Probab. Theory Relat. Fields 115, 325–355 (1999)
The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes
K. D. Elworthy1, Xue-Mei Li2,M.Yor3 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. e-mail: [email protected] 2 Department of Mathematics, University of Connecticut, 196 Auditorium road, Storrs, CT 06269, USA. e-mail: [email protected] 3 Laboratoire de probabilites,´ Universite´ Pierre et Marie Curie, Tour 56, 3e etage,´ 4 place Jussieu, F-75252 Paris Cedex 05, France
Received: 6 August 1996 / Revised version: 27 July 1998
Abstract. For a wide class of local martingales (Mt ) there is a default func- tion, which is not identically zero only when (Mt ) is strictly local, i.e. not a true martingale. This ‘default’ in the martingale property allows us to char- acterize the integrability of functions of sups≤t Ms in terms of the integrabil- ity of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times.
Mathematics Subject Classification (1991): 60G44, 60J60, 60H15
1. Introduction
In this paper we encounter a number of examples of strictly local martin- gales, i.e. local martingales which are not martingales. A local martingale (Mt ) is a true martingale if and only if $MT = $M0 for every bounded stopping time T . The quantity γM (T ) = $M0 −$MT quantifies the ‘strict’ local property of the local martingale and shall be called the ‘default’. The corresponding γM shall be called the default function. This ‘default’ in the martingale property allows us to characterize the exponents p for which 326 K. D. Elworthy et al. p $ sups≤t |Ms| < ∞ and more generally, those p for which certain fam- ilies of semi-martingales (X(α) : t ≥ 0) parameterized by α ∈ I satisfy: t (α) p sup $ sup |Xs | < ∞ . α∈I s≤t
In §2.1 a representation for the default γM (T ) is given in terms of the weak tail of sups≤T Ms, which shall be called the default formula. From this we also see that a local martingale whose negative part (M−) belongs to class DL is a true martingale if and only if γM (t) ≡ 0. In §3 the default formula is generalized to perturbations of local martingales, especially to semi-martingales. In particular we discuss under which conditions on a local martingale (Mt ), γM (T ) equals the weak tail of sups≤T |Ms|. Strictly local martingales appear naturally in applications: for example, the local martingale components found in many applications of Ito’sˆ for- mula are often strictly local martingales, especially when we are working on noncompact spaces. On the other hand, the expectation of a stochastic integral with respect to a martingale is often assumed to be zero, e.g. in a number of probabilistic discussions related to partial differential equations. A natural question is, then, whether the mean value function of a strictly local martingale has any special property? In §3.2, local martingales with mean value function m(t) given by an arbitrary continuous positive non- increasing function m : 1+ → 1+ are constructed. Similarly we give examples of strictly local sub-martingales, i.e. the sum of a strictly local martingale and an increasing process, which are mean decreasing. (A local sub-martingale (Xt ) is said to be mean decreasing if $Xt ≤ $Xs whenever t>s, and $X < $X for at least one pair of numbers t >s .) t∗ s∗ ∗ ∗ These results are applied to study the integrability properties of function- als of Bessel processes and radial Ornstein–Uhlenbeck (O.–U.) processes. First we calculate the default function γ for radial O.–U. processes, in terms of the law of T0, the first time the process hits zero. Using the Girsanov the- orem for last passage times, the law of T0, for a (4 − δ) dimensional O.–U. process starting from a is in turn given by the last hitting time of a by a δ-dimensional O.–U. process starting from 0. There is a brief analogous discussion for O.–U. processes with non-linear drift. In §3.4 we give a re- lated non-integrability result for general diffusions using techniques from stochastic flows. We would like to mention other topics where strictly local martingales play an essential role, namely; asymptotics for the Wiener sausage (Spitzer [18], Le Gall [11], see also section 2), and probabilistic proofs of the two main theorems in Nevanlinna theory (K. Carne [3], A. Atsuji [1]). Strictly local martingales have also come to play a role in mathematical finance as discussed in Delbaen and Schachermayer [4]. See also Sin [17] who gives The importance of strictly local martingales 327 a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures yield only strictly local martin- gales, and Ornstein–Uhlenbeck processes are given as volatility processes in, e.g., Stein-Stein [19] and Heston [10]. See also Elworthy-Li-Yor [5], Galtchouk-Novikov[7] and Takaoka [20] for related work. In fact Takaoka 1 2 [20] relates the weak tails of supt≤T |MT | and hMiT under ‘almost’ minimal assumptions. Finally, in order to avoid some possible confusion, let us stress that our strictly local martingales do not bear any relation with Le Jan’s strict martingales.
2. Strictly local martingales
Here we define a strictly local martingale to be a local martingale which is not a true martingale. Here are a couple of examples:
2−δ 1. Let {Rt } be a δ-dimensional Bessel process, δ>2. Then {Rt } is a 2−δ strictly local martingale. In fact $Rt is a strictly decreasing function in t by direct calculation. Alternatively see (34). 2. More generally, let (Xt ,t ≤ζ)be a regular transient diffusion on (0, ∞) and s its speed measure such that (i) ζ = inf{t>0:Xt− =0,or ∞} (ii) s(0) =−∞,s(∞)<∞. (iii). 0 is an entrance point for the diffusion X. Then {s(Xt)} is a strictly local martingale. See Elworthy-Li-Yor [5].
2.1. The default formula
A. Let Xt be a stochastic process adapted to an underlying filtration (Ft ). Recall that X belongs to class D if the family of random variables XS where S ranges through all stopping times S is uniformly integrable. It is in class DL if for each a>0, {XS} is uniformly integrable over all bounded T stopping times S ≤ a.IfT is a stopping time, we write X· = XT ∧·. Below it will be convenient to decompose Xt into its positive and negative parts, + − Xt = Xt − Xt . We have the following maximal and limiting maximal equalities.
Lemma 2.1. Let Mt be a continuous local martingale with $|M0| < ∞. T − Let T be a finite stopping time such that (M ) is of class D. Then MT ∧· is bounded in L1 and h i $ M + xP sup M ≥ x + $[M − x]+ = $M . (1) T 1(sup Mt Furthermore, lim xP sup Mt ≥ x = $M0 − $MT . (2) x→∞ t≤T If limt→∞ Mt exists and is finite, e.g. for a positive local martingale (Mt ), the stopping time T does not need to be finite. Note. A special case of the default formula appeared in Carne [3]. See also Atsuji [1]. A complement to the default formula in terms of hXiT is presented in Elworthy-Li-Yor [5]. Proof. Let (Sn,n∈-)be a reducing sequence of stopping times for M·, i.e. (S ) increases to ∞ as n →∞and is such that each {M } is a uniformly n t∧Sn integrable martingale. Set Tx = inf{t ≥ 0:Mt ≥x}. First, assume M0 is a constant. If x>M0then Tx = inf{t>0:Mt =x}and $ M = $M . (3) Tx ∧T ∧Sn 0 Letting n →∞, we obtain: $ M = $M . (4) Tx ∧T 0 This is (1): h i $ M + xP sup M ≥ x = $M . T 1(sup Mt On the other hand (1) is clearly true for x ≤ M0. In general for M0 an integrable random variable, we take conditional expectations with respect to F0 to get (1). Now MT is integrable by (4): $|M |≤ lim $ M + 2M− ≤ $M + 2$M− . (5) T T ∧Tx T ∧T 0 T x→∞ x Writeh i h i h i + − $ MT = $ M − $ M 1(supt≤T Mt $MT + lim xP sup Mt ≥ x = $M0 . (6) x→∞ t≤T If limt→∞ Mt exists and is finite, the argument above holds for non-finite T . B. For a stopping time T , set γM (T ) = $M0 − $MT . (7) The importance of strictly local martingales 329 We call the quantity defined in (7) the default of the process (MT ) and the limiting maximal equality (2) the default formula. Proposition 2.2. A local martingale {Mt } such that $|M0| < ∞ and its negative part belongs to class DL is a super-martingale. It is a martingale if and only if $Mt = $M0, i.e. γM (t) = 0, for all t>0. Proof. The second statement is well known and follows from the above discussions: {Mt } is a martingale if and only if γM (T ) = 0 for all bounded stopping times T , the latter is equivalent to γM (t) = 0 for all t>0. The equivalence of the latter comes from formula (2): γM (T ) = lim xP sup Mt ≥ x ≤ lim xP sup Mt ≥ x , x→∞ x→∞ t≤T t≤t0 where t0 is an upper bound for T . To prove {Mt } is a super-martingale, take two bounded stopping times S ≤ T . We only need to prove $MS ≥ $MT . Let Nt = Mt+S, then {Nt } is a local martingale with respect to the filtration {Ft+S}. So (2) gives: lim xP sup Nt ≥ x = $N0 − $NT −S = $MS − $MT . x→∞ t≤T −S Thus $MS ≥ $MT and {M·} is a super-martingale. 2.2. Integrability of functionals R + + x A. Let F : 1 → 1 be defined by F(x) = 0 dyf(y), for f a nonnega- tive Borel function. Proposition 2.3. Let {Mt } be a local martingale and T a stopping time suchR that the default formula (2) holds with γM (T ) = $M0 − $MT finite. ∞ f(y)dy < ∞, If y then $ F(sup Ms) < ∞ . s≤T R γ (T ) > 0 ∞ f(y)dy =∞ Furthermore if also M , then y if and only if $ F(sup Ms) =∞ . s≤T 330 K. D. Elworthy et al. Proof. This follows from Z ∞ $ F(sup Ms) = f(y)P sup Ms ≥ y dy s≤T 0 s≤T and (2). B. As an example we look at the integrability of certain standard Brown- + ian and Bessel functionals. For δ ∈ 1 , let {Rt } be a δ-dimensional Bessel process. The expectation of a δ-dimensional Bessel process starting from (δ) a will be denoted by Ea . The subscript may be omitted if no initial value is specified. For δ ≥ 2, {Rt } is the unique non-negative solution to the stochastic differential equation: δ − 1 dρt = dβt + dt , 2ρt where {βt } is a 1-dimensional Brownian motion. For δ<2 the situation is different because the Bessel process (Rt : t ≥ 0) has a non-trivial set of zeros. Moreover there is an increasing process Lt (R) whose support is 2−δ precisely this set of zeros, and such that Rt − Lt (R), t ≥ 0, is a local martingale, in fact a martingale with moments of all orders. Thus for our purposes the case δ<2 is uninteresting and we shall restrict discussions 1 to δ ≥ 2. Furthermore { δ−2 } is a local martingale for δ>2 and when Rt δ = 2, log r is a scale function for Rt and (log Rt ) is therefore again a local martingale. This leads to: Corollary 2.4. Let I = [r1,r2]be an interval with r1 > 0. Let t>0and p>0we have 1. For δ>2, 1 p sup E(δ) sup < ∞. p<1 . (8) a δ−2 if and only if a∈I s≤t Rs This also holds for t ≡∞. 2. For δ>2, Z p t 1 2 sup E(δ) ds < ∞, p<1 . a 2(δ−1) if and only if a∈I 0 Rs 3. a 6= 0 For , (2) 1 p Ea sup | log | < ∞, if and only if p<1 s≤t Rs and Z p t ds 2 E(2) < ∞, p<1 . a 2 if and only if 0 Rs The importance of strictly local martingales 331 Since { 1 } is a strictly local martingale, (8) is a direct consequence Proof. Rδ−2 of Propositiont 2.3 if I contains a single point. It also holds for t =∞ by the last sentence of Lemma 2.1, since lim 1 = 0. For genuine t→∞ R intervals I using a corresponding result for a family oft local martingales, −(δ−2) c.f. Proposition 3.5. For part 2, note that for δ 6= 2, (Rt ) satisfies: Z t dβ (R )−(δ−2) = (R )−(δ−2) − (δ − 2) s (9) t 0 δ−1 0 Rs The required result follows from the Burkholder-Davis-Gundy inequality and (8) for each 0 ≤ t ≤∞. For the case of δ = 2, we apply Proposition 2.3 to log 1 . First note that Rt the negative part of log 1 is uniformly integrable over finite times: Rt 1 − log ≡ log R ≤ (R − 1)+ ≤ R . t 1(Rt ≥1) t t Rt Next we observe that {log 1 } is a strictly local martingale, since E(2)R R a t is strictly increasing in t and sot (2) 1 (2) 1 Ea log ≤−log Ea [Rt ] < log . Rt a The required result follows. From Lemma 2.1 we also have, for a 2-dimensional Bessel process, (2) lim rP inf log Rs ≤−r =E log Rt − log a. r→∞ s≤t This is closely related to Spitzer’s asymptotics [18], although Spitzer presents a different quantity on the right hand side; see also the estimate (0a) for Wiener sausages in Le Gall[11], and (12) in Elworthy, Li and Yor[5]. Using Girsanov transform, the result of Corollary 2.4 extends to radial Ornstein–Uhlenbeck processes of integer dimensions, as below. Let n ≥ 2 be an integer. An n-dimensional radial Ornstein–Uhlenbeck process {Rt } with parameter λ can be realized as the radial part of an n- dimensional Ornstein–Uhlenbeck process {Ut } with parameter λ as the so- lution to the stochastic differential equation: Z t n 1 Ut = x + Bt − λ Us ds, x ∈ 1 ,λ∈1 , (10) 0 R −λt t λs which can be solved explicitly to give: Ut = e x + 0 e dBs .For λ ≥ 0, the reciprocal { 1 } is a local sub-martingale as seen by Ito’sˆ Rn−2 formula: t 332 K. D. Elworthy et al. Z Z 1 1 t hU ,dB i t λ = − (n − 2) s s +(n − 2) ds (11) n−2 n−2 n n−2 |Ut | |x| 0 |Us| 0 |Us| for x 6= 0 (when x = 0, this is true with x replaced by U and the lower limit 0 in the integrals replaced by , for any >0 and so we have a local sub-martingale for t>0). Proposition 2.5. Let n>2, λ>0, and Rt =|Ut|be the radial part of an n-dimensional O.–U. process, then for K a compact subset of 1n −{0}, p −λ (n) 1 sup E|x| sup < ∞ if and only if p 3. The default formula for general processes In complete generality note that if ξ,η,ρ are real valued random variables with ξ ≤ η + ρ and $ρ<∞, then lim xP(ξ ≥ x) ≤ lim xP(η ≥ x) . x→∞ x→∞ This will give us a perturbation result: Lemma 3.1. Let Xt = Mt + Dt be the sum of two stochastic processes. Then if $ sups≤t |Ds| < ∞, lim xP(sup Xs ≥ x) = lim xP(sup Ms ≥ x) x→∞ s≤t x→∞ s≤t and lim xP(sup Xs ≥ x) = lim xP(sup Ms ≥ x) . x→∞ s≤t x→∞ s≤t Consequently the weak tails of sups≤t Xs and sups≤t Ms are equal if they exist and they exist at the same time. (By the weak tail, of a random variable ξ, we mean limx→∞ xP (ξ ≥ x) if it exists.) In particular if (Mt ) is a local martingale, the weak tail of sups≤t Xs is related to the default function γM . However the assumption on Dt can be much weakened when (Dt ) is of bounded variation. Recall that a continuous semi-martingale (Xt : t ≥ 0) is a stochastic process with canonical decomposition Xt = Mt + At into the sum of a continuous local martingale Mt and a continuous adapted process At of bounded variation (starting from 0). It is a local sub(super)martingale if At is increasing (decreasing) in t.Itisintegrable if Xt is integrable for each t. Lemma 3.2. Let Xt = Mt + At be a continuous semi-martingale with T − $|X0| < ∞ and A0 = 0. Let T be a finite stopping time such that (X ) is of class D. 1. Suppose At is a monotone process. Then h i + $ XT + xP sup Xt ≥ x + $[X0 − x] 1(supt≤T Xt where Tx = inf{t ≥ 0:Xt ≥x}. Furthermore if $AT < +∞, then 1 XT ∧· is bounded in L and 334 K. D. Elworthy et al. lim xP sup Xt ≥ x = $X0 − $XT + $AT . (14) x→∞ t≤T Let γX(T ) = $X0 − $XT + $AT . Then γX(T ) is finite for local super- martingales with (XT )− in class D. 1 2. For a general A of finite variation if its increasing part At satisfies 1 $AT < ∞, then $(AT )<∞and both (13) and (14) hold. Proof. Let (Sn,n∈-)be a reducing sequence of stopping times for (Mt ). As in the proof of Lemma 2.1, we have: $ X = $X + $ A , (15) Tx ∧T ∧Sn 0 Tx ∧T ∧Sn T first assuming M0 is a constant. Now the assumption on X allows us to take n →∞to get: $ X = $X + $ A , (16) Tx ∧T 0 Tx ∧T which gives (13). Both (16) and (13) hold for general M0 by taking condi- tional expectation with respect to F0.Now,XT is integrable if $AT < +∞; indeed by (16):