conclude that ( M s , s ≤ t ) is L bounded and lim λ →∞ λ P (cid:0) sup s ≤ t M s ≥ λ (cid:1) = E M − E M t < ∞ . Since M t ≥ ,the argument in the proof for part (1) shows that E (cid:0) sup s ≤ t | M s | α (cid:1) is finite for any α ∈ ( , ).The construction below for strict local martingales depends crucially on the Burkholder-Davies-Gundy inequality for small values p < . For a c´adl´ag local martingale, BDGinequalities are only known to hold with values p greater or equal to one. The samplecontinuity assumption is hence essential for the statement below, on the other hand ac´adl´ag local martingale with a.s. finite total jumps is the sum of a continuous local mar-tingale and a finite variation process, and one can construct strict local martingales withjump by adding such a jump process. Our method does not seem to apply to stochasticintegrals driven by a L´evy process, for which please see the papers by Philip Prot-ter [Pro15] and that by Constantinos Kardaras, D¨orte Kreher, and Ashkan Nikeghbali[KKN15]. Corollary 4.
Let f be a progressively measurable function with values in R d . If E (cid:16)R t | f ( s ) | ds (cid:17) α/ = ∞ for some number α ∈ ( , ) then R t h f ( s ) , dB s i is a strictlocal martingale.Proof. Set M t = R t h f ( s ) , dB s i . By Burkholder-Davis-Gundy inequality, E (cid:18) sup s ≤ t | M s | α (cid:19) ≥ C α E (cid:16) ( h M, M i t ) α/ (cid:17) , which is infinite by the assumption, apply part (1) of Proposition 1 to conclude.Let ( W t ) be a one dimensional Brownian motion. Then stochastic integrals of theform R t g ( s ) dW s is a strict local martingale if E (cid:16)R t | g ( s ) | α ds (cid:17) = ∞ for some number α < . This follows from H¨older’s inequality: k g k L α ( Ω ,L ([ ,t ]) ≥ k g k L α ( Ω; L α ([ ,t ])) . Itis immediate that on [ , t ] where t < / , R t e | W s | dW s is an L martingale (Fer-nique’s theorem); while R t e | W s | dW s is a strict local martingale on any interval. XAMPLES Finally we remark on the characterisation of a continuous local martingale being amartingale. Assume M ∈ L . We have seen that if ℓ ( t ) := lim sup λ →∞ λ P (cid:18) sup s ≤ t | M s | ≥ λ (cid:19) is infinite then ( M t ) is a strict local martingale. On the other hand if ℓ ( t ) = 0 and if( M t ) is L bounded, then ( M t ) is a uniformly integrable martingale. See K. MuraliRao [Rao69], J. Az´ema, R. F. Gundy and M. Yor [AGY80]. From the point of viewof mathematical finance the class of strict local martingales with ℓ ( t ) finite has beenoften studied. Also, suppose that the negative part of ( M t ) is of class D and M t ∈ L ,then γ ( t ) := lim sup λ →∞ λ P (cid:0) sup s ≤ t M s ≥ λ (cid:1) is always a finite number that couldvanish in which case we have a martingale or that is a non-zero number in whichcase we have a strict local martingale. Moreover γ ( t ) = E [ M − M t ], see [ELY97],and [ELY99]. If ( M s , s ≤ t ), where t is a finite number, is a martingale then it isuniformly integrable and so ℓ ( t ) = γ ( t ) = 0 . This may be confusing, after all wehave been warned a martingale may not be L bounded, L bounded martingales maynot be uniformly integrable! Such caution needs only be taken if the martingale isdefined on an infinite time horizon or on an open interval [ , t ). (However we shouldpay attention to the problem whether the supremum of the martingale is integrable.)Recently, Hardy Hulley and Johannas Ruf [HJ15] studied a necessary and sufficientcondition for a suitable class of local martingales with jumps to be a martingale. Weconclude this paper with the following open question. Open Question.
What is a suitbale analogous result for local martingales withjumps?
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