Existence of equivalent local martingale deflators in semimartingale market models
aa r X i v : . [ q -f i n . M F ] J un EXISTENCE OF EQUIVALENT LOCAL MARTINGALEDEFLATORS IN SEMIMARTINGALE MARKET MODELS
ECKHARD PLATEN AND STEFAN TAPPE
Abstract.
This paper offers a systematic investigation on the existence ofequivalent local martingale deflators, which are multiplicative special semi-martingales, in financial markets given by positive semimartingales. In par-ticular, it shows that the existence of such deflators can be characterized bymeans of the modified semimartingale characteristics. Several examples illus-trate our results. Furthermore, we provide interpretations of the deflators froman economic point of view. Introduction
For a given model of a financial market an important issue is the absence ofarbitrage opportunities. For continuous time models this has been characterized bymeans of the existence of an equivalent local martingale measure (ELMM); see, inparticular, the papers [13] and [15], the textbook [16], and also the paper [28]. Theabsence of arbitrage has also been characterized by means of the existence of anequivalent local martingale deflator (ELMD); see [42], and also the earlier papers[8] and [31]. In certain situations, results about criteria for the absence of arbitragehave been derived, for example, in [32, 11, 12].Moreover, if a financial market is free of arbitrage opportunities, it arises thequestion how to perform pricing and hedging of contingent claims. For example, ifthere are several ELMMs, one has to choose a suitable pricing measure. Work inthis direction has been done in the risk-neutral setting, for example, in [22, 40, 26,10, 41, 14], and beyond the risk-neutral approach, for example, in [19, 35, 39, 23].For the aforementioned results concerning the absence of arbitrage it is implicitlyassumed that the market consists of discounted price processes of risky assets withrespect to some savings account. In the recent paper [37] we have characterizedthe absence of arbitrage opportunities for semimartingale models which do notneed to have a savings account that could be used as numéraire. More precisely,let S = { S , . . . , S d } be a financial market consisting of positive semimartingaleswhich does not need to have a savings account. Provided we are allowed to adda savings account B to the market, we have shown in [37] that the market isfree of arbitrage if and only if there exists an ELMD Z which is a multiplicativespecial semimartingale, and that in this case the savings account has to fit into themultiplicative decomposition Z = DB − of the deflator. In this context, a savingsaccount is a predictable, strictly positive process B of locally finite variation.Motivated by this result, the goal of this paper is to provide a systematic in-vestigation on the existence of ELMDs which are multiplicative semimartingales,and to interpret these deflators from an economic point of view. We will study this Date : 2 June, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Equivalent local martingale deflator, multiplicative special semi-martingale, market price of risk, jump-diffusion model.We are grateful to Claudio Fontana, Alexander Melnikov and Martin Schweizer for valuablediscussions.
ECKHARD PLATEN AND STEFAN TAPPE problem for an arbitrary market S = { S i : i ∈ I } with an arbitrary nonempty indexset I = ∅ , identifying the tradeable securities. As noted in [37], the existence of anELMD, which is a multiplicative special semimartingale, is then still sufficient forthe absence of arbitrage. Note that later on in this paper we will often consider afinite market S = { S , . . . , S d } . Since we consider a market with strictly positivesemimartingales, for each i ∈ I we have S i = S i E ( X i ) , (1.1)where S i > , the process X i is a semimartingale such that X = 0 and ∆ X i > − ,and E denotes the stochastic exponential. We will assume that for each i ∈ I thesemimartingale X i (or, equivalently, the semimartingale S i ) is a special semimartin-gale with canonical decomposition X i = M i + A i , where M i ∈ M loc with M i = 0 is the local martingale part, and the predictableprocess A i ∈ V is the finite variation part. Here M loc denotes the space of localmartingales, and V denotes the space of all adapted processes with locally finitevariation starting at zero. As already mentioned, we are looking for multiplicativespecial ELMDs of the form Z = DB − with a local martingale D and a, so-called, virtual savings account B , which would be the savings account when it wereincluded as a traded asset in the market. Hence, we consider a multiplicative specialsemimartingale Z = DB − , where the local martingale D ∈ M loc and the virtual savings account B are givenby D = E ( − Θ) and B = E ( R ) for some Θ ∈ M loc with Θ = 0 and ∆Θ < , and some predictable R ∈ V with ∆ R > − . This candidate for an ELMD can be written as Z = E ( − e Θ − e R ) with uniquely determined processes e Θ ∈ M loc and e R ∈ V such that e Θ = 0 and ∆ e Θ + ∆ e R < , where we refer to Appendix D for further details. Our first mainresult (Theorem 4.5) states that the multiplicative special semimartingale Z is anELMD if and only if for each i ∈ I we have the drift condition A i = e R + [ A i , e R ] + [ M i , Θ] p (1.2)satisfied, where [ M i , Θ] p denotes the predictable compensator of the quadratic co-variation [ M i , Θ] . Note that (1.2) provides a decomposition of the return A i of theasset S i , and that the quantities appearing in this decomposition have the followinginterpretations: • The process e R + [ A i , e R ] is the locally risk-free return of the asset S i if M i were zero. • The process Θ is a market price of risk . • Furthermore, the process R is the general locally risk-free return or virtualshort rate of the savings account B .The arguments are as follows. Assume that D is the density process of an equivalentprobability measure Q ≈ P . Then the first term e R + [ A i , e R ] appearing in (1.2) is thedrift of the semimartingale X i under the measure Q ; see Remark 4.12 for further XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 3 details. The second term [ M i , Θ] p explains why we call Θ a market price of risk.At this stage we point out that condition (1.2) is satisfied if and only if A i = e R + [ A i , e R ] + [ M i , e Θ] p . We prefer to call Θ , and not e Θ , a market price of risk because it also shows up inthe density process D = E ( − Θ) of the measure change, provided it exists.The drift condition (1.2) also provides us with the following insight. If we considerthe stocks S = { S i : i ∈ I } , that is, the tradeable productive units of the economy,then a potential market price of risk Θ and also a respective virtual short rate R are visible, and provided by an ELMD Z = E ( − Θ) E ( R ) − . Of course, the processes Θ and R are generally not unique. If a central bank decidesto choose a lower short rate R , then the market price of risk Θ increases, whichcan potentially stimulate the economy. Accordingly, if a central bank decides tochoose a higher short rate, then the market price of risk decreases, which canpotentially thwart the economy. This follows from the drift condition (1.2) andprovides interpretations for different choices of the deflator.The situation simplifies if for each i ∈ I the semimartingale X i (or, equivalently,the semimartingale S i ) is locally square-integrable and quasi-left-continuous. Thenwe have R = e R and Θ = e Θ , and the ELMD Z can be expressed as Z = E ( − Θ) exp( R ) − = E ( − Θ − R ) . Furthermore, the drift condition (1.2) simplifies to A i = R + h M i , Θ i , (1.3)where h M i , Θ i denotes the predictable quadratic covariation of M i and Θ ; see The-orem 4.14 below. The quantities on the right-hand side of (1.3) have the followinginterpretations: • The process R is the virtual short rate and simultaneously for every i ∈ I the locally risk-free return of the asset S i if M i were zero. • The process Θ is a market price of risk .In order to investigate the existence of ELMDs further, consider a finite market S = { S , . . . , S d } consisting of locally square-integrable, quasi-left-continuous semi-martingales. Then the existence of an ELMD can be characterized on the basis of themodified integral characteristics of the R d -valued semimartingale X = ( X , . . . , X d ) appearing in (1.1). More precisely, agreeing on the notation R d = (1 , . . . , ∈ R d ,the existence of an ELMD Z , which is a multiplicative semimartingale, is essentiallyequivalent to the existence of an R d -valued process x and an R -valued process r satisfying the R d -valued linear equation c mod x = a − r R d , (1.4)where the R d -valued process a denotes the first integral characteristic, and the R d × d -valued process c mod denotes the modified second integral characteristic of X ;we refer to Section 5 for further details.For illustration, we consider the particular situation with jump-diffusion models,where for each i ∈ I the process X i appearing in (1.1) is of the form X i = a i · λ + σ i · W + γ i ∗ ( p − q ) with an R m -valued standard Wiener process and a homogeneous Poisson randommeasure p with compensator of the form q = λ ⊗ F , where F is a σ -finite measureon the mark space ( E, E ) . As we will show, the existence of an ELMD Z , whichis a multiplicative special semimartingale, is essentially equivalent to the existence ECKHARD PLATEN AND STEFAN TAPPE of an R m -valued process θ , an L ( F ) -valued process ψ and an R -valued process r such that h σ i , θ i R m + h γ i , ψ i L ( F ) = a i − r, i ∈ I. (1.5)For a finite market S = { S , . . . , S d } we can regard σ and γ as multidimensionallinear functionals, and then equation (1.5) can be expressed as the R d -valued linearequation σθ + γψ = a − r R d . (1.6)Furthermore, we will show that the existence of a solution to (1.6) is equivalent tothe existence of an R d -valued process x and an R -valued process r satisfying the R d -valued linear equation (1.4), where in this particular situation with a jump-diffusionmodel the modified second integral characteristic c mod is given by c ij mod = h σ i , σ j i R m + h γ i , γ j i L ( F ) for all i, j = 1 , . . . , d .We refer to Section 6 for further details.We provide several examples of jump-diffusion models, including Heath-Jarrow-Morton and Brody-Hughston interest rate term structure models. These two modelshave the interesting feature that the savings account B in the multiplicative decom-position of an ELMD Z = DB − , provided the latter exists, is unique; see Section3 for more details.The remainder of this paper is organized as follows. In Section 2 we introduce thefinancial market. In Section 3 we present several examples, where we already utilizethe results which we have indicated above. Afterwards, we proceed with the system-atic investigation on the existence of ELMDs, which are special semimartingales.In Section 4 we derive criteria when a multiplicative special semimartingale is anELMD. In Section 5 we treat the existence of ELMDs, and in Section 6 we focus onjump-diffusion models. Section 7 concludes. For convenience of the reader, severalauxiliary results concerning stochastic processes, matrices and linear operators aregathered in Appendices A–G.2. The financial market
In this section we introduce the financial market. Let (Ω , F , ( F t ) t ∈ R + , P ) be astochastic basis satisfying the usual conditions, see [27, Def. I.1.3]. Furthermore,we assume that F = { Ω , ∅} . Then every F -measurable random variable is P -almost surely constant. Let I = ∅ be an arbitrary nonempty index set, and let S = { S i : i ∈ I } be a financial market consisting of positive semimartingales. Moreprecisely, for each i ∈ I we assume that S i , S i − > . Then for each i ∈ I we have S i = S i E ( X i ) (2.1)with a semimartingale X i such that X = 0 and ∆ X i > − .2.1. Definition.
We call a semimartingale Z with Z, Z − > an equivalent localmartingale deflator (ELMD) for S if S i Z ∈ M loc for all i ∈ I . Definition.
We call a semimartingale ¯ Z with ¯ Z, ¯ Z − > an equivalent localmartingale numéraire (ELMN) for S if S i ¯ Z ∈ M loc for all i ∈ I . Note that a semimartingale Z with Z, Z − > is an ELMD if and only if ¯ Z = Z − is an ELMN. XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 5
Definition.
We call an equivalent probability measure Q ≈ P on (Ω , F ∞− ) an equivalent local martingale measure (ELMM) for S if S i is a Q -local martingalefor all i ∈ I . Definition.
We call every predictable process B of locally finite variation with B = 1 and B, B − > a savings account (or a locally risk-free asset ). The following result shows why we are interested in the existence of an ELMDwhich is a multiplicative special semimartingale. We denote by P +sf , ( S ) the convexset of all nonnegative, self-financing portfolios with initial value one evaluated at afixed terminal time T ∈ (0 , ∞ ) . Considering this set is equivalent to looking at allnonnegative, self-financing portfolios with strictly positive initial values; see [38] and[37] for more details. Furthermore, the following no-arbitrage concepts are NUPBR(No Unbounded Profit with Bounded Risk), NAA (No Asymptotic Arbitrage ofthe 1st Kind) and NA (No Arbitrage of the 1st Kind), which are known to beequivalent in the present situation; see, for example [29], [38] or [37] for furtherdetails.2.5. Theorem.
Suppose there exists an ELMD Z for S which is a multiplicativespecial semimartingale, and let Z = DB − be a multiplicative decomposition witha savings account B . Then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA .Proof. This is a consequence of [37, Thm. 7.5 and Remark 7.10]. (cid:3)
If the market S is finite, that is the index set I is finite, then the existence of suchan ELMD is equivalent to the existence of a savings account B such that P +sf , ( S ∪{ B } ) satisfies NUPBR, NAA and NA ; see [37, Thm. 7.5]. In view of Theorem 2.5,we are interested in the existence of an ELMD Z which is a multiplicative specialsemimartingale because this ensures the absence of arbitrage.3. Examples
Before we proceed with the systematic investigation on the existence of ELMDs,which are multiplicative special semimartingales, for the purpose of illustration wepresent concrete examples of financial models, where we discuss the existence of suchdeflators. In the upcoming examples, we utilize the results which we will develop inSections 4–6 later on. In each of the following examples the market S = { S i : i ∈ I } is given by a jump-diffusion model, where for each i ∈ I the process X i appearingin (2.1) is of the form X i = a i · λ + σ i · W + γ i ∗ ( p − q ) with λ denoting the Lebesgue measure, an R m -valued standard Wiener process W and a homogeneous Poisson random measure p with compensator of the form q = λ ⊗ F , where F is a σ -finite measure on the mark space ( E, E ) . In order tolook for ELMDs which are multiplicative special semimartingales, we consider amultiplicative special semimartingale Z = DB − , where D = E (cid:0) − θ · W − ψ ∗ ( p − q ) (cid:1) and B = E ( r · λ ) = exp( r · λ ) (3.1)with appropriate processes θ , ψ and r such that ψ < . As already mentioned inSection 1, for the absence of arbitrage we have to find a solution to the linear equa-tion (1.5), and for a finite market S = { S , . . . , S d } this equation can be expressedas the R d -valued linear equation (1.6). We refer to Section 6 for further details. ECKHARD PLATEN AND STEFAN TAPPE
Pure diffusion models
Consider a finite market S , where the R d -valued semimartingale X is an Itô processof the form X = a · λ + σ · W. Here we consider a multiplicative special semimartingale Z = DB − , where thelocal martingale D and the savings account B are of the form D = E ( − θ · W ) and B = E ( r · λ ) = exp( r · λ ) , (3.2)and the R d -valued linear equation (1.6) reads σθ = a − r R d , (3.3)where σ is regarded as an R d × m -valued process. The existence of a solution ( θ, r ) to(3.3) gives rise to an ELMD Z = DB − for the market S with the local martingale D and the savings account B given by (3.2).3.2. Examples of pure diffusion models
A particular situation of the pure diffusion model from Section 3.1 arises if the R d -valued semimartingale X is of the form X = a · λ + σ R d · W with constants a ∈ R d , σ > and an R -valued standard Wiener process W . Thenthe R d -valued linear equation (3.3) reads a = ( σθ + r ) R d , and this equation has a solution if and only if a ∈ lin { R d } , where lin { R d } denotesthe one-dimensional linear space generated by R d = (1 , . . . , ∈ R d .3.3. The Black-Scholes model
In the Black-Scholes model, which goes back to [3] and [33], the market is given by S = { S } , where the stock price satisfies dS t = aS t dt + σS t dW t with constants a ∈ R , σ > and an R -valued standard Wiener process W . This isa particular case of the pure diffusion models considered in Section 3.1. For a fixedconstant r ∈ R the linear equation (3.3) is the R -valued equation σθ = a − r, and it has the unique solution θ = a − rσ . Therefore, choosing the ELMD Z = DB − with the local martingale D and thesavings account B given by (3.2), we deduce that P +sf , ( S ∪ { B } ) = P +sf , ( { S, B } ) satisfies NUPBR, NAA and NA . This is in accordance with our findings from [37,Sec. 9]. XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 7
The Heston model
In the Heston model, which goes back to [25], the market is given by S = { S } ,where the stock price satisfies dS t = aS t dt + √ v t S t dW t with a constant a ∈ R and an R -valued standard Wiener process W . The varianceprocess v is a Cox-Ingersoll-Ross process dv t = κ ( ϑ − v t ) dt + ξ √ v t d f W t with initial value v > and constants κ, ϑ, ξ > . The process f W is another R -valued standard Wiener process. We assume that κϑ > ξ , which ensures that thevariance process v is strictly positive. The Heston model is also a particular case ofthe pure diffusion models considered in Section 3.1. For a fixed constant r ∈ R , thelinear equation (3.3) is the R -valued equation √ vθ = a − r, and it has the unique solution θ = a − r √ v . By the continuity of v we have θ ∈ L ( W ) . Therefore, choosing the ELMD Z = DB − with the local martingale D and the savings account B given by (3.2), wededuce that P +sf , ( S ∪ { B } ) = P +sf , ( { S, B } ) satisfies NUPBR, NAA and NA .3.5. The Merton model
In the Merton model, which goes back to [34], the market is given by S = { S } ,where the stock price satisfies dS t = µS t dt + σS t dW t + S t − dQ t with constants µ ∈ R , σ > , an R -valued standard Wiener process W and acompound Poisson process Q such that ∆ Q > − . More precisely, the jump sizedistribution is that of Y − , where Y has a lognormal distribution. We consider amore general situation, namely a jump-diffusion model with one asset, where themark space is given by ( E, E ) = ( R , B ( R )) , and where the measure F satisfies F ( R ) < ∞ , supp( F ) ⊂ ( − , ∞ ) and Z ( − , ∞ ) | x | F ( dx ) < ∞ . Furthermore, we define the mapping γ : Ω × R + × R → R as γ ( ω, t, x ) = x { x> − } , ( ω, t, x ) ∈ Ω × R + × R . Note that this model covers the Merton model, because the lognormal distributionadmits second order moments. The linear equation (1.6) is the R -valued equation σθ + Z R xψ ( x ) F ( dx ) = a − r. (3.4)Note that equation (3.4) admits several solutions. For example, choose a constant r ∈ R and a function ψ ∈ L ( F ) such that ψ < . Then the solution to the linearequation (3.4) is given by θ = 1 σ (cid:18) a − r − Z R xψ ( x ) F ( dx ) (cid:19) . ECKHARD PLATEN AND STEFAN TAPPE
Therefore, choosing the ELMD Z = DB − with the local martingale D and thesavings account B given by (3.1), we deduce that P +sf , ( S ∪ { B } ) = P +sf , ( { S, B } ) satisfies NUPBR, NAA and NA .3.6. Jump-diffusion models with finitely many jumps
Consider a finite market S and assume that the measure F on the mark space ( E, E ) is concentrated on finitely many points. More precisely, we assume there arepairwise different elements x , . . . , x n ∈ E such that { x j } ∈ E for each j = 1 , . . . , n ,and that the measure F is of the form F = n X j =1 c j δ x j with finite numbers c , . . . , c n > . Here δ x j denotes the Dirac measure at point x j for each j = 1 , . . . , n . Then the space L ( p ) can be identified with the space of alloptional processes ρ : Ω × R + → R n such that k ρ k R n · λ ∈ V + . Indeed, for each ψ ∈ L ( p ) the corresponding process ρ is given by ρ = (cid:0) ψ ( x ) , . . . , ψ ( x n ) (cid:1) . With this identification we have ψ < if and only if ρ j < for all j = 1 , . . . , n , andfor each ψ ∈ L ( p ) we have γψ = Γ ρ, where the R d × n -valued process Γ is given by Γ ij = c j γ i ( x j ) for all i = 1 , . . . , d and j = 1 , . . . , n .Therefore, for θ ∈ L ( W ) and an optional processes ρ : Ω × R + → R n suchthat k ρ k R n · λ ∈ V + and ρ j < for all j = 1 , . . . , n the multiplicative specialsemimartingale Z = DB − with the local martingale D and the savings account B given by (3.1), where ψ ∈ L ( p ) is defined as ψ := P nj =1 ρ j { x j } , is an ELMD for S if and only if σθ + Γ ρ = a − r R d . (3.5)Recall that in the linear equation (3.5) the process σ is R d × m -valued, and that theprocess Γ is R d × n -valued.3.7. The Black-Scholes model with an additional Poisson process
Consider a market S = { S } with one asset, where the stock price satisfies dS t = µS t dt + σS t dW t + S t − dN t with constants µ ∈ R , σ > , an R -valued standard Wiener process W and a Poissonprocess N with intensity c > . Note that this is a particular case of the Mertontype model considered in Section 3.5. This model is also a jump-diffusion model asin Section 3.6, and the linear equation (3.5) is the R -valued equation σθ + cρ = a − r. (3.6)Clearly, there are several solutions ( θ, ρ, r ) with ρ < to the linear equation (3.6),ensuring the absence of arbitrage. For example, for constants r ∈ R and ρ < thesolution to (3.6) is given by θ = a − r − cρσ . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 9
Heath-Jarrow-Morton interest rate term structure models
In this section we consider Heath-Jarrow-Morton (HJM) interest term structuremodels for modeling markets of zero coupon bonds. Under risk-neutral pricing werefer to [24] for the classical HJM model driven by Wiener processes. Furthermore,we refer, for example, to [18, 17] for risk-neutral HJM models driven by Lévy pro-cesses, and, for example, to [2, 1, 21] for risk-neutral HJM models driven by Wienerprocesses and Poisson random measures. In the framework of the Benchmark Ap-proach (see [35]) HJM models driven by Wiener processes and Poisson randommeasures have been studied in [9, 7]. We assume that for each T ∈ R + the forwardrate is given by f ( T ) = f ( T ) + α ( T ) · λ + σ ( T ) · W + γ ( T ) ∗ ( p − q ) with a starting value f ( T ) ∈ R and suitable integrands α ( T ) ∈ L ( λ ) , σ ( T ) ∈ L ( W ) and γ ( T ) ∈ L ( p ) . By a monotone class argument, the short rate f · ( · ) given by f t ( t ) for all t ∈ R + has an optional version. for each T ∈ R + we define the process F ( T ) as F t ( T ) := − Z Tt f t ( s ) ds, t ∈ R + , and the bond price process P ( T ) := exp( F ( T )) . In the sequel, we are interested in the bond market S = { P ( T ) : T ∈ R + } . Let T ∈ R + be arbitrary. We define the processes A ( T ) , Σ( T ) and Γ( T ) as follows.For t ≤ T we set A t ( T ) := − Z Tt α t ( s ) ds, Σ t ( T ) := − Z Tt σ t ( s ) ds, Γ t ( T ) := − Z Tt γ t ( s ) ds, and for t > T we set A t ( T ) := A T ( T ) , Σ t ( T ) := Σ T ( T ) , Γ t ( T ) := Γ T ( T ) . Of course, this requires appropriate integrability conditions on α , σ and γ . Underfurther suitable regularity conditions, a standard calculation using stochastic Fubinitheorems shows that F ( T ) = F ( T ) + ( f · ( · ) + A ( T )) · λ + Σ( T ) · W + Γ( T ) ∗ ( p − q ) . By [27, Thm. II.8.10] we have P ( T ) = P ( T ) E ( X ( T )) , where the semimartingale X ( T ) is given by X ( T ) = (cid:18) f · ( · ) + A ( T ) + 12 k Σ( T ) k R m + Z E (cid:0) e Γ( T ) − − Γ( T ) (cid:1) F ( dx ) (cid:19) · λ + Σ( T ) · W + (cid:0) e Γ( T ) − (cid:1) ∗ ( p − q ) . Now, we consider a multiplicative special semimartingale Z = DB − , where thelocal martingale D and the savings account B are given by (3.1) with θ ∈ L ( W ) and ψ ∈ L ( p ) such that ψ < , as well as an optional process r ∈ L ( λ ) .3.1. Theorem.
We assume that the processes f · ( · ) , r , θ and ψ are càd (right-continuous) or càg (left-continuous), and that for each T ∈ R + the processes A ( T ) , Σ( T ) and Γ( T ) are càd or càg. Then the following statements are equivalent: (i) Z is an ELMD for the bond market S . (ii) Z is an ELMD for the extended bond market S ∪ { B } . (iii) We have up to an evanescent set r = f · ( · ) (3.7) and for each T ∈ R + we have up to an evanescent set (3.8) − A ( T ) = 12 k Σ( T ) k R m − h Σ( T ) , θ i R m + Z E (cid:16) (1 − ψ ( x ))( e Γ( T,x ) − − Γ( T, x ) (cid:17) F ( dx ) . If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA .Proof. Noting the assumed regularity conditions, by Theorem 6.2 the process Z isan ELMD for S , or equivalently for S ∪ { B } , if and only if for each T ∈ R + we haveup to an evanescent set h Σ( T ) , θ i R m + h Γ( T ) , ψ i L ( F ) = f · ( · ) + A ( T ) + 12 k Σ( T ) k R m + Z E (cid:0) e Γ( T,x ) − − Γ( T, x ) (cid:1) F ( dx ) − r. Evaluating this equation at t = T for every T ∈ R + we obtain that Z is an ELMDfor S if and only if we have (3.7) up to an evanescent set, and for each T ∈ R + wehave (3.8) up to an evanescent set. The additional statement is a consequence ofTheorem 2.5. (cid:3) Consequently, if an ELMD Z for the bond market, which is a multiplicativespecial semimartingale, exists, then the savings account B in the multiplicativedecomposition Z = DB − is unique, and it is given by B t = exp (cid:18) Z t f s ( s ) ds (cid:19) , t ∈ R + . Remark.
Differentiating equation (3.8) with respect to T yields the drift con-dition α ( T ) = −h σ ( T ) , Σ( T ) − θ i R m − Z E γ ( T, x ) (cid:0) (1 − ψ ( x )) e Γ( T,x ) − (cid:1) F ( dx ) . This drift condition also appears in the framework of the Benchmark Approach; see [9] and [7] . Brody-Hughston interest rate term structure models
In this section we investigate Brody-Hughston interest rate term structure models.Such term structure models driven by Wiener processes have been introduced in[5, 6] in the risk-neutral setting; see also [20] for such models driven by Wienerprocesses and Poisson random measures in the risk-neutral setting. Let ρ = ( ρ t ) t ∈ R + be a stochastic process consisting of strictly positive probability densities on R + .For each T ∈ R + we define the bond prices P t ( T ) := Z ∞ T − t ρ t ( u ) du, t ∈ [0 , T ] . (3.9)We are interested in the bond market S = { P ( T ) : T ∈ R + } . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 11
Let T ∈ R + be arbitrary. Since the bond prices given by (3.9) are strictly positive,we may assume that P ( T ) = P ( T ) E (cid:0) α ( T ) · λ + σ ( T ) · W + γ ( T ) ∗ ( p − q ) (cid:1) with suitable integrands α ( T ) ∈ L ( λ ) , σ ( T ) ∈ L ( W ) and γ ( T ) ∈ L ( p ) . Thenwe have P ( T ) = P ( T ) + P ( T ) α ( T ) · λ + P ( T ) σ ( T ) · W + P − ( T ) γ ( T ) ∗ ( p − q ) . Now we switch to the Musiela parametrization p t ( x ) = P t ( t + x ) , t, x ∈ R + . Subject to appropriate regularity conditions, we obtain that p is a solution to thestochastic partial differential equation (SPDE) p ( x ) = p ( x ) + (cid:0) ∂ x p ( x ) + p ( x )ˆ α ( x ) (cid:1) · λ + p ( x )ˆ σ ( x ) · W + p − ( x )ˆ γ ( x ) ∗ ( p − q ) , x ∈ R + , where the new coefficients ˆ α, ˆ σ, ˆ γ are given by ˆ α t ( x ) := α ( t, t + x ) , t, x ∈ R + , ˆ σ t ( x ) := σ ( t, t + x ) , t, x ∈ R + , ˆ γ t ( x ) := γ ( t, t + x ) , t, x ∈ R + . By (3.9) we have p t ( x ) = Z ∞ x ρ t ( u ) du, t, x ∈ R + , (3.10)and hence ρ t ( x ) = − ∂ x p t ( x ) , t, x ∈ R + . Therefore, subject to appropriate regularity conditions, which allow us to inter-change differentiation and integration, we obtain that ρ satisfies the SPDE ρ ( x ) = ρ ( x ) + (cid:0) ∂ x ρ ( x ) − ∂ x ( p ( x )ˆ α ( x )) (cid:1) · λ − ∂ x ( p ( x )ˆ σ ( x )) · W − ∂ x ( p − ( x )ˆ γ ( x )) ∗ ( p − q ) , x ∈ R + . Since ρ is strictly positive, we can define the coefficients ¯ α, ¯ σ, ¯ γ as ¯ α ( x ) := − ∂ x ( p ( x )ˆ α ( x )) ρ ( x ) , x ∈ R + , ¯ σ ( x ) := − ∂ x ( p ( x )ˆ σ ( x )) ρ ( x ) , x ∈ R + , ¯ γ ( x ) := − ∂ x ( p − ( x )ˆ γ ( x )) ρ ( x ) , x ∈ R + . Therefore, we obtain ρ ( x ) = ρ ( x ) + (cid:0) ∂ x ρ ( x ) + ρ ( x )¯ α ( x ) (cid:1) · λ + ρ ( x )¯ σ ( x ) · W + ρ − ( x )¯ γ ( x ) ∗ ( p − q ) , x ∈ R + . Consequently, noting that ρ ( x ) ∂ x ln( ρ ( x )) = ∂ x ρ ( x ) , x ∈ R + , for each x ∈ R + we have the representation ρ ( x ) = ρ ( x ) E (cid:0) ( ∂ x ln( ρ ( x )) + ¯ α ( x )) · λ + ¯ σ ( x ) · W + ¯ γ ( x ) ∗ ( p − q ) (cid:1) . Since the process ρ leaves the convex set of probability densities invariant, we haveup to an evanescent set Z ∞ (cid:0) ∂ x ρ ( x ) + ρ ( x )¯ α ( x ) (cid:1) dx = 0 , (3.11) Z ∞ ¯ σ ( x ) ρ ( x ) dx = 0 , (3.12) Z ∞ ¯ γ ( x ) ρ − ( x ) dx = 0 . (3.13)Note that condition (3.11) is satisfied if and only if we have up to an evanescent set ρ (0) = Z ∞ ρ ( x )¯ α ( x ) dx λ -a.e. P -a.e.(3.14)Now, we consider a multiplicative special semimartingale Z = DB − , where thelocal martingale D and the savings account B are given by (3.1) with θ ∈ L ( W ) and ψ ∈ L ( p ) such that ψ < , as well an an optional process r ∈ L ( λ ) .3.3. Theorem.
We assume that the following conditions are fulfilled: • The processes r , θ and ψ are càd or càg, and for each x ∈ R + the process ρ ( x ) is càd or càg. • For each T ∈ R + the processes α ( T ) , σ ( T ) and γ ( T ) are càd or càg, andthe processes α , σ and γ are continuous in the second argument T . • For each x ∈ R + the processes ˆ α ( x ) , ˆ σ ( x ) and ˆ γ ( x ) are càd or càg, and theprocesses ˆ α , ˆ σ and ˆ γ are continuous in the second argument x . • For each x ∈ R + the processes ¯ α ( x ) , ¯ σ ( x ) and ¯ γ ( x ) are càd or càg.Then the following statements are equivalent: (i) Z is an ELMD for the bond market S . (ii) Z is an ELMD for the extended bond market S ∪ { B } . (iii) We have up to an evanescent set r = ρ (0) , (3.15) and for each x ∈ R + we have up to an evanescent set ¯ α ( x ) = r + h ¯ σ ( x ) , θ i R m + h ¯ γ ( x ) , ψ i L ( F ) . (3.16) If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA .Proof. In view of the assumed regularity, by Theorem 6.2 the process Z is an ELMDfor S , or equivalently for S ∪ { B } , if and only if for each T ∈ R + we have up to anevanescent set α ( T ) = r + h σ ( T ) , θ i R m + h γ ( T ) , ψ i L ( F ) . (3.17)By the assumed continuity in the second argument, this is satisfied if and only iffor each x ∈ R + we have up to an evanescent set ˆ α ( x ) = r + h ˆ σ ( x ) , θ i R m + h ˆ γ ( x ) , ψ i L ( F ) . (3.18)Let x ∈ R + be arbitrary. If condition (3.18) is satisfied, then we have ¯ α ( x ) = − ∂ x ( p ( x )ˆ α ( x )) ρ ( x ) = − ∂ x ( p ( x )( r + h ˆ σ ( x ) , θ i R m + h ˆ γ ( x ) , ψ i L ( F ) )) ρ ( x )= r + h ¯ σ ( x ) , θ i R m + h ¯ γ ( x ) , ψ i L ( F ) , showing (3.16). Conversely, suppose that condition (3.16) is satisfied. Then we have ∂ x ( p ( x )ˆ α ( x )) = ∂ x (cid:0) p ( x )( r + h ˆ σ ( x ) , θ i R m + h ˆ γ ( x ) , ψ i L ( F ) ) (cid:1) . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 13
Noting (3.11)–(3.13) and (3.10), integrating gives us p ( x )ˆ α ( x ) = − Z ∞ x ∂ u ( p ( u )ˆ α ( u )) du = − Z ∞ x ∂ u (cid:0) p ( u )( r + h ˆ σ ( u ) , θ i R m + h ˆ γ ( u ) , ψ i L ( F ) ) (cid:1) du = p ( x )( r + h ˆ σ ( x ) , θ i R m + h ˆ γ ( x ) , ψ i L ( F ) ) , showing (3.18). Therefore, for each x ∈ R + condition (3.16) is satisfied up to anevanescent set if and only if for each T ∈ R + condition (3.17) is satisfied up toan evanescent set. Consequently, the process Z is an ELMD for S , or equivalentlyfor S ∪ { B } , if and only if for each x ∈ R + we have (3.16) up to an evanescentset. Inserting (3.16) into (3.14) and noting (3.12), (3.13) as well as R ∞ ρ ( x ) dx = 1 ,we obtain that Z is an ELMD for S if and only if we have (3.15) and (3.16). Theadditional statement is a consequence of Theorem 2.5. (cid:3) Note that the situation with the Brody-Hughston model is similar to that withthe HJM model from the previous section. Namely, if an ELMD Z , which is amultiplicative special semimartingale, exists, then the savings account B in themultiplicative decomposition Z = DB − is unique, and it is given by B t = exp (cid:18) Z t ρ s (0) ds (cid:19) , t ∈ R + . Equivalent local martingale deflators
After these examples, we proceed with the systematic investigation on the ex-istence of ELMDs, which are special semimartingales. In this section we derivecriteria when a multiplicative special semimartingale is an ELMD, and draw someconsequences. Let S = { S i : i ∈ I } be a financial market of the form (2.1) as inSection 2. As motivated there, we are interested in the existence of an ELMD Z which is a multiplicative special semimartingale because this ensures the absenceof arbitrage. Let Z be a semimartingale of the form Z = E ( − Y ) (4.1)with a semimartingale Y such that Y = 0 and ∆ Y < .4.1. Proposition.
The following statements are equivalent: (i) Z is an ELMD for S . (ii) For each i ∈ I we have X i − Y − [ X i , Y ] ∈ M loc . (4.2) Proof.
Let i ∈ I be arbitrary. By Yor’s formula (see [27, II.8.19]) we have S i Z = S i E ( X i ) E ( − Y ) = S i E ( X i − Y − [ X i , Y ]) , which proves the stated equivalence. (cid:3) In the upcoming results S p denotes the space of all special semimartingales, and A loc denotes the space of all elements from V which are locally integrable; cf. [27].4.2. Corollary.
Suppose that Z is an ELMD for S , and that S i ∈ S p for some i ∈ I . Then the following statements are equivalent: (i) Z is a multiplicative special semimartingale. (ii) We have Z ∈ S p . (iii) We have [ X i , Y ] ∈ A loc . Proof. (i) ⇔ (ii): This equivalence is a consequence of [27, Thm. II.8.21].(ii) ⇔ (iii): By Lemma C.1 we have X i ∈ S p , and we have Z ∈ S p if and onlyif Y ∈ S p . By Proposition 4.1 we have (4.2). Since X i ∈ S p , we deduce that Y + [ X i , Y ] ∈ S p . Therefore, we have Y ∈ S p if and only if [ X i , Y ] ∈ A loc . (cid:3) Corollary.
Suppose that Z is an ELMD for S , which is a multiplicative specialsemimartingale. Then for each i ∈ I the following statements are equivalent: (i) We have S i ∈ S p . (ii) We have [ X i , Y ] ∈ A loc .Proof. According to [27, Thm. II.8.21] we have Z ∈ S p . Hence, by Lemma C.1 wehave Y ∈ S p , and we have S i ∈ S p if and only if X i ∈ S p . By Proposition 4.1 wehave (4.2). Since Y ∈ S p , we deduce that X i − [ X i , Y ] ∈ S p . Therefore, we have X i ∈ S p if and only if [ X i , Y ] ∈ A loc . (cid:3) From now on, we assume that for each i ∈ I the semimartingale X i appearingin (2.1) is a special semimartingale with canonical decomposition X i = M i + A i , (4.3)where M i is the local martingale part and A i is the finite variation part. Further-more, let R ∈ V be a predictable process with ∆ R > − , and let Θ ∈ M loc be alocal martingale with Θ = 0 and ∆Θ < . Let e R ∈ V be the predictable processwith ∆ e R < according to Proposition D.2, and let e Θ ∈ M loc be the local mar-tingale with e Θ = 0 and ∆ e Θ + ∆ e R < according to Proposition D.3. We assumethat the semimartingale Y appearing in (4.1) is the special semimartingale withcanonical decomposition Y = e Θ + e R. (4.4)Then by Proposition D.3 we have the multiplicative decomposition Z = DB − , where D = E ( − Θ) and B = E ( R ) . We call R the locally risk-free return of thesavings account B .4.4. Lemma.
For each i ∈ I the following statements are equivalent: (i) We have [ X i , Y ] ∈ A loc . (ii) We have [ M i , e Θ] ∈ A loc . (iii) We have [ M i , Θ] ∈ A loc .In either case we have [ X i , Y ] p = [ A i , e R ] + [ M i , e Θ] p = [ A i , e R ] + [ M i , Θ] p . Proof.
This is an immediate consequence of Lemma B.3. (cid:3)
Theorem.
The following statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } . (iii) For each i ∈ I we have [ X i , Y ] ∈ A loc and up to an evanescent set A i = e R + [ X i , Y ] p . (4.5)(iv) For each i ∈ I we have [ M i , Θ] ∈ A loc and up to an evanescent set A i = e R + [ A i , e R ] + [ M i , Θ] p . (4.6) If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 15
Proof. (i) ⇒ (ii): This implication follows, because BZ = D ∈ M loc .(ii) ⇒ (i): This implication is obvious.(i) ⇔ (iii): Let i ∈ I be arbitrary. Taking into account the canonical decompositions(4.3) and (4.4), we have condition (4.2) if and only if M i + A i − e Θ − e R − [ X i , Y ] ∈ M loc , which is equivalent to A i − e R − [ X i , Y ] ∈ M loc ∩ V . Since M loc ∩ V ⊂ A loc , this is satisfied if and only if [ X i , Y ] ∈ A loc and A i − e R − [ X i , Y ] p ∈ M loc ∩ V . (4.7)Note that the process on the left-hand side of (4.7) is predictable. Hence, accordingto [27, Cor. I.3.16], condition (4.7) is satisfied if and only if we have (4.5) up to anevanescent set. Consequently, applying Proposition 4.1 the process Z is an ELMDfor S if and only if we have [ X i , Y ] ∈ A loc and (4.5) up to an evanescent set.(iii) ⇔ (iv): Let i ∈ I be arbitrary. By Lemma 4.4 we have [ X i , Y ] ∈ A loc if andonly if [ M i , Θ] ∈ A loc , and in this case, using Lemmas B.1 and B.2 we obtain [ X i , Y ] p = [ M i + A i , e Θ + e R ] p = [ M i , e Θ] p + [ A i , e R ] p = [ M i , Θ − [Θ , e R ]] p + [ A i , e R ] = [ M i , Θ] p + [ A i , e R ] . The additional statement is a consequence of Theorem 2.5. (cid:3)
Remark.
In the situation of Theorem 4.5 we can also formally check that thedrift conditions (4.6) are satisfied for the extended market S ∪ { B } . Indeed, thenthe additional drift condition R = e R + [ R, e R ] is just equation (D.7) from Proposition D.2. In principle, for a fixed predictable process R ∈ V it is possible to change thelocal martingale Θ ∈ M loc in order to obtain another ELMD. More precisely, wehave the following result.4.7. Corollary.
Suppose that the equivalent conditions from Theorem 4.5 are ful-filled. Let T ∈ M loc be a local martingale with T = 0 and ∆Θ + ∆ T < suchthat [ M i , T ] ∈ A loc for each i ∈ I . We define the process ˆ Z := ˆ DB − , where ˆ D := E ( − Θ − T ) . Then the following statements are equivalent: (i) ˆ Z is an ELMD for S . (ii) ˆ Z is an ELMD for S ∪ { B } . (iii) For each i ∈ I we have up to an evanescent set [ M i , T ] p = 0 . Proof.
This is an immediate consequence of Theorem 4.5. (cid:3)
On the other hand, for a fixed Θ ∈ M loc it is not possible to change the pre-dictable process R ∈ V in order to obtain another ELMD. More precisely, we havethe following result, which is in accordance with [37, Prop. 7.9].4.8. Corollary.
Suppose that the equivalent conditions from Theorem 4.5 are ful-filled. Let V ∈ V be a predictable process with ∆ V > − , and let e V ∈ V be thecorresponding predictable process with ∆ e V < according to Proposition D.2. Wedefine the process ˆ Z := D ˆ B − , where ˆ B := E ( V ) . Then the following statementsare equivalent: (i) ˆ Z is an ELMD for S . (ii) ˆ Z is an ELMD for S ∪ { ˆ B } . (iii) We have B = ˆ B up to an evanescent set. (iv) We have R = V up to an evanescent set. (v) We have e R = e V up to an evanescent set.Proof. (i) ⇔ (ii): This follows from Theorem 4.5.(i) ⇔ (v): Since Z is an ELMD for S , by Theorem 4.5 for each i ∈ I we have (4.5)up to an evanescent set. Furthermore, by Theorem 4.5 the process ˆ Z is an ELMDfor S if and only if for each i ∈ I we have up to an evanescent set A i = e V + [ X i , Y ] p , which by (4.5) is equivalent to e R = e V up to an evanescent set.(iii) ⇔ (iv): This equivalence is evident.(iv) ⇔ (v): This equivalence follows from Proposition D.2. (cid:3) In the next result we determine the dynamics of the ELMN, provided it exists.4.9.
Proposition.
Suppose that the equivalent conditions from Theorem 4.5 arefulfilled, and define the ELMN ¯ Z := Z − . Let ν be the predictable compensator ofthe random measure µ Y . Then we have the representation ¯ Z = E (cid:18) e R + e Θ + h Θ c , Θ c i + y − y ∗ µ Y (cid:19) , and the following statements are equivalent: (i) ¯ Z is a special semimartingale. (ii) We have y − y ∗ ν ∈ A +loc . If the previous conditions are fulfilled, then we have ¯ Z = E ( ¯ N + ¯ B ) , where the local martingale ¯ N ∈ M loc and the predictable process ¯ B ∈ V are givenby ¯ N = e Θ + y − y ∗ ( µ Y − ν ) , ¯ B = e R + h Θ c , Θ c i + y − y ∗ ν. Proof.
This is an immediate consequence of Proposition C.4. (cid:3)
We define the new market with discounted assets S B − := { S i B − : i ∈ I } .4.10. Proposition.
Suppose that D ∈ M with P ( D ∞ > , and let Q ≈ P be theprobability measure on (Ω , F ∞− ) with density process D relative to P . Then thefollowing statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } . (iii) Q is an ELMM for S B − .Proof. (i) ⇔ (ii): This equivalence follows from Theorem 4.5.(i) ⇔ (iii): Z is an ELMD for S if and only if D is an ELMD for S B − . By [37,Lemma 4.7] this is the case if and only if Q is an ELMM for S B − . (cid:3) XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 17
Remark.
Suppose that D ∈ M with P ( D ∞ > , and let Q ≈ P be theprobability measure on (Ω , F ∞− ) with density process D relative to P . Let i ∈ I be such that [ M i , Θ] ∈ A loc . By Proposition E.4 the process X i is also a specialsemimartingale under Q , and its canonical decomposition X i = ( M i ) ′ + ( A i ) ′ isgiven by ( M i ) ′ = M i + [ M i , Θ] p and ( A i ) ′ = A i − [ M i , Θ] p , where the predictable compensators are computed under P . By Yor’s formula (see [27, II.8.19] ) we have S i B − = S i E ( X i ) E ( − e R ) = S i E (cid:0) X i − e R − [ X i , e R ] (cid:1) = S i E (cid:0) ( M i ) ′ + ( B i ) ′ (cid:1) , where the predictable process ( B i ) ′ ∈ V is given by ( B i ) ′ = ( A i ) ′ − e R − [ A i , e R ] . Therefore, S i B − is a Q -local martingale if and only if ( B i ) ′ = 0 , which is equivalentto (4.6), confirming Theorem 4.5 and Proposition 4.10. Remark.
Suppose that the equivalent conditions from Proposition 4.10 arefulfilled. In view of Remark 4.11, condition (4.6) reads ( A i ) ′ = e R + [ A i , e R ] , and we see that the process e R + [ A i , e R ] in (4.6) can be regarded as the locally risk-free return of the asset S i if M i were zero, and that the process Θ in (4.6) can beregarded as a market price of risk . Remark.
Assume I = { , . . . , d } for some d ∈ N , and that the equivalent con-ditions from Proposition 4.10 are fulfilled. Then X = ( X , . . . , X d ) is an R d -valuedspecial semimartingale. We denote by ( A, C, ν ) its characteristics. By Remark 4.11and Proposition E.4 the process X is also a special martingale under Q , and itscharacteristics ( A ′ , C ′ , ν ′ ) are given by ( A i ) ′ = e R + [ A i , e R ] , i = 1 , . . . , d,C ′ = C,ν ′ = (cid:0) − M P µ X (∆Θ | f P ) (cid:1) · ν. For the rest of this section, we assume that the special semimartingales ( X i ) i ∈ I appearing in (4.3) and the special semimartingale Y appearing in (4.4) are locallysquare-integrable and quasi-left-continuous. Then we have M i ∈ H for each i ∈ I ,and we have e Θ ∈ H , where H denotes the space of all locally square-integrablemartingales. Furthermore, by Lemma A.5, for each i ∈ I the local martingale M i is quasi-left-continuous and the process A i is continuous, and the local martingale e Θ is quasi-left-continuous and the process e R is continuous. Using Propositions D.2and D.3 we have Θ = e Θ , R = e R and B = exp( R ) .4.14. Theorem.
The following statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } . (iii) For each i ∈ I we have up to an evanescent set A i = R + h M i , Θ i . (4.8) If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA .Proof. Taking into account Lemma B.4, this is an immediate consequence of The-orem 4.5. (cid:3)
Now we determine the dynamics of the ELMN in the present setting, providedit exists.4.15.
Proposition.
Suppose that the equivalent conditions from Theorem 4.14 arefulfilled, and define the ELMN ¯ Z := Z − . Let ν be the predictable compensator ofthe random measure µ Θ . Then the following statements are equivalent: (i) ¯ Z is a locally square-integrable semimartingale. (ii) We have θ − θ ∗ ν, (cid:18) θ − θ (cid:19) ∗ ν ∈ A +loc . (4.9)(iii) We have θ − θ ∗ ν, (cid:18) θ − θ (cid:19) ∗ ν ∈ A +loc . (4.10)(iv) There exists a quasi-left-continuous local martingale ¯ N ∈ H with ¯ N = 0 such that ¯ N − Θ ∈ V , ¯ N c = Θ c and ∆ ¯ N = ∆Θ1 − ∆Θ . (4.11) If the previous conditions are fulfilled, then the process ¯ Z admits the representation ¯ Z = E (cid:0) R + h Θ , ¯ N i + ¯ N (cid:1) , (4.12) and the local martingale ¯ N ∈ M loc is given by ¯ N = Θ + θ − θ ∗ ( µ Θ − ν ) . Proof.
This is an immediate consequence of Proposition C.5. (cid:3) Existence of equivalent local martingale deflators
In this section we treat the existence of ELMDs. We consider the frameworkof Section 4 with I = { , . . . , d } for some d ∈ N ; that is, we have finitely manyassets. We introduce the R d -valued special semimartingale X := ( X , . . . , X d ) . Asat the end of Section 4, we assume that X is locally square-integrable and quasi-left-continuous. In view of Theorem 4.5, we are interested in finding a continuousprocess R ∈ V and a quasi-left-continuous local martingale Θ ∈ H with Θ = 0 and ∆Θ < such that up to an evanescent set A i = R + h M i , Θ i for all i = 1 , . . . , d ,(5.1)because then Z = E ( − Θ) E ( R ) − = E ( − Θ) exp( R ) − is an ELMD for S . By Proposition F.12 there exist a continuous process Γ ∈ A +loc and modified integral characteristics ( a, c mod , K ) of X with respect to Γ . Further-more, by Proposition F.13 there exist integral characteristics ( a, c, K ) and a purelydiscontinuous second integral characteristic v of X with respect to Γ , and we have c mod = c + v Γ -a.e. P -a.e.(5.2)5.1. Proposition.
The following statements are equivalent: (i)
There exist an optional R -valued process r and an optional S ( d +1) × ( d +1)+ -valued process ˆ c mod such that ˆ c ij mod = c ij mod for all i, j = 1 , . . . , d , and wehave (cid:0) ˆ c i,d +1mod (cid:1) i =1 ,...,d = a − r R d Γ -a.e. P -a.e. (5.3) XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 19 (ii)
There exist an optional R -valued process r and optional S ( d +1) × ( d +1)+ -valuedprocesses ˆ c and ˆ v such that ˆ c ij = c ij and ˆ v ij = v ij for all i, j = 1 , . . . , d ,and we have (cid:0) ˆ c i,d +1 (cid:1) i =1 ,...,d + (cid:0) ˆ v i,d +1 (cid:1) i =1 ,...,d = a − r R d Γ -a.e. P -a.e. (5.4)(iii) There exist an optional R -valued process r and an optional R d -valued pro-cess x such that c mod x = a − r R d Γ -a.e. P -a.e. (5.5)(iv) There exist an optional R -valued process r and optional R d -valued processes x and y such that cx + vy = a − r R d Γ -a.e. P -a.e. (5.6) Proof.
Taking into account (5.2), this is a consequence of Proposition G.7. Notethat the corresponding processes can indeed be chosen to be optional, which followsfrom Lemma G.1 and the additional statements from Lemma G.3 and PropositionG.4. (cid:3)
Remark.
Note that the equivalent conditions from Proposition 5.1 are fulfilledif c mod ∈ S d × d ++ Γ -a.e. P -a.e. The following results show that the existence of a continuous process R ∈ V and a quasi-left-continuous local martingale Θ ∈ H satisfying (5.1) is essentiallyequivalent to the existence of an optional R -valued process r and an optional R d -valued process x satisfying (5.5).Let R ∈ V be a continuous process, and let Θ ∈ H be a quasi-left-continuouslocal martingale with Θ = 0 and ∆Θ < . Denoting by L (Γ) the space of alloptional processes r : Ω × R + → R such that | r | · Γ ∈ V + , we assume there isan optional process r ∈ L (Γ) such that R = r · Γ , and that the R d +1 -valuedsemimartingale ˆ X := ( X, Θ) admits modified integral characteristics (ˆ a, ˆ c mod , ˆ K ) with respect to Γ , where of course ˆ a = ( a, .5.3. Proposition.
If condition (5.1) is satisfied, then condition (5.3) is satisfied aswell.Proof.
By (5.1), for all i = 1 , . . . , d we have ˆ c i,d +1mod · Γ = ˆ C i,d +1mod = h M i , Θ i = A i − R = ( a i − r ) · Γ , showing that (5.3) is fulfilled. (cid:3) Now, we assume that the equivalent conditions from Proposition 5.1 are fulfilled.By (5.2), (5.3) and (5.4) we may assume that ˆ c mod = ˆ c + ˆ v Γ -a.e. P -a.e.5.4. Lemma.
There is a transition kernel ˆ K from (Ω × R + , O ) into ( R d +1 , B ( R d +1 )) such that on Ω × R + we have ˆ K ( { } ) = 0 and Z R d +1 | ˆ x | ˆ K ( d ˆ x ) < ∞ , (5.7) for every nonnegative, measurable function f : R d → R + we have Z R d f ( x ) K ( dx ) = Z R d +1 f ( x ) ˆ K ( d ˆ x ) , (5.8) and for all i, j = 1 , . . . , d + 1 with i ≤ d or j ≤ d we have ˆ v ij = Z R d +1 ˆ x i ˆ x j ˆ K ( d ˆ x ) . (5.9) Proof.
This is a consequence of Lemma G.8 and Fubini’s theorem for transitionkernels, where we note Lemma G.1. (cid:3)
By adjusting ˆ v d +1 ,d +1 if necessary, we even have (5.9) for all i, j = 1 , . . . , d + 1 .Note that this does not affect equation (5.4). We assume that r ∈ L (Γ) anddefine the continuous process R ∈ V as R := r · Γ . Furthermore, we assume thereexists a quasi-left-continuous local martingale Θ ∈ H with Θ = 0 and ∆Θ < such that the R d +1 -valued semimartingale ˆ X = ( X, Θ) has the modified integralcharacteristics (ˆ a, ˆ c mod , ˆ K ) with respect to Γ , where ˆ a = ( a, . Note that the lattercondition is related to the martingale problem (see [27, Sec. III.2]), which can besolved in many situations.5.5. Proposition.
Under the previous assumptions, condition (5.1) is fulfilled.Proof.
Using (5.3), for all i = 1 , . . . , d we have up to an evanescent set A i − R = ( a i − r ) · Γ = ˆ c i,d +1mod · Γ = ˆ C i,d +1mod = h M i , Θ i , showing that condition (5.1) is fulfilled. (cid:3) Summing up, Propositions 5.3 and 5.5 show that the existence of an ELMD Z ,which is a multiplicative special semimartingale, is essentially, up to a solution tothe martingale problem, equivalent to the existence of optional processes r and x satisfying (5.5).5.6. Example.
Assume that c mod = (cid:18) (cid:19) Γ -a.e. P -a.e.Then equation (5.5) has a solution if and only if a ∈ lin { R } Γ -a.e. P -a.e.where lin { R } denotes the linear spaces generated by the vector R = (1 , . Jump-diffusion models
In this section we study the existence of ELMDs for jump-diffusion models. Let λ be the Lebesgue measure on ( R + , B ( R + )) , and let W be an R m -valued standardWiener process for some m ∈ N . Furthermore, let p be a homogeneous Poissonrandom measure on some mark space ( E, E ) , which we assume to be a Blackwellspace. Then its compensator is of the form q = λ ⊗ F with some σ -finite measure F on the mark space ( E, E ) . Let L ( λ ) be the space of all optional processes a : Ω × R + → R such that | a | · λ ∈ V + , let L ( W ) be the space of all optionalprocesses σ : Ω × R + → R m such that k σ k R m · λ ∈ V + , and let L ( p ) be the spaceof all optional processes γ : Ω × R + × E → R such that | γ | ∗ q ∈ V + .6.1. Remark.
In view of our upcoming results such as condition (6.2) below, weemphasize that we may assume that the processes from L ( W ) and L ( p ) areoptional. For example, for each σ ∈ L ( W ) we have σ = p σ λ -a.e. P -a.e. XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 21 where p σ denotes the predictable projection of σ . Indeed, by Fubini’s theorem wehave E (cid:20) Z t σ s ds (cid:21) = Z t E [ σ s ] ds = Z t E [ E [ σ s | F s − ]] ds = Z t E [( p σ ) s ] ds = E (cid:20) Z t ( p σ ) s ds (cid:21) for each t ∈ R + . As in the previous sections, we suppose that the market is given by S = { S i : i ∈ I } , where for each i ∈ I the asset S i is given by a stochastic exponential (2.1). Herewe assume that for each i ∈ I the semimartingale X i in (2.1) is given by X i = a i · λ + σ i · W + γ i ∗ ( p − q ) with a i ∈ L ( λ ) , σ i ∈ L ( W ) and γ i ∈ L ( p ) such that γ i > − . Then for each i ∈ I the semimartingale X i is locally square-integrable and quasi-left-continuous,and hence we are in the framework considered at the end of Section 4. In orderto look for ELMDs which are multiplicative special semimartingales, we consider amultiplicative special semimartingale Z = DB − , where D = E (cid:0) − θ · W − ψ ∗ ( p − q ) (cid:1) and B = E ( r · λ ) = exp( r · λ ) (6.1)with θ ∈ L ( W ) and ψ ∈ L ( p ) such that ψ < , and an optional process r ∈ L ( λ ) . Note that for each i ∈ I the process γ i can be considered as an L ( F ) -valued process, and analogously ψ can be considered as an L ( F ) -valued process.6.2. Theorem.
The following statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } . (iii) For each i ∈ I we have h σ i , θ i R m + h γ i , ψ i L ( F ) = a i − r λ -a.e. P -a.e. (6.2) If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NUPBR, NAA and NA .Proof. This is an immediate consequence of Theorem 4.14. (cid:3)
In the upcoming result we consider the situation where the deflator admits ameasure change.6.3.
Proposition.
Suppose that D ∈ M with P ( D ∞ > , and let Q ≈ P bethe probability measure on (Ω , F ∞− ) with density process D relative to P . Thenthe process W ′ := W + θ · λ is an R m -valued Q -standard Wiener process, therandom measure p is a Q -integer valued random measure with compensator givenby q ′ = (1 − ψ ) · q , and the following statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } . (iii) Q is an ELMM for S B − . (iv) For each i ∈ I we have up to an evanescent set X i = r · λ + σ i · W ′ + γ i ∗ ( p − q ′ ) , Proof.
The statements about W ′ and p follow from Proposition E.4 combined withLévy’s theorem (see [27, Thm. II.4.4]).(i) ⇔ (ii) ⇔ (iii): This is a consequence of Proposition 4.10. (iii) ⇔ (iv): For each i ∈ I we have r · λ + σ i · W ′ + γ i ∗ ( p − q ′ )= r · λ + σ i · W + σ i · ( θ · λ ) + γ i ∗ ( p − q ) + γ i ∗ ( q − q ′ )= (cid:0) r + h σ i , θ i R m + h γ i , ψ i L ( F ) (cid:1) · λ + σ i · W + γ i ∗ ( p − q ) , and hence, this equivalence follows from Theorem 6.2. (cid:3) In the next result we investigate when the corresponding ELMN is locally square-integrable, and derive its dynamics in this case.6.4.
Proposition.
Suppose that the equivalent conditions from Theorem 6.2 arefulfilled, and define the ELMN ¯ Z := Z − . Then the following statements are equiv-alent: (i) ¯ Z is a locally square-integrable semimartingale. (ii) We have ψ √ − ψ , ψ − ψ ∈ L ( p ) . (iii) We have ψ √ − ψ , ψ − ψ ∈ L ( p ) . If the previous conditions are fulfilled, then the process ¯ Z admits the representation ¯ Z = E (cid:18)(cid:18) r + k θ k R m + (cid:28) ψ, ψ − ψ (cid:29) L ( F ) (cid:19) · λ + θ · W + ψ − ψ ∗ ( p − q ) (cid:19) . (6.3) Proof.
This is an immediate consequence of Proposition 4.15. (cid:3)
Remark.
The representation (6.3) has been derived in earlier works as thestructure of a growth optimal portfolio; see, for example, the articles [9] , [7] and [36] . Now, we consider the situation with finitely many assets; that is I = { , . . . , d } for some d ∈ N . We define the R d -valued semimartingale X := ( X , . . . , X d ) andthe R d -valued process a := ( a , . . . , a d ) . Furthermore, we define the optional S d × d + -valued process as c ij = h σ i , σ j i R m for all i, j = 1 , . . . , d .We define the transition kernel K from (Ω × R + , O ) into ( R d , B ( R d )) as the imagemeasure K := F ◦ γ , and we define the S d × d + -valued process v as v ij := h γ i , γ j i L ( F ) for all i, j = 1 , . . . , d .Furthermore, we define the S d × d + -valued process c mod := c + v. The following obvious auxiliary result shows that we are in the framework of Sec-tion 5.6.6.
Lemma.
The following statements are true: (1)
The triplet ( a, c, K ) consists of integral characteristics of X with respect to λ . (2) The triplet ( a, c mod , K ) consists of modified integral characteristics of X with respect to λ . (3) The process v is a purely discontinuous second integral characteristic of X with respect to λ . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 23
By identification, we may regard σ as the L ( R m , R d ) -valued process σ = (cid:0) h σ i , ·i R m (cid:1) i =1 ,...,d , (6.4)where we note that L ( R m , R d ) ∼ = R d × m .Similarly, we may regard γ as the L ( L ( F ) , R d ) -valued process γ = (cid:0) h γ i , ·i L ( F ) (cid:1) i =1 ,...,d . (6.5)6.7. Proposition.
The following statements are equivalent: (i)
There exist an optional R -valued process r , an optional R m -valued processes θ and an optional L ( F ) -valued process ψ such that for each i = 1 , . . . , d we have (6.2). (ii) There exist an optional R -valued process r , an optional R m -valued process θ and an optional L ( F ) -valued process ψ such that σθ + γψ = a − r R d λ -a.e. P -a.e. (6.6)(iii) There exist an optional R -valued process r and optional R d -valued processes x and y such that cx + vy = a − r R d λ -a.e. P -a.e. (6.7)(iv) There exist an optional R -valued process r and a optional R d -valued process x such that c mod x = a − r R d λ -a.e. P -a.e. (6.8) Proof. (i) ⇔ (ii): Using the identifications (6.4) and (6.5), this equivalence is obvi-ous.(ii) ⇔ (iii): Let T ∈ L ( R m ⊕ L ( F ) , R d ) be the continuous linear operator givenby T ( θ, ψ ) := σθ + γψ, θ ∈ R m and ψ ∈ L ( F ) . Then the linear equation (6.6) can equivalently be written as T ( θ, ψ ) = a − r R d λ -a.e. P -a.e.and hence this equivalence follows from Lemmas G.9–G.11.(iii) ⇔ (iv): This equivalence follows from Proposition 5.1. (cid:3) Note that Theorem 6.2 and Proposition 6.7 have the following consequences. Ifan ELMD Z = DB − of the form (6.1) exists, then the linear equation (6.8) has asolution ( r, x ) . Conversely, if the linear equation (6.8) has a solution, then – subjectto the conditions r ∈ L ( λ ) , θ ∈ L ( W ) and ψ ∈ L ( p ) with ψ < – an ELMD Z = DB − of the form (6.1) exists as well. This is in accordance with the findingsof Section 5, but here we do not have to deal with the martingale problem.7. Conclusion
In this paper we have provided a systematic investigation on the existence ofELMDs, which are multiplicative special semimartingales, for a given market S .There are connected questions which give rise to future research projects. One issueis the tradeability of the deflator Z = DB − ; that is, whether the correspondingELMN ¯ Z = Z − can be realized as a self-financing portfolio constructed in theextended market S ∪ { B } . Even if it cannot be replicated, it arises the question howa central bank can approximate the ELMN ¯ Z , which gives rise to diversification. Appendix A. Semimartingales
In this appendix we provide the required results about semimartingales.A.1.
Definition.
An adapted càdlàg process X is called quasi-left-continuous if ∆ X T = 0 almost surely on { T < ∞} for every predictable time T . A.2.
Lemma.
Let X and Y be two adapted càdlàg processes such that X is pre-dictable and Y is quasi-left-continuous. Then we have { ∆ X = 0 } ∩ { ∆ Y = 0 } = ∅ up to an evanescent set.Proof. Since X is predictable, by [27, Prop. I.2.24] there exists an exhausting se-quence ( S n ) n ∈ N of predictable times such that { ∆ X = 0 } = [ n ∈ N [[ S n ]] . Since Y is quasi-left-continuous, by [27, Prop. I.2.26] there exists an exhaustingsequence ( T m ) m ∈ N of totally inaccessible stopping times such that { ∆ Y = 0 } = [ m ∈ N [[ T m ]] . Therefore, we obtain up to an evanescent set { ∆ X = 0 } ∩ { ∆ Y = 0 } = (cid:18) [ n ∈ N [[ S n ]] (cid:19) ∩ (cid:18) [ m ∈ N [[ T m ]] (cid:19) = [ n,m ∈ N (cid:0) [[ S n ]] ∩ [[ T m ]] (cid:1) = ∅ , completing the proof. (cid:3) A.3.
Definition.
A semimartingale X is called a special semimartingale if thereexists a semimartingale decomposition X = X + M + A such that A is predictable. Let X be a special semimartingale. Then the decomposition X = X + M + A with a predictable process A ∈ V is unique up to an evanescent set (see [27, I.3.16])and we call X = X + M + A the canonical decomposition of X .A.4. Definition.
A semimartingale X is called locally square-integrable if it is aspecial semimartingale with canonical decomposition X = X + M + A satisfying M ∈ H . A.5.
Lemma.
For a special semimartingale X = X + M + A the following state-ments are equivalent: (i) X is quasi-left-continuous. (ii) M is quasi-left-continuous and A is continuous.Proof. (i) ⇒ (ii): By [27, Cor. I.2.31] we have p (∆ X ) = ∆ A . Since X is quasi-left-continuous, we have ∆ A T = E [∆ X T | F T − ] = 0 almost surely on { T < ∞} for every predictable time T . Therefore, A is quasi-left-continuous, and hence M is quasi-left-continuous as well. Since A is also predictable,by [27, Prop. I.2.18.b] we deduce that A is continuous.(ii) ⇒ (i): This implication is obvious. (cid:3) XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 25
Appendix B. The quadratic variation
In this appendix we provide the required results about the quadratic variationof semimartingales.B.1.
Lemma.
Let M ∈ M loc be a local martingale, and let A ∈ V be a predictableprocess. Then the following statements are true: (1) We have [ M, A ] ∈ M loc . (2) We have [ M, A ] ∈ A loc and [ M, A ] p = 0 Proof.
The first statement follows from [27, Prop. I.4.49.c], and the second state-ment is a consequence of [27, Lemma I.3.11 and I.3.22]. (cid:3)
B.2.
Lemma.
Let
M, N ∈ M loc be local martingales, and let A ∈ V be a predictableprocess. Then the following statements are true: (1) We have [ M, [ N, A ]] ∈ M loc . (2) We have [ M, [ N, A ]] ∈ A loc and [ M, [ N, A ]] p = 0 .Proof. By [27, Prop. 4.49.c] we have [ N, A ] ∈ M loc . Therefore, by [27, Prop. 4.49.a]we obtain [ M, [ N, A ]] = ∆ M · [ N, A ] ∈ M loc . The second statement is a consequence of [27, Lemma I.3.11 and I.3.22]. (cid:3)
B.3.
Lemma.
Let X and Y be two special semimartingales with canonical decom-positions X = M + A and Y = N + B . Then the following statements are equivalent: (i) We have [ X, Y ] ∈ A loc . (ii) We have [ M, N ] ∈ A loc . (iii) We have [ M, e N ] ∈ A loc , where e N := N − [ N, B ] .In either case, we have [ X, Y ] p = [ A, B ] + [
M, N ] p = [ A, B ] + [ M, e N ] p , (B.1) and the quadratic variation [ A, B ] is given by [ A, B ] = X s ≤• ∆ A s ∆ B s . (B.2) Proof.
Note the decomposition [ X, Y ] = [
M, N ] + [
M, A ] + [
N, B ] + [
A, B ] . Furthermore, by [27, Thm. I.4.52] we have (B.2). Therefore, the quadratic variation [ A, B ] is predictable, and hence, by [27, Lemma I.3.10] we have [ A, B ] ∈ A loc with [ A, B ] p = [ A, B ] . Consequently, the equivalences (i) ⇔ (ii) ⇔ (iii) and the formula(B.1) follow from Lemmas B.1 and B.2. (cid:3) B.4.
Lemma.
Let X and Y be two locally square-integrable, quasi-left-continuoussemimartingales with canonical decompositions X = M + A and Y = N + B . Thenwe have [ X, Y ] ∈ A loc and [ X, Y ] p = h M, N i = h M c , N c i + (cid:20) X s ≤• ∆ M s ∆ N s (cid:21) p . Proof.
By Lemma A.5 the processes A and B are continuous. Therefore, the state-ment is an immediate consequence of Lemma B.3 and [27, Prop. I.4.50.b and Thm.I.4.52]. (cid:3) Appendix C. The stochastic exponential
In this appendix we provide the required results about the stochastic exponentialof a semimartingale.C.1.
Lemma.
Let X be a semimartingale with X = 0 and ∆ X > − , and set Z := E ( X ) . Then the following statements are true: (1) X is a special semimartingale if and only if Z is a special semimartingale. (2) X is a locally square-integrable semimartingale if and only if Z is a locallysquare-integrable semimartingale. (3) X is quasi-left-continuous if and only if Z is quasi-left-continuous.Proof. Noting that Z = 1 + Z − · X and X = ( Z − ) − · Z , the proof is immediate. (cid:3) C.2.
Definition.
Let S be a semimartingale with S, S − > . (a) S is called inversely special if S − is a special semimartingale. (b) S is called inversely locally square-integrable if S − is a locally square-integrable semimartingale. In the definition (C.1) below we follow the convention from [27, II.1.5] to putthe integral equal to + ∞ if it diverges.C.3. Lemma.
Let D ⊂ R d be a subset containing zero, and let X be a D -valuedcàdlàg, adapted process. Furthermore, let ϕ : D → R be a measurable mapping with ϕ (0) = 0 , and set A := ϕ ( x ) ∗ µ X = X s ≤• ϕ (∆ X s ) . (C.1) Let ν be the predictable compensator of the random measure µ X . Then the followingstatements are equivalent: (i) We have A ∈ A loc . (ii) We have ϕ ( x ) ∗ ν ∈ A loc .In either case, the following statements are true: (1) We have A p = ϕ ( x ) ∗ ν . (2) We have A − A p = ϕ ( x ) ∗ ( µ X − ν ) . (3) We have ∆( A p ) = p [ ϕ (∆ X )] . (4) If X is quasi-left-continuous, then we have ∆( A p ) = 0 .Proof. We have A ∈ A loc if and only if | ϕ ( x ) | ∗ µ X ∈ A +loc , and we have ϕ ( x ) ∗ ν ∈ A loc if and only if | ϕ ( x ) | ∗ ν ∈ A +loc . Hence, the equivalence (i) ⇔ (ii) follows from[27, Thm. II.1.8.i]. Now assume that A ∈ A loc . By [27, Thm. II.1.8.ii] we have A p = ϕ ( x ) ∗ ν , and by [27, Prop. II.1.28] we have A − A p = ϕ ( x ) ∗ ( µ X − ν ) .Furthermore, by [27, I.3.21] we have ∆( A p ) = p (∆ A ) . Since ϕ (0) = 0 , we have ∆ A = ϕ (∆ X ) , and hence we obtain ∆( A p ) = p [ ϕ (∆ X )] . Now assume that X isquasi-left-continuous. Then we have ∆ A T = ϕ (∆ X T ) = 0 almost surely on the set { T < ∞} for every predictable time T . Hence, by thedefinition of the predictable projection (see [27, Thm. I.2.28.a]) and [27, Prop.I.2.18.b] we deduce that ∆( A p ) = p (∆ A ) = 0 . (cid:3) C.4.
Proposition.
Let X = M + A be a special semimartingale with ∆ X > − ,denote by ν the predictable compensator of µ X , and set S := E ( X ) . Then we have S − = E (cid:18) − X + h X c , X c i + x x ∗ µ X (cid:19) , (C.2) and the following statements are equivalent: XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 27 (i) S is inversely special. (ii) We have x x ∗ ν ∈ A +loc . (C.3) If the previous conditions are fulfilled, then we have S − = E ( N + B ) , (C.4) where the local martingale N ∈ M loc and the predictable process B ∈ V are givenby N = − M + x x ∗ ( µ X − ν ) ,B = − A + h M c , M c i + x x ∗ ν. Proof.
The identity (C.2) follows from [30, Lemma 3.4]. Noting (C.2), by [27, Prop.I.4.23] and Lemma C.1 the semimartingale S − is a special semimartingale if andonly we have x x ∗ µ X ∈ A +loc , which is equivalent to (C.3) according to Lemma C.3. Now, assume that (C.3) isfulfilled. Using Lemma C.3, we arrive at the representation (C.4). (cid:3) C.5.
Proposition.
Let X = M + A be a locally square-integrable and quasi-left-continuous semimartingale with ∆ M > − (or equivalently ∆ X > − ), denoteby ν the predictable compensator of µ M , and set S := E ( X ) . Then the followingstatements are equivalent: (i) S is inversely locally square-integrable. (ii) We have x x ∗ ν, (cid:18) x x (cid:19) ∗ ν ∈ A +loc . (C.5)(iii) We have x x ∗ ν, (cid:18) x x (cid:19) ∗ ν ∈ A +loc . (C.6)(iv) There exists a quasi-left-continuous local martingale N ∈ H with N = 0 such that M + N ∈ V , N c = − M c and ∆ N = − ∆ M M . (C.7)
If the previous conditions are fulfilled, then we have the representation S − = E (cid:0) − A − h M, N i + N (cid:1) , (C.8) and the local martingale N ∈ M loc is given by N = − M + x x ∗ ( µ M − ν ) . (C.9) Proof. (i) ⇔ (ii): By Proposition C.4 the process S is inversely special if and onlyif we have (C.3), and in this case we have the representation S − = E ( N + B ) , (C.10) where the local martingale N ∈ M loc and the predictable process B ∈ V are givenby N = − M + x x ∗ ( µ M − ν ) , (C.11) B = − A + h M c , M c i + x x ∗ ν. (C.12)Using Lemma C.1, the process S − is locally square-integrable if and only if N ∈ H . Since M ∈ H , this is the case if and only if x x ∗ ( µ M − ν ) ∈ H . (C.13)Since M is quasi-left-continuous, by [27, Cor. II.1.19] there exists a version of ν that satisfies ν ( ω ; { t } × R ) = 0 for all ( ω, t ) ∈ Ω × R + , and hence, by [27, Thm.II.1.33.a] we have (C.13) if and only if (cid:18) x x (cid:19) ∗ ν ∈ A +loc . (ii) ⇔ (iii): By [27, Prop. II.2.29.b] we have x ∗ ν ∈ A +loc . Therefore, the statedequivalence follows from the identity x x = x − x x for all x ∈ ( − , ∞ ) .(ii) ⇒ (iv): By virtue of [27, Prop. II.1.28] we can define N ∈ M loc as (C.9). Using[27, Thm. II.1.33.a] we have N ∈ H , and noting the identity − x + x x = − x x for all x ∈ ( − , ∞ ) ,we immediately see that all conditions in (C.7) are fulfilled.(iv) ⇒ (ii): Noting (C.7), we have ∆ M + ∆ N = ∆ M − ∆ M M = (∆ M ) M .
Since M ∈ H , we also have M + N ∈ H ∩ V , and hence, by [27, Thm. I.4.56.aand b] and Lemma C.3 we deduce (C.5).It remains to prove the representation (C.8). Indeed, taking into account (C.10),(C.12) and (C.7), by Lemmas C.3 and B.4 we obtain S − = E ( N + B )= E (cid:18) − A + h M c , M c i + x x ∗ ν + N (cid:19) = E (cid:18) − A + h M c , M c i + (cid:20) X s ≤• (∆ M s ) M s (cid:21) p + N (cid:19) = E (cid:18) − A − h M c , N c i − (cid:20) X s ≤• ∆ M s ∆ N s (cid:21) p + N (cid:19) = E (cid:0) − A − h M, N i + N (cid:1) , completing the proof. (cid:3) XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 29
Appendix D. Multiplication of stochastic exponentials
In this appendix we provide the required results about the multiplication ofstochastic exponentials. These results are required for the analysis of the structureof an ELMD in Section 4. In particular, we introduce the transformations R e R and Θ e Θ . The following auxiliary result is elementary.D.1. Lemma.
The mapping ϕ : ( − , ∞ ) → ( −∞ , given by ϕ ( x ) = x x , x ∈ ( − , ∞ ) is bijective with inverse ϕ − : ( −∞ , → ( − , ∞ ) given by ϕ − ( x ) = x − x , x ∈ ( −∞ , . Now, we introduce the transformation R e R .D.2. Proposition.
There is a bijection between the set of all predictable processes R ∈ V with ∆ R > − and the set of all predictable processes e R ∈ V with ∆ e R < ,which is given as follows: (i) For each predictable processes R ∈ V with ∆ R > − we assign R e R := R − X s ≤• (∆ R s ) R s . (D.1)(ii) For each predictable processes e R ∈ V with ∆ e R < we assign e R R := e R + X s ≤• (∆ e R s ) − ∆ e R s . (D.2) Furthermore, for every predictable processes R ∈ V with ∆ R > − and the corre-sponding predictable processes e R ∈ V with ∆ e R < we have E ( − e R ) = E ( R ) − , (D.3) e R c = R c and ∆ e R = ∆ R R , (D.4) R c = e R c and ∆ R = ∆ e R − ∆ e R , (D.5) [ R, e R ] = X s ≤• ∆ R s ∆ e R s , (D.6) R = e R + [ R, e R ] , (D.7) and R is continuous if and only if e R is continuous, and in this case we have R = e R and E ( R ) = exp( R ) . (D.8) Proof.
Let R ∈ V be a predictable processes with ∆ R > − , and let the predictableprocess e R ∈ V be given by (D.1). Noting the equation x − x x = x x for all x ∈ ( − , ∞ ) ,we arrive at (D.4). Now, let e R ∈ V be a predictable processes with ∆ e R < , andlet the predictable process R ∈ V be given by (D.2). Noting the equation x + x − x = x − x for all x ∈ ( −∞ , , we arrive at (D.5). Therefore, by Lemma D.1 the mapping induced by (D.1) and(D.2) is a bijection. Furthermore, by [27, Thm. I.4.52] we have (D.6), and hence,by (D.2) and (D.5) we obtain R = e R + X s ≤• (∆ e R s ) − ∆ e R s = e R + X s ≤• ∆ R s ∆ e R s = e R + [ R, e R ] , showing (D.7). Therefore, by Yor’s formula (see [27, II.8.19]) we obtain E ( R ) E ( − e R ) = E ( R − e R − [ R, e R ]) = 1 , proving (D.3). Finally, from (D.4) and (D.5) we immediately see that ∆ R = 0 ifand only if ∆ e R = 0 , and that in this case we have (D.8). (cid:3) Next, we introduce the transformation Θ e Θ . For this purpose, we fix a pre-dictable process R ∈ V with ∆ R > − , and denote by e R ∈ V the correspondingpredictable process with ∆ e R < from Proposition D.2.D.3. Proposition.
There is a bijection between the set of all local martingales Θ ∈ M loc with Θ = 0 and ∆Θ < and the set of all local martingales e Θ ∈ M loc with e Θ = 0 and ∆ e Θ + ∆ e R < , which is given as follows: (i) For each local martingale Θ ∈ M loc with Θ = 0 and ∆Θ < we assign Θ e Θ := Θ − [Θ , e R ] . (D.9)(ii) For each local martingale e Θ ∈ M loc with e Θ = 0 and ∆ e Θ + ∆ e R < weassign e Θ Θ := e Θ c + 11 − ∆ e R · e Θ d . (D.10) Furthermore, for every local martingale Θ ∈ M loc with Θ = 0 and ∆Θ < andthe corresponding local martingales e Θ ∈ M loc with e Θ = 0 and ∆ e Θ + ∆ e R < wehave E ( − Θ) E ( R ) − = E ( − e Θ − e R ) , (D.11) e Θ c = Θ c and ∆ e Θ = (1 − ∆ e R )∆Θ , (D.12) Θ c = e Θ c and ∆Θ = ∆ e Θ1 − ∆ e R . (D.13)
Furthermore, we have
Θ = e Θ if and only if up to an evanescent set { ∆ e R = 0 } ∩ { ∆Θ = 0 } = ∅ , (D.14) or equivalently, up to an evanescent set { ∆ R = 0 } ∩ { ∆Θ = 0 } = ∅ , (D.15) and Θ is quasi-left-continuous if and only if e Θ is quasi-left-continuous, and in thiscase we have Θ = e Θ .Proof. Let Θ ∈ M loc be a local martingale with Θ = 0 and ∆Θ < , and let e Θ be the process given by (D.9). By Lemma B.1 we have e Θ ∈ M loc . Furthermore, wehave e Θ = 0 and the jumps are given by ∆ e Θ = ∆Θ − ∆[Θ , e R ] = ∆Θ − ∆Θ∆ e R = (1 − ∆ e R )∆Θ , showing (D.12). Furthermore, since ∆ e R < and − ∆Θ > , we have ∆ e Θ + ∆ e R = (1 − ∆ e R )∆Θ + ∆ e R = ∆ e R (1 − ∆Θ) + ∆Θ < (1 − ∆Θ) + ∆Θ = 1 . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 31
Now, let e Θ ∈ M loc be a local martingale with e Θ = 0 and ∆ e Θ + ∆ e R < , and let Θ be the process given by (D.10). Then we have Θ ∈ M loc with Θ = 0 , and (D.13)is satisfied. Since ∆ e Θ < − ∆ e R and − ∆ e R > , we obtain ∆Θ = ∆ e Θ1 − ∆ e R < . Moreover, by (D.12) and (D.13) the mapping induced by (D.9) and (D.10) is abijection. Using Proposition D.2, Yor’s formula (see [27, II.8.19]) and (D.9) weobtain E ( − Θ) E ( R ) − = E ( − Θ) E ( − e R ) = E ( − Θ − e R + [Θ , e R ]) = E ( − e Θ − e R ) , showing (D.11). By (D.12) we see that Θ = e Θ if and only if we have (D.14) upto an evanescent set, and by Proposition D.2 this is equivalent to (D.15) up to anevanescent set. Since − ∆ e R > , by (D.12) and (D.13) we see that Θ is quasi-left-continuous if and only if e Θ is quasi-left-continuous, and in this case, by Lemma A.2we have Θ = e Θ . (cid:3) D.4.
Remark.
The formula (D.11) from Proposition D.3 can also be obtained byusing the multiplicative decomposition theorem. Indeed, the process X = E ( − e R − e Θ) has the canonical decomposition X = 1 + M + A with M = − X − · e Θ and A = − X − · e R. Therefore, by [27, Thm. II.8.21] we have the multiplicative decomposition X = LD ,where the local martingale L is given by L = E (cid:18) X − + ∆ A · M (cid:19) = E (cid:18) − X − − X − ∆ e R · ( X − · e Θ) (cid:19) = E (cid:18) − − ∆ e R · e Θ (cid:19) = E (cid:18) − e Θ c − − ∆ e R · e Θ d (cid:19) = E ( − Θ) , and where the process D with locally finite variation is given by D = E (cid:18) − X − + ∆ A · A (cid:19) − = E (cid:18) X − − X − ∆ e R · ( X − · e R ) (cid:19) − = E (cid:18) − ∆ e R · e R (cid:19) − = E (cid:18) e R c + X s ≤• ∆ e R s − ∆ e R s (cid:19) − = E ( R ) − . For the last step, we note (D.2) and the equation x − x = x − x − x for all x ∈ ( −∞ , . Appendix E. A version of Girsanov’s theorem
In this section we establish a version of Girsanov’s theorem for the particularsituation with a special semimartingale and an equivalent measure change. Let Θ ∈ M loc be a local martingale such that Θ = 0 and ∆Θ < . We define the localmartingale D ∈ M loc as the stochastic exponential D := E ( − Θ) . We assume that D ∈ M with P ( D ∞ >
0) = 1 . Let Q ≈ P be the probability measure on (Ω , F ∞− ) with density process D relative to P .E.1. Proposition.
Let M ∈ M loc with M = 0 be such that [ M, Θ] ∈ A loc . Thenthe process M ′ := M + [ M, Θ] p is a Q -local martingale, where the predictable compensator [ M, Θ] p is computedunder P .Proof. Using [27, Thm. I.3.18] we have D − · [ M, D ] p = (cid:18) D − · [ M, D ] (cid:19) p = (cid:20) M, D − · D (cid:21) p = [ M, L ( D )] p = − [ M, Θ] p . Hence, the assertion follows from [27, Thm. III.3.11]. (cid:3)
E.2.
Corollary.
Let X be a special semimartingale with canonical decomposition X = X + M + A such that [ M, Θ] ∈ A loc . Then X is also a special semimartingaleunder Q , and its canonical decomposition X = X + M ′ + A ′ is given by M ′ = M + [ M, Θ] p and A ′ = A − [ M, Θ] p . Proof.
This is an immediate consequence of Proposition E.1. (cid:3)
E.3.
Lemma.
Let X be a special semimartingale with canonical decomposition X = X + M + A , and let N ∈ M loc be a local martingale. Then the following statementsare equivalent: (i) We have [ X, N ] ∈ A loc . (ii) We have [ M, N ] ∈ A loc .In either case, we have [ X, N ] p = [ M, N ] p .Proof. By [27, Prop. I.4.49.c] we have [ A, N ] ∈ M loc . Hence, by [27, Lemma I.3.11]we have [ A, N ] ∈ A loc , which proves the equivalence (i) ⇔ (ii). Furthermore, by[27, I.3.22] we have [ A, N ] p = 0 , which concludes the proof. (cid:3) E.4.
Proposition.
Let X be an R d -valued special semimartingale with canonicaldecomposition X = X + M + A and characteristics ( A, C, ν ) . Then the followingstatements are equivalent: (i) We have [ X i , Θ] ∈ A loc for all i = 1 , . . . , d . (ii) We have [ M i , Θ] ∈ A loc for all i = 1 , . . . , d .In either case, we have [ X i , Θ] p = [ M i , Θ] p for all i = 1 , . . . , d , and the process X isa special semimartingale under Q with canonical decomposition X = X + M ′ + A ′ given by ( M ′ ) i = M i + [ M i , Θ] p , i = 1 , . . . , d, (E.1) ( A ′ ) i = A i − [ M i , Θ] p , i = 1 , . . . , d, (E.2) and characteristics ( A ′ , C ′ , ν ′ ) given by ( A ′ ) i = A i − [ X i , Θ] p , i = 1 , . . . , d, (E.3) C ′ = C, (E.4) ν ′ = (cid:0) − M P µ X (∆Θ | f P ) (cid:1) · ν. (E.5) Proof.
The stated equivalence (i) ⇔ (ii) follows from Lemma E.3. Furthermore, byCorollary E.2 the process X is a special semimartingale under Q with canonicaldecomposition X = X + M ′ + A ′ given by (E.1) and (E.2). Concerning the char-acteristics ( A ′ , C ′ , ν ′ ) , we immediately see that the first characteristic A ′ is givenby (E.3). According to [27, Thm. III.3.24] the second characteristic C ′ is given by(E.4), and there exists a predictable nonnegative function Y : e Ω → R + such thatthe third characteristic is given by ν ′ = Y · ν, XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 33 and the function Y satisfies Y D − = M P µ X ( D | f P ) . Noting that D = 1 − D − · Θ , we have ∆ D = − D − ∆Θ . Therefore, we obtain DD − = D − + ∆ DD − = 1 − ∆Θ , and hence Y = M P µ X (cid:18) DD − (cid:12)(cid:12)(cid:12)(cid:12) f P (cid:19) = M P µ X (1 − ∆Θ | f P ) = 1 − M P µ X (∆Θ | f P ) , showing (E.5). (cid:3) Appendix F. Integral characteristics of semimartingales
In this appendix we provide the results about integral characteristics of semi-martingales, which we require in Section 5. We start with an auxiliary result. Let S d × d + the convex cone of all symmetric, positive semidefinite d × d -matrices.F.1. Lemma.
Let A ∈ S d × d + be arbitrary. Then the following statements are true: (1) For all x, y ∈ R d we have |h Ax, y i R d | ≤ (cid:0) h Ax, x i R d + h Ay, y i R d (cid:1) . (2) In particular, for all i, j = 1 , . . . , d we have | A ij | ≤ (cid:0) A ii + A jj (cid:1) . Proof.
For all x, y ∈ R d we have by polarization |h Ax, y i R d | = 14 |h A ( x + y ) , x + y i R d − h A ( x − y ) , x − y i R d |≤ (cid:0) h A ( x + y ) , x + y i R d + h A ( x − y ) , x − y i R d (cid:1) = 12 (cid:0) h Ax, x i R d + h Ay, y i R d (cid:1) , proving the first statement. The second statement is an immediate consequence bytaking x = e i and y = e j for all i, j = 1 , . . . , d . (cid:3) Now, let X be an R d -valued locally square-integrable, quasi-left-continuous semi-martingale with canonical decomposition X = X + M + A. F.2.
Definition.
We introduce the following notions: (1)
Let C ∈ V d × d be the continuous S d × d + -valued process given by C ij = h M i,c , M j,c i , i, j = 1 , . . . , d. (2) Let C mod ∈ V d × d be the continuous S d × d + -valued process given by C ij mod = h M i , M j i , i, j = 1 , . . . , d. (3) Let V ∈ V d × d be the continuous S d × d + -valued process given by V ij = h M i,d , M j,d i , i, j = 1 , . . . , d. (4) Let ν is the predictable compensator of the random measure µ X associatedto the jumps of X . (5) We call the triplet ( A, C, ν ) the characteristics of X . (6) We call C mod the modified second characteristic of X . (7) We call the triplet ( A, C mod , ν ) the modified characteristics of X . (8) We call V the purely discontinuous second characteristic of X . F.3.
Remark.
According to [27, II.2.12] we may assume that for all ≤ s ≤ t wehave C t − C s , V t − V s , C mod ,t − C mod ,s ∈ S d × d + . F.4.
Lemma.
We have C mod = C + V .Proof. This is clear, because h M, N i = 0 for two local martingales M, N ∈ H such that M is continuous and N is purely discontinuous. (cid:3) F.5.
Lemma.
We have V ij = ( x i x j ) ∗ ν for all i, j = 1 , . . . , d .Proof. Since X is quasi-left-continuous, this is a consequence of Lemma F.4 and[27, Prop. II.2.17]. (cid:3) F.6.
Lemma.
Let C be an optional S d × d + -valued process. Furthermore, let Γ ∈ V + be such that dC ii ≪ d Γ for all i = 1 , . . . , d . Then there exists a S d × d + -valued optionalprocess c such that C = c · Γ .Proof. We have dC ij ≪ d Γ for all i, j = 1 , . . . , d . Indeed, let ≤ s ≤ t be such that Γ t − Γ s = 0 . Then we have C iit − C iis = 0 for all i = 1 , . . . , d . Since C t − C s ∈ S d × d + ,by Lemma F.1 it follows that C ijt − C ijs = 0 for all i, j = 1 , . . . , d . Therefore, by [27,Prop. I.3.13] there exists an R d × d -valued, optional process c ∈ L (Γ) such that C = c · Γ . Proceeding as in step (c) of the proof of [27, Prop. II.2.9] we obtain, afterchanging c on an evanescent set if necessary, that c is S d × d + -valued. (cid:3) F.7.
Definition.
Let Γ ∈ A +loc be a continuous process. We call a triplet ( a, c, K ) integral characteristics of X with respect to Γ if the following conditions are fulfilled: (1) a is an optional R d -valued process such that a i ∈ L (Γ) for all i = 1 , . . . , d . (2) c is an optional S d × d + -valued process such that c ii ∈ L (Γ) for all i =1 , . . . , d . (3) K is a transition kernel from (Ω × R + , O ) into ( R d , B ( R d )) such that on Ω × R + we have K ( { } ) = 0 and Z R d | x | K ( dx ) < ∞ . (4) We have B = b · Γ , C = c · Γ and ν = K ⊗ Γ . F.8.
Remark.
Since Γ is continuous, it suffices that a , c and K are optional ratherthan predictable. F.9.
Remark.
Let c be an optional S d × d + -valued process such that c ii ∈ L (Γ) for all i = 1 , . . . , d . Then, by Lemma F.1 we also have c ij ∈ L (Γ) for all i, j =1 , . . . , d . F.10.
Definition.
Let Γ ∈ A +loc be a continuous process. We call a process c mod modified second integral characteristic of X with respect to Γ if the following con-ditions are fulfilled: XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 35 (1) c mod is an optional S d × d + -valued process such that c ii mod ∈ L (Γ) for all i = 1 , . . . , d . (2) We have C mod = c mod · Γ .In this case, we call the triplet ( a, c mod , K ) modified integral characteristics of X with respect to Γ . F.11.
Definition.
Let Γ ∈ A +loc be a continuous process. We call a process v purelydiscontinuous second integral characteristic of X with respect to Γ if the followingconditions are fulfilled: (1) v is an optional S d × d + -valued process such that v ii ∈ L (Γ) for all i =1 , . . . , d . (2) We have V = v · Γ . F.12.
Proposition.
There exist a continuous process Γ ∈ A +loc and modified integralcharacteristics ( a, c mod , K ) of X with respect to Γ .Proof. The proof is analogous to that of [27, Prop. II.2.9]. (cid:3)
F.13.
Proposition.
Let Γ ∈ A +loc be a continuous process. Then the followingstatements are equivalent: (i) There exist modified integral characteristics ( a, c mod , K ) of X with respectto Γ . (ii) There exist integral characteristics ( a, c, K ) and a purely discontinuous sec-ond integral characteristic v of X with respect to Γ .If the previous conditions are fulfilled, then we have c mod = c + v Γ -a.e. P -a.e. (F.1) and for all i, j = 1 , . . . , d we have v ij = Z R d x i x j K ( dx ) Γ -a.e. P -a.e. (F.2) Proof. (i) ⇒ (ii): By [27, Prop. I.3.5] we have dC ii mod ≪ d Γ for all i = 1 , . . . , d .By Lemma F.4 we have C mod = C + V , and hence it follows that dC ii ≪ d Γ and V ii ≪ d Γ for all i = 1 , . . . , d . Hence, by Lemma F.6 there exist optional S d × d + -valuedprocesses c, v ∈ L (Γ) such that C = c · Γ and V = v · Γ .(ii) ⇒ (i): We define the optional process c mod := c + v . Then we have c mod ∈ L (Γ) , and by Lemma F.4 we obtain C mod = C + V = c · Γ + v · Γ = ( c + v ) · Γ = c mod · Γ , completing the proof of this implication. The additional statement (F.2) followsfrom Lemma F.5. (cid:3) Appendix G. Matrices and linear operators
In this appendix we provide the required results about matrices and linear op-erators. We denote by S d × d ⊂ R d × d the subspace of all symmetric, real-valuedmatrices. Furthermore, we denote by S d × d + ⊂ S d × d the convex cone of all symmet-ric, positive semidefinite matrices, and we denote by S d × d ++ ⊂ S d × d + the subset ofall symmetric, positive definite matrices. For a matrix A ∈ R d × m we denote by A † ∈ R m × d the Moore-Penrose inverse; see [4, page 649].G.1. Lemma.
The Moore-Penrose inverse R d × m → R m × d , A A † is measurable.Proof. This follows from the representation A † = lim ǫ → ( A ⊤ A + ǫ Id) − A ⊤ , see [4, page 649]. (cid:3) G.2.
Lemma.
Let ˆ A ∈ S ( d +1) × ( d +1) be a symmetric matrix of the form ˆ A = (cid:18) A bb ⊤ c (cid:19) (G.1) with a symmetric, positive semidefinite matrix A ∈ S d × d + , a vector b ∈ R d and areal number c ∈ R . Then the following statements are equivalent: (i) ˆ A is positive semidefinite; that is ˆ A ∈ S ( d +1) × ( d +1)+ . (ii) We have (Id − AA † ) b = 0 and the Schur complement satisfies c − h A † b, b i R d ≥ . Proof.
See [4, page 651]. (cid:3)
G.3.
Lemma.
For a matrix A ∈ R d × m and a vector b ∈ R d the following statementsare equivalent: (i) The system of linear equations Ax = b, x ∈ R m (G.2) has a solution. (ii) We have AA † b = b ; that is (Id − AA † ) b = 0 .In either case, a solution to (G.2) is given by x = A † b .Proof. (i) ⇒ (ii): According to [4, page 649] have AA † = Π ran( A ) , the orthogonalprojection on the range of A . Since b ∈ ran( A ) , we obtain AA † b = Π ran( A ) b = b. (ii) ⇒ (i): By hypothesis, a solution to (G.2) is given by x = A † b . (cid:3) G.4.
Proposition.
Let A ∈ S d × d + be a symmetric, positive semidefinite matrix, andlet b ∈ R d be a vector. Then the following statements are equivalent: (i) There exists a symmetric, positive semidefinite matrix ˆ A ∈ S ( d +1) × ( d +1)+ such that ˆ A ij = A ij for all i, j = 1 , . . . , d and ˆ A i,d +1 = b i for all i = 1 , . . . , d . (ii) The system of linear equations Ax = b, x ∈ R d has a solution.If the previous conditions are fulfilled, then for every c ≥ h A † b, b i R d the symmetricmatrix (G.1) is positive semidefinite.Proof. This is a consequence of Lemmas G.2 and G.3. (cid:3)
G.5.
Lemma.
Let A ∈ R d × m and B ∈ R d × n be matrices, and let b ∈ R d be avector. Then the following statements are equivalent: (i) The system of linear equations Ax + By = b, x ∈ R m and y ∈ R n (G.3) has a solution. (ii) There exists c ∈ R d such that the system of linear equations (cid:26) Ax = c, x ∈ R m By = b − c, y ∈ R n has a solution. (iii) There exists c ∈ R d such that (Id − AA † ) c = 0 and (Id − BB † )( b − c ) = 0 . XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 37
Proof. (i) ⇔ (ii): This equivalence is obvious.(ii) ⇔ (iii): This equivalence follows from Lemma G.3. (cid:3) G.6.
Proposition.
Let
A, B ∈ S d × d + be symmetric, positive semidefinite matrices,and let b ∈ R d be a vector. Then the following statements are equivalent: (i) There exist symmetric, positive semidefinite matrices ˆ A, ˆ B ∈ S ( d +1) × ( d +1)+ such that ˆ A ij = A ij and ˆ B ij = B ij for all i, j = 1 , . . . , d as well as ˆ A i,d +1 +ˆ B i,d +1 = b i for all i = 1 , . . . , d . (ii) The system of linear equations Ax + By = b, x, y ∈ R d (G.4) has a solution.Proof. The first statement is satisfied if and only if there exist c ∈ R d and sym-metric, positive semidefinite matrices ˆ A, ˆ B ∈ S ( d +1) × ( d +1)+ such that ˆ A ij = A ij and ˆ B ij = B ij for all i, j = 1 , . . . , d as well as ˆ A i,d +1 = c i and ˆ B i,d +1 = b i − c i for all i = 1 , . . . , d . By Proposition G.4 this is the case if and only if there exists c ∈ R d such that the system of linear equations (cid:26) Ax = c, x ∈ R d By = b − c, y ∈ R d has a solution. According to Lemma G.5 this is equivalent to the existence of asolution to the system of linear equations (G.4). (cid:3) G.7.
Proposition.
Let
A, B ∈ S d × d + be arbitrary, and set C := A + B ∈ S d × d + .Furthermore, let b ∈ R d be arbitrary. Then the following statements are equivalent: (i) There exists a symmetric, positive semidefinite matrix ˆ C ∈ S ( d +1) × ( d +1)+ such that ˆ C ij = C ij for all i, j = 1 , . . . , d and ˆ C i,d +1 = b i for all i =1 , . . . , d . (ii) There exist symmetric, positive semidefinite matrices ˆ A, ˆ B ∈ S ( d +1) × ( d +1)+ such that ˆ A ij = A ij and ˆ B ij = B ij for all i, j = 1 , . . . , d as well as ˆ A i,d +1 +ˆ B i,d +1 = b i for all i = 1 , . . . , d . (iii) The system of linear equations Cx = b, x ∈ R d has a solution. (iv) The system of linear equations Ax + By = b, x, y ∈ R d has a solution.Proof. (i) ⇔ (iii): This equivalence follows from Proposition G.4.(ii) ⇔ (iv): This equivalence follows from Proposition G.6.(iii) ⇒ (iv): Since C = A + B , this implication is obvious.(ii) ⇒ (i): The matrix ˆ C := ˆ A + ˆ B has the desired properties. (cid:3) G.8.
Lemma.
Let K be a measure on ( R d , B ( R d )) such that K ( { } ) = 0 and Z R d | x | K ( dx ) < ∞ . Furthermore, let ˆ A ∈ S ( d +1) × ( d +1)+ be a symmetric, positive semidefinite matrix ofthe form (G.1) such that A ij = Z R d x i x j K ( dx ) for all i, j = 1 , . . . , d . Then there exists a measure ˆ K on ( R d +1 , B ( R d +1 )) with ˆ K ( { } ) = 0 and Z R d +1 | ˆ x | ˆ K ( d ˆ x ) < ∞ (G.5) such that for every nonnegative, measurable function ˆ f : R d +1 → R + we have Z R d +1 ˆ f (ˆ x ) ˆ K ( d ˆ x ) = Z R d ˆ f ( x, h A † b, x i R d ) K ( dx ) , (G.6) for every nonnegative, measurable function f : R d → R + we have Z R d f ( x ) K ( dx ) = Z R d +1 f ( x ) ˆ K ( d ˆ x ) , (G.7) and for all i, j = 1 , . . . , d + 1 such that i ≤ d or j ≤ d we have ˆ A ij = Z R d +1 ˆ x i ˆ x j ˆ K ( d ˆ x ) . (G.8) Proof.
By Proposition G.4 there exists a solution y ∈ R d to the system of linearequations Ay = b, y ∈ R d , and according to Lemma G.3 one such solution is given by y = A † b . We define thelinear mapping ℓ : R d → R d +1 as ℓ ( x ) := ( x, h y, x i R d ) , x ∈ R d and the image measure ˆ K := K ◦ ℓ . Since ℓ is one-to-one, we obtain ˆ K ( { } ) = K ( ℓ − ( { } )) = K ( { } ) = 0 . Moreover, we have Z R d +1 | ˆ x | ˆ K ( d ˆ x ) = Z R d | ℓ ( x ) | K ( dx ) = Z R d (cid:0) | x | + |h y, x i R d | (cid:1) K ( dx ) < ∞ , showing (G.5). Furthermore, we have (G.6). Let f : R d → R + be a nonnegative,measurable function, and let ˆ f : R d +1 → R + be the extension given by ˆ f (ˆ x ) = f ( x ) for each ˆ x = ( x, y ) ∈ R d +1 .Then by (G.6) we have Z R d f ( x ) K ( dx ) = Z R d ˆ f ( x, h y, x i R d ) K ( dx ) = Z R d +1 ˆ f (ˆ x ) ˆ K ( d ˆ x ) = Z R d +1 f ( x ) ˆ K ( d ˆ x ) , showing (G.7). Furthermore, since Ay = b , for each i = 1 , . . . , d we have Z R d +1 ˆ x i ˆ x d +1 ˆ K ( d ˆ x ) = Z R d x i h y, x i R d K ( dx ) = d X j =1 y j Z R d x i x j K ( dx )= d X j =1 A ij y j = b i = ˆ A i,d +1 , proving (G.8). (cid:3) G.9.
Lemma.
Let X be a Hilbert space. Then for every continuous linear operator T ∈ L ( X, R d ) and every y ∈ R d following statements are equivalent: (i) There exists x ∈ X such that T x = y. (ii) There exists η ∈ R d such that T T ∗ η = y. XISTENCE OF EQUIVALENT LOCAL MARTINGALE DEFLATORS 39
Proof. (i) ⇒ (ii): The range ran( T ∗ ) is a finite dimensional subspace of X , andhence it is closed. Therefore, we have the direct sum decomposition X = ran( T ∗ ) ⊕ ker( T ) . Let x = x + x be the corresponding decomposition of x . Then we have T x = y .Since x ∈ ran( T ∗ ) , there exists η ∈ R d such that T ∗ η = x , and hence T T ∗ η = y .(ii) ⇒ (i): Taking x = T ∗ η we have T x = y . (cid:3) G.10.
Lemma.
Let X and Y be Hilbert spaces, and let T ∈ L ( X, R d ) and S ∈ L ( Y, R d ) be continuous linear operators. We define R ∈ L ( X ⊕ Y, R d ) as R ( x, y ) := T x + Sy, x ∈ X and y ∈ Y. Then the following statements are true: (1)
We have R ∗ = ( T ∗ , S ∗ ) . (2) We have RR ∗ = T T ∗ + SS ∗ .Proof. For all x ∈ X , y ∈ Y and z ∈ R d we have h R ( x, y ) , z i R d = h T x + Sy, z i R d = h x, T ∗ z i X + h y, S ∗ z i Y = h ( x, y ) , ( T ∗ z, S ∗ z ) i X ⊕ Y , proving the first statement. Now, the second statement is an immediate conse-quence. (cid:3) G.11.
Lemma.
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