Stochastic Exponentials and Logarithms on Stochastic Intervals -- A Survey
aa r X i v : . [ m a t h . P R ] N ov Stochastic Exponentials and Logarithms on StochasticIntervals — A Survey ∗ Martin Larsson † Johannes Ruf ‡ November 16, 2018
Abstract
Stochastic exponentials are defined for semimartingales on stochastic intervals, andstochastic logarithms are defined for semimartingales, up to the first time the semi-martingale hits zero continuously. In the case of (nonnegative) local supermartingales,these two stochastic transformations are inverse to each other. The reciprocal of astochastic exponential on a stochastic interval is again a stochastic exponential on astochastic interval.
Keywords:
Involution, stochastic exponential, stochastic interval, stochastic log-arithm.
MSC2010 subject classification:
Primary 60G99; secondary: 60H10, 60H99.
The exponential and logarithmic functions are essential building blocks of classical calculus.As is emphasized by Itˆo’s formula, in stochastic calculus, second-order terms appear; theappropriate modifications of exponentials and logarithms lead to stochastic exponentialsand stochastic logarithms.This note collects results for the calculus of stochastic exponentials and logarithms ofsemimartingales, possibly defined on stochastic intervals only. While the results of this noteare no doubt well known, we were not able to find a suitable reference. We have foundthese results rather useful in a number of situations, mostly in the context of measurechanges, where it is often convenient to switch between stochastic logarithms and stochas-tic exponentials. For example, a change of probability measure is given by a nonnegative ∗ Parts of this note appeared in the unpublished manuscript Larsson and Ruf (2014). We thank twoanonymous referees and Robert Stelzer for very helpful comments that led to an improvement of the paper. † Department of Mathematics, ETH Zurich, R¨amistrasse 101, CH-8092, Zurich, Switzerland. E-mail:[email protected] ‡ Department of Mathematics, London School of Economics and Political Science, Columbia House,Houghton St, London WC2A 2AE, United Kingdom. E-mail: [email protected] Z . On the other hand, Girsanov’s theo-rem, which describes the semimartingale characteristics of some semimartingale under thenew measure, is more conveniently stated in terms of the stochastic logarithm of Z .As elaborated, within stochastic calculus, stochastic exponentials and logarithms ap-pear naturally in the context of absolutely continuous changes of measures. If this changeof measure is not equivalent, but only absolutely continuous, the corresponding Radon-Nikodym derivative hits zero. Depending on whether it hits zero by a jump or continu-ously, the corresponding stochastic logarithm may or may not be defined on [0 , ∞ ). Thiscomplication motivated us to formulate precise statements concerning the interplay be-tween nonnegative semimartingales and their stochastic logarithms. The price to pay isthat these stochastic logarithms may only be defined on stochastic intervals and not on allof [0 , ∞ ).The reciprocal of a Radon-Nikodym derivative also bears an important interpretation.Provided the original change of measure is equivalent, this reciprocal serves again, underthe new measure, as a Radon-Nikodym derivative; indeed it yields exactly the originalmeasure. For this reason, it is convenient to have a description of the dynamics of thereciprocal at hand.In general semimartingale theory, which in particular allows for jumps, the notion ofstochastic exponential dates back to at least Dol´eans-Dade (1976). Nowadays, basicallyany textbook on stochastic calculus introduces this notion. We highlight the survey arti-cle Rheinl¨ander (2010), which reviews well known properties of stochastic exponentials ofsemimartingales. In particular, this survey also collects classical conditions for the mar-tingale property of the stochastic exponential. The article of Kallsen and Shiryaev (2002)provides further interesting identities, especially relating to exponential and logarithmictransforms, a subject which we do not discuss in this note. In contrast to these articles,we especially discuss the definition of stochastic logarithms of general semimartingales,without the assumption of strict positivity.We proceed as follows. In Section 2, we establish notation and introduce the conceptof processes on stochastic intervals. In Section 3, we define stochastic exponentials andlogarithms, discuss their basic properties, and prove that they are inverse to each other. InSection 4, we describe the stochastic logarithm of the reciprocal of a stochastic exponential.Finally, in Section 5, we provide some examples. These examples illustrate that stochasticexponentials of semimartingales, defined on stochastic intervals only, arise naturally. The following definitions are consistent with those in Jacod and Shiryaev (2003), to whichthe reader is referred for further details. We work on a stochastic basis (Ω , F , F , P ), wherethe filtration F = ( F t ) t ≥ is right-continuous but not necessarily augmented with the2 -nullsets. Relations between random quantities are understood in the almost sure sense.Given a process X = ( X t ) t ≥ , write X − for the left limit process (limit inferior ifa limit does not exist) and ∆ X = X − X − for its jump process, using the convention X − = X . The corresponding jump measure is denoted by µ X , and for any (random)function F : Ω × R + × R → R , the stochastic integral of F with respect to µ X is the process F ∗ µ X given by F ∗ µ Xt = (P s ≤ t F ( s, ∆ X s ) { ∆ X s =0 } , if P s ≤ t | F ( s, ∆ X s ) | { ∆ X s =0 } < ∞ , + ∞ , otherwise, t ≥ . Semimartingales are required by definition to be right-continuous, almost surely admit-ting left limits. If X is a semimartingale, H · X is the stochastic integral of an X –integrableprocess H with respect to X .For a stopping time τ , we let X τ denote the process X stopped at τ , and we define thestochastic interval [[0 , τ [[= { ( ω, t ) ∈ Ω × R + : 0 ≤ t < τ ( ω ) } . Note that stochastic intervals are disjoint from Ω × {∞} by definition.A process X on a stochastic interval [[0 , τ [[, where τ is a stopping time, is the restrictionto [[0 , τ [[ of some process. In this paper, τ will be a foretellable time; that is, a [0 , ∞ ]–valuedstopping time that admits a nondecreasing sequence ( τ n ) n ∈ N of stopping times, with τ n < τ almost surely for all n ∈ N on the event { τ > } , and lim n ↑∞ τ n = τ almost surely. Sucha sequence is called an announcing sequence. Every predictable time is foretellable, andif the stochastic basis is complete the converse also holds; see Jacod and Shiryaev (2003,Theorem I.2.15 and I.2.16). If τ is a foretellable time and X is a process on [[0 , τ [[, we say that X is a semimartingale(local martingale / local supermartingale) on [[0 , τ [[ if there exists an announcing sequence( τ n ) n ∈ N for τ such that X τ n is a semimartingale (martingale / supermartingale) for each n ∈ N . Basic notions for semimartingales carry over by localization to semimartingales onstochastic intervals. For instance, if X is a semimartingale on [[0 , τ [[, its quadratic variationprocess [ X, X ] and the continuous version [
X, X ] c are defined as the processes on [[0 , τ [[ thatsatisfy [ X, X ] τ n = [ X τ n , X τ n ] and ([ X, X ] c ) τ n = [ X τ n , X τ n ] c , respectively, for each n ∈ N .The jump measure µ X of X is defined analogously, as are stochastic integrals with respectto X (or µ X ). In particular, H is called X –integrable if it is X τ n –integrable for each n ∈ N ,and H · X is defined as the semimartingale on [[0 , τ [[ that satisfies ( H · X ) τ n = H · X τ n foreach n ∈ N . We refer to Maisonneuve (1977), Jacod (1979), and Appendix A in Carr et al.(2014) for further details on local martingales on stochastic intervals. In general, the converse implication does not hold. For example, consider the canonical space ofc`adl`ag paths, equipped with the Skorohod topology and the Wiener measure. Then the first time that thecoordinate process crosses a given level is foretellable but not predictable, given the canonical filtration notaugmented by the Wiener nullsets.
Proposition 2.1.
Let τ be a foretellable time, and let X be a local supermartingale on [[0 , τ [[ bounded from below. Then lim t ↑ τ X t exists in R and [ X, X ] τ = lim t ↑ τ [ X, X ] t is finite.Proof. Without loss of generality, we shall assume X ≥
0. We define X ′ = X [[0 ,τ [[ . We nowargue that X ′ is a supermartingale, which in particular implies that X ′ allows for a modi-fication with left limits almost surely. This observation then implies the assertion since theclassical supermartingale convergence theorem (see Problem 1.3.16 in Karatzas and Shreve(1991)) yields that X can be closed. This directly implies the existence of a limit at infinityand the convergence of [ X, X ]; in particular then [
X, X ] τ < ∞ .To prove that X ′ is a supermartingale, let ( τ n ) n ∈ N be an announcing sequence for τ such that X τ n is a supermartingale. Fix s, t ≥ s < t . Then, on the event { s ≥ τ } ,we have E [ X ′ t | F s ] = 0 = X ′ s . On the event { s < τ } , Fatou’s lemma implies E [ X ′ t | F s ] = E h lim n →∞ X τ n t { τ>t } (cid:12)(cid:12)(cid:12) F s i ≤ lim inf n →∞ E [ X τ n t | F s ] ≤ lim n →∞ X τ n s = X s = X ′ s , yielding the claim.The following corollary will be used below. Lemma 2.2.
Let τ be a foretellable time, and let X be a local supermartingale on [[0 , τ [[ with ∆ X ≥ − . Then we have, almost surely, the set identity (cid:26) lim t ↑ τ X t does not exist in R (cid:27) = (cid:26) lim t ↑ τ X t = −∞ (cid:27) ∪ { [ X, X ] τ = ∞} . (2.1) Proof.
This statement is proven in Corollary 4.4 of Larsson and Ruf (2018). For sake ofcompleteness, we provide a proof here of the inclusion “ ⊃ ”. For an arbitrary m ∈ N , definethe stopping time ρ = inf { t ≥ X t ≤ − m } . Then X ρ is a local supermartingale on[[0 , τ [[ bounded from below by − m −
1, whence [ X ρ , X ρ ] τ < ∞ by Proposition 2.1. Since X coincides with X ρ on { X ≥ − m } , we deduce that (cid:26) lim t ↑ τ X t exists in R (cid:27) ⊂ [ m ∈ N { X ≥ − m } ⊂ { [ X, X ] τ < ∞} ;hence the inclusion follows. For the inclusion “ ⊂ ”, see Remark 3.8 below. In this section, we define stochastic exponentials and logarithms, develop some of theirproperties, and show that they are inverse to each other.4 .1 Stochastic exponentials
Definition 3.1 (Stochastic exponential) . Let τ be a foretellable time, and let X be asemimartingale on [[0 , τ [[. The stochastic exponential of X is the process E ( X ) defined by E ( X ) t = exp (cid:18) X t −
12 [
X, X ] ct (cid:19) Y
Note that for each n ∈ N , on the interval [[0 , τ n [[ there are only finitely manytimes t such that ∆ X t < −
1. Moreover, whenever ∆ X t ≥ − t ∈ [0 , τ ), then thecorresponding factor in the infinite product in (3.1) lies in [0 , , τ n [[ for each n ∈ N . We also emphasize that the stochasticexponential E ( X ) of a semimartingale X on [[0 , τ [[ need not be a semimartingale on [[0 , ∞ [[,but only on [[0 , τ [[. Remark 3.3.
Whenever we have ∆ X t < − t ∈ [0 , τ ) the resulting stochas-tic exponential changes sign at t . Such stochastic exponentials appear in the context ofsigned measures such as in the study of mean-variance hedging strategies; see, for example,ˇCern´y and Kallsen (2007).The process E ( X ) is sometimes also called generalized stochastic exponential; see, forexample, Mijatovi´c et al. (2012). If ( τ n ) n ∈ N is an announcing sequence for τ , then E ( X )of Definition 3.1 coincides on [[0 , τ n ]] with the usual (Dol´eans-Dade) stochastic exponentialof X τ n . This shows that E ( X ) coincides with the classical notion when τ = ∞ . Manyproperties of stochastic exponentials thus remain valid. For instance, if ∆ X > − E ( X ) is strictly positive on [[0 , τ [[. If ∆ X t = − t ∈ [0 , τ ) then E ( X ) jumps tozero at time t and stays there. Also, on [[0 , τ [[, E ( X ) is the unique solution to the equation Z = e X + Z − · X on [[0 , τ [[; (3.2)see Dol´eans-Dade (1976). We also record the alternative expression E ( X ) = [[0 ,τ [[ exp (cid:18) X −
12 [
X, X ] c − ( x − log | x | ) ∗ µ X (cid:19) ( − P t ≤· { ∆ Xt< − } , (3.3)where we use the convention − log(0) = ∞ and e −∞ = 0.The following results relate the convergence of E ( X ) to zero to the behavior of X . Proposition 3.4.
The following set inclusion holds almost surely: (cid:26) lim t ↑ τ E ( X ) t = 0 (cid:27) ⊂ (cid:26) lim t ↑ τ X t = −∞ (cid:27) ∪ { [ X, X ] τ = ∞} ∪ { ∆ X t = − for some t ∈ [0 , τ ) } . Moreover, if we additionally have ∆ X ≥ − and lim sup t ↑ τ X t < ∞ , then the reverse setinclusion also holds. roof. Assume we are on the event (cid:26) lim t ↑ τ E ( X ) t = 0 (cid:27) ∩ { [ X, X ] τ < ∞} ∩ (cid:8) ∆ X t = − t ∈ [0 , τ ) (cid:9) . We need to argue that lim t ↑ τ X t = −∞ on this event. To this end, observe that theinequality x − log | x | ≤ x for all x ≥ − −∞ = lim t ↑ τ (cid:18) X t −
12 [
X, X ] ct − ( x − log | x | ) ∗ µ Xt (cid:19) ≥ lim t ↑ τ (cid:0) X t − [ X, X ] t − ( x − log | x | ) x< − / ∗ µ Xt (cid:1) . By assumption, [
X, X ] τ < ∞ . In particular, X can only have finitely many jumps boundedaway from zero. We deduce that the second and third terms on the right-hand side converge,and therefore lim t ↑ τ X t = −∞ . This yields the first set inclusion.We now assume that ∆ X ≥ − t ↑ τ X t < ∞ , and prove the reverse setinclusion. On the event { ∆ X t = − t ∈ [0 , τ ) } , X jumps to zero before τ andstays there, so that clearly lim t ↑ τ E ( X ) t = 0. If ∆ X t > − t ∈ [0 , τ ), then (3.3) andthe inequality x − log(1 + x ) ≥ ( x ∧ / x > − ≤ E ( X ) t ≤ exp (cid:18) X t −
12 [
X, X ] ct −
14 ( x ∧ ∗ µ Xt (cid:19) , t ∈ [0 , τ ) . On the event { lim t ↑ τ X t = −∞} , the right-hand side converges to zero. The same thinghappens on the event { [ X, X ] τ = ∞} , thanks to the assumption that lim sup t ↑ τ X t < ∞ and the observation that [ X, X ] cτ + ( x ∧ ∗ µ Xτ = ∞ if and only if [ X, X ] τ = ∞ . Thisconcludes the proof of the reverse set inclusion. To be able to discuss stochastic logarithms, recall that for a stopping time ρ and an event A ∈ F , the restriction of ρ to A is given by ρ ( A ) = ρ A + ∞ A c . Here ρ ( A ) is a stopping time if and only if A ∈ F ρ . Define now for a progressivelymeasurable process Z the running infimum of its absolute value by Z = inf t ≤· | Z t | and thestopping times τ = inf { t ≥ Z t = 0 } ; As the filtration might not be augmented by the nullsets, the following definitions might not be stoppingtimes. However, there exist appropriate modifications of these random times which turns them into stoppingtimes; see Appendix A, in particular, Lemma A.3, in Perkowski and Ruf (2015). We shall always work withthese modifications. C = τ ( A C ) , A C = (cid:8) Z τ − = 0 (cid:9) ; (3.4) τ J = τ ( A J ) , A J = (cid:8) Z τ − > (cid:9) . These stopping times correspond to the two ways in which Z can reach zero: either con-tinuously or by a jump. We have the following well known property of τ C ; see, e.g.,Exercise 6.11.b in Jacod (1979). Lemma 3.5.
Fix a progressively measurable process Z . The stopping time τ C of (3.4) isforetellable.Proof. We claim that an announcing sequence ( σ n ) n ∈ N for τ C is given by σ n = n ∧ σ ′ n ( A n ) , σ ′ n = n ∧ inf (cid:26) t ≥ Z t ≤ n (cid:27) , A n = n Z σ ′ n > o . To prove this, we first observe that σ n = n < ∞ = τ C on A cn for all n ∈ N . Moreover,we have σ n = σ ′ n < τ C on A n for all n ∈ N , where we used that Z τ C − = 0 on theevent { τ C < ∞} . We need to show that lim n ↑∞ σ n = τ C . On the event A C , see (3.4),we have τ C = τ = lim n ↑∞ σ ′ n = lim n ↑∞ σ n since A C ⊂ A n for all n ∈ N . On the event A cC = S ∞ n =1 A cn , we have τ C = ∞ = lim n ↑∞ n = lim n ↑∞ σ n . Hence ( σ n ) n ∈ N is an announcingsequence of τ C , as claimed.If a semimartingale Z reaches zero continuously, the process H = Z − { Z − =0 } explodesin finite time, and is therefore not left-continuous. In fact, it is not Z –integrable. However,if we view Z as a semimartingale on the stochastic interval [[0 , τ C [[, then H is Z –integrablein the sense of stochastic integration on stochastic intervals, as introduced in Section 2.Thus H · Z exists as a semimartingale on [[0 , τ C [[, which we call the stochastic logarithm of Z . Definition 3.6 (Stochastic logarithm) . Let τ be a foretellable time and Z be a progres-sively measurable process such that τ ≤ τ C and such that Z is a semimartingale on [[0 , τ [[.The semimartingale L ( Z ) on [[0 , τ [[ defined by L ( Z ) = 1 Z − { Z − =0 } · Z on [[0 , τ [[is called the stochastic logarithm of Z (on [[0 , τ [[ ) . Theorem 3.7.
Let τ be a foretellable time. We then have the following two statements. (i) Let Z be a progressively measurable process with Z = 1 such that τ ≤ τ C , Z is asemimartingale on [[0 , τ [[ , and Z = 0 on [[ τ, ∞ [[ . Then Z = E ( L ( Z )) . Let X be a semimartingale on [[0 , τ [[ with X = 0 such that X stays constant afterits first jump by − . Then E ( X ) is a semimartingale on [[0 , τ [[ , does not hit zerocontinuously strictly before τ , and satisfies X = L ( E ( X )) on [[0 , τ [[ . Proof.
We start by proving (i). By assumption and by Definition 3.1, both sides are zeroon [[ τ, ∞ [[. Moreover, Z satisfies the equation Z = 1 + Z − ( Z − ) − { Z − =0 } · Z = 1 + Z − · L ( Z ) on [[0 , τ [[,whose unique solution is E ( L ( Z )) on [[0 , τ [[.We next prove (ii). Note that E ( X ) is clearly a semimartingale on [[0 , τ [[, which also doesnot hit zero continuously strictly before τ , thanks to Proposition 3.4. Then the definitionof stochastic logarithm along with (3.2) yield L ( E ( X )) = 1 E ( X ) − { E ( X ) − =0 } · E ( X ) = 1 E ( X ) − { E ( X ) − =0 } E ( X ) − · X = { E ( X ) − =0 } · X = X on [[0 , τ [[ , where the last equality follows from the fact that X stays constant after it jumps by − Consider now the case where X is a local supermartingale on [[0 , τ [[ with ∆ X ≥ −
1. Then E ( X ) is also a local supermartingale on [[0 , τ [[ due to its positivity and (3.2). Moreover,the same argument as in the proof of Proposition 2.1 yields that E ( X ) is in fact a super-martingale globally, i.e. on [[0 , ∞ [[. Remark 3.8.
We can now provide an alternative proof of the inclusion “ ⊂ ” in Lemma 2.2under the additional assumption that ∆ X > −
1. Thanks to Proposition 3.4, it suffices toshow (cid:26) lim t ↑ τ X t does not exist in R (cid:27) ⊂ (cid:26) lim t ↑ τ E ( X ) t = 0 (cid:27) . As in the proof of the inclusion “ ⊃ ”, we deduce (cid:26) lim t ↑ τ X t does not exist in R (cid:27) ⊂ \ m ∈ N { X ≥ − m } c ⊂ (cid:26) lim inf t ↑ τ X t = −∞ (cid:27) . Since (3.3) yields 0 ≤ E ( X ) ≤ [[0 ,τ [[ e X and the limit lim t ↑ τ E ( X ) t exists by Proposition 2.1,the inclusion follows. 8onnegative supermartingales Z can be associated to a probability measure; see Chap-ter 11 in Chung and Walsh (2005) in the context of so-called h -transforms, or Perkowski and Ruf(2015) in the general context. Girsanov then provides the drift correct correction for a pro-cess Y as the quadratic covariation of Y and L ( X ), namely [ Y, L ( X )]. Hence, it ishelpful to understand well the connection between a nonnegative supermartingale and itsstochastic logarithm.To this end, we now want to make Theorem 3.7 more concrete, namely to work out therelationship of stochastic exponentials and logarithms in the local supermartingale case.The following definition will be helpful. Definition 3.9 (Maximality) . Let τ be a foretellable time, and let X be a semimartingaleon [[0 , τ [[. We say that τ is X –maximal if the inclusion { τ < ∞} ⊂ (cid:26) lim t ↑ τ X t does not exist in R (cid:27) holds almost surely.Let now Z be the set of all nonnegative supermartingales Z with Z = 1. Any such pro-cess Z automatically satisfies Z = Z τ . Furthermore, let L denote the set of all stochasticprocesses X satisfying the following conditions:(i) X is a local supermartingale on [[0 , τ [[ for some foretellable, X –maximal time τ .(ii) X = 0, ∆ X ≥ − , τ [[, and X is constant after the first time ∆ X = − −
1. The reader isreferred to Proposition I.5 in L´epingle and M´emin (1978) and Appendix A of Kardaras(2008) for related results. In both of these references, the local martingale is not allowedto hit zero continuously.
Theorem 3.10 (Relationship of stochastic exponential and logarithm) . The stochasticexponential E is a bijection from L to Z , and its inverse is the stochastic logarithm L .Suppose Z = E ( X ) for some Z ∈ Z and X ∈ L . Then τ = τ C , where τ is the foretellable X –maximal time corresponding to X , and τ C is given by (3.4) .Proof. By Theorem 3.7, we have E ◦ L = id and L ◦ E = id.Next, E maps each X ∈ L to some Z ∈ Z with τ C = τ . Since Z = E ( X ) is a nonnegativesupermartingale with Z = 1, we have Z ∈ Z and only need to argue that τ C = τ . ByTheorem 3.7(ii), we have τ ≤ τ C . The reverse inequality follows from Proposition 3.4,Lemma 2.2, and the X –maximality of τ .Further, L maps each Z ∈ Z to some X ∈ L with τ = τ C . Indeed, X = L ( Z ) is alocal supermartingale on [[0 , τ C [[ with X = 0, ∆ X = Z/Z − − ≥ − , τ C [[, and X is9onstant after the first time ∆ X = −
1. It remains to check that τ = τ C is X –maximal.To this end, observe that Z = E ( L ( Z )) = E ( X ). Thus on { τ < ∞} we have τ J = ∞ andhence ∆ X > −
1. It follows from Proposition 3.4 and Lemma 2.2 that { τ < ∞} ⊂ (cid:26) lim t ↑ τ Z t = 0 (cid:27) ∩ { ∆ X > − } ⊂ (cid:26) lim t ↑ τ X t = −∞ (cid:27) ∪ { [ X, X ] τ = ∞}⊂ (cid:26) lim t ↑ τ X t does not exist in R (cid:27) , hence τ is X –maximal as claimed. Reciprocals of stochastic exponentials appear naturally in connection with changes of prob-ability measures. We now develop some identities related to such reciprocals. The followingfunction plays an important role: φ : ( − , ∞ ) → ( − , ∞ ) , φ ( x ) = − x . Note that φ is an involution, that is, φ ( φ ( x )) = x . The following notation is convenient:Given functions F : Ω × R + × R → R and f : R → R , we write F ◦ f for the function( ω, t, x ) F ( ω, t, f ( x )). We now identify the reciprocal of a stochastic exponential or,more precisely, the stochastic logarithm of this reciprocal. Part of the following result iscontained in Lemma 3.4 of Karatzas and Kardaras (2007). Theorem 4.1 (Reciprocal of a stochastic exponential) . Let τ be a foretellable time, andlet X be a semimartingale on [[0 , τ [[ . Define the semimartingale Y = − X + [ X, X ] c + x x x = − ∗ µ X on [[0 , τ [[ . (4.1) Then E ( X ) E ( Y ) = 1 on [[0 , τ ∧ τ J [[ . Furthermore, for any nonnegative function G :Ω × R + × R → R + we have G ∗ µ Y = ( G ◦ φ ) ∗ µ X on [[0 , τ ∧ τ J [[ . (4.2)For an alternative, systematic proof of Theorem 4.1, see also ˇCern´y and Ruf (2018). Remark 4.2.
Since | x / (1 + x ) | ≤ x for | x | ≤ /
2, the process x / (1 + x ) ∗ µ X appearingin (4.1) is finite-valued on [[0 , τ [[. Remark 4.3.
Since φ is an involution, the identity (4.2) is equivalent to F ∗ µ X = ( F ◦ φ ) ∗ µ Y on [[0 , τ ∧ τ J [[for the nonnegative function F = G ◦ φ . 10 roof of Theorem 4.1. Note that, in view of (4.1), we have∆ Y = − ∆ X + (∆ X ) X = φ (∆ X ) on [[0 , τ ∧ τ J [[ . This implies (4.2). Now, applying (4.2) to the function G ( y ) = y − log | y | yields( y − log | y | ) ∗ µ Y = (cid:18) − x + log | x | (cid:19) ∗ µ X on [[0 , τ ∧ τ J [[ . A direct calculation then gives E ( Y ) = 1 / E ( X ) on [[0 , τ ∧ τ J [[. This completes the proof. In this section, we collect some examples to put this note’s results into context. We beginwith two examples that are rather standard and concern geometric Brownian motion andthe stochastic exponential of a one-jump martingale.
Example 5.1.
Let X ∈ L be Brownian motion. Then the stopping time τ = ∞ is X –maximal, and E ( X ) t = e X t − t/ for all t ≥ Y t = t − X t for all t ≥ E ( X ) E ( Y ) = 1. Example 5.2.
Let E be a standard exponentially distributed random variable and assumethat F is the smallest right-continuous filtration such that [[0 ,E [[ is adapted. Let now X be given by X t = t ∧ E − { E ≤ t } for all t ≥
0. Then X ∈ L , X is a martingale, τ = ∞ is X –maximal, and E ( X ) t = e t [[0 ,E [[ for all t ≥ Example 5.3.
Let (Θ n ) n ∈ N denote a sequence of independent random variables with P (Θ n = 1) = P (Θ n = −
1) = 1 / X t = P [ t ] n =1 Θ n for all t ≥ t ] denotes the integer part of t . Assume that F is the smallest right-continuousfiltration such that X is adapted. Define the stopping time ρ = inf { t ≥ X t = − } . Then E ( X ) t = E ( X ρ ) t = 2 [ t ] [[0 ,ρ [[ for all t ≥ X ρ = L ( E ( X )) ∈ L . Example 5.4.
Similarly to Example 5.3, let (Θ n ) n ∈ N denote a sequence of independentrandom variables with P (Θ n = 1 /
2) = P (Θ n = − /
2) = 1 /
2. Let now X be a localmartingale on [0 , t n ) n ∈ N where t n = 1 − n − . That is, X t = P n : t n ≤ t Θ n for all t ∈ [0 , F isthe smallest right-continuous filtration such that X is adapted. Then the deterministic11topping time τ = 1 is X –maximal; hence X ∈ L . Moreover, the stochastic exponential Z = E ( M ) ∈ Z is given by Z t = exp X n : t n ≤ t log(1 + Θ n ) = Y n : t n ≤ t (1 + Θ n ) . Theorem 3.10 yields that τ C = 1, where τ C was defined in (3.4). Alternatively, thestrong law of large numbers and the fact that E [log(1 + Θ )] < P (lim t ↑ Z t = 0) =1. This illustrates that τ C in (3.4) can be finite, even if the local martingale Z has nocontinuous component.Define next the semimartingale Y on [[0 , Y t = − X t + x x ∗ µ Xt = (cid:18) − x + x x (cid:19) ∗ µ Xt = − x x ∗ µ Xt for all t ∈ [0 , . Theorem 4.1 yield that E ( X ) E ( Y ) = 1 on [[0 , E ( Y ) = [[0 , exp (cid:0) log(1 + y ) ∗ µ Y (cid:1) = [[0 , exp (cid:18) log (cid:18) − x x (cid:19) ∗ µ X (cid:19) = [[0 , exp (cid:0) − log (1 + x ) ∗ µ X (cid:1) , which confirms this claim.The last example interprets Brownian motion starting in one and stopped when hittingzero as a stochastic exponential and discusses the corresponding stochastic logarithm. Example 5.5.
Let B be Brownian motion starting in zero, define the stopping time ρ = inf { t ≥ B t = − } and the nonnegative martingale Z = 1 + B ρ ∈ Z . That is, Z is Brownian motion startedin one and stopped as soon as it hits zero. We now compute X = L ( Z ) = 1 Z { Z> } · Z = 11 + B · B on [[0 , ρ [[ . Note that ρ is indeed X –maximal by Theorem 3.10. References
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