Explicit description of all deflators for market models under random horizon with applications to NFLVR
aa r X i v : . [ q -f i n . M F ] O c t Explicit description of all deflators for markets under random horizon
Tahir Choulli and Sina Yansori
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
October 31, 2018
Abstract
This paper considers an initial market model, specified by its underlying assets S and its flowof information F , and an arbitrary random time τ which might not be an F -stopping time. In thissetting, our principal goal resides in describing as explicit as possible the set of all deflators, whichconstitutes the dual set of all “admissible” wealth processes, for the stopped model S τ . Since thedeath time and the default time (that τ might represent) can be seen when they occur only, theprogressive enlargement of F with τ sounds tailor-fit for modelling the new flow of informationthat incorporates both F and τ . Thanks to the deep results of Choulli et al. [8], on martingalesclassification and representation for progressive enlarged filtration, our aim is fully achieved for bothcases of local martingale deflators and general supermartingale delators. The results are illustratedon several particular models for ( τ, S, F ) such as the discrete-time and the jump-diffusion settingsfor ( S, F ), and the case when τ avoids F -stopping times. This paper considers an initial market model represented by the pair ( S, F ), where S represents thediscounted stock prices for d -stocks, and F is the flow of “public” information which is available toall agents. To this initial market model, we add a random time τ that might not be seen through F when it occurs. Mathematically speaking, this means that τ might not be an F -stopping time. Thus,for modelling the new flow of information, we adopt the progressive enlargement of F with τ , thatwe denote throughout the paper by G . Hence, our resulting informational market model is the pair( S τ , G ). This information modelling allows us to apply our obtained results to credit risk theory andlife insurance (mortality and/or longevity risk), where the progressive enlargement of filtration soundstailor-fit, and the initial enlargement of filtration –as in the insider trading framework– is totally in-adequate. In fact the death time of an agent can not be seen with certainty before its occurrence, andthere is no single financial literature that models the information in the default of a firm τ as fullyseen from the beginning as in the case of insider trading.For this new market model ( S τ , G ), which includes the two important settings of default and mortality,many challenging questions arise in finance (both theoretical and empirical) and mathematical finance.Most of these questions are still open problems nowadays and are essentially concerned with measuringthe impact of τ on the financial and economical concepts, theories, rules, models, methodologies, ....,etcetera. Among these we cite the (consumption-based) capital asset pricing model (s), equilibrium,arbitrage theory, market’s viability, the fundamental theorem of asset pricing, the optimal portfolios(e.g. the log-optimal portfolio, the num´eraire portfolio and other types of portfolios to cite few),1tility maximization, the various pricing rules, ..., etcetera. The first fundamental question to allthese aforementioned problems, lies in the impact of the random time on the market’s viability andthe corresponding no-arbitrage concept(s). In virtue of [11], see also [22, 23] for related discussions,the market’s viability in its various weakest form, the existence of the num´eraire portfolio, and theno-unbounded-profit-with-bounded-risk (NUPBR) concept are equivalent or intimately related. Inthis spirit, there were an upsurge interest in studying first the effect of the random time on NUPBRin a series of papers, see [1, 2, 3, 4, 9, 10, 24] for details. This very recent literature answers fully thequestion when NUPBR is altered for ( S τ , G ). Some of these papers, especially [2, 3, 4, 24], construct examples of deflators for special and very particular cases (such as when ( S, F ) is local martingaleunder the physical probability), while the following question remains open and beyond reach up tonow. How can we describe the set of all deflators for the model ( S τ , G )? (1)The importance of this set and its numerous roles in optimization problems intrinsic to financial prob-lems sound clear and without reproach. Indeed, the set of deflators represents somehow the dual set ofall “admissible” wealth processes. No matter what is the optimization criterion, any optimal portfoliocorresponds uniquely to an optimal deflator, and they are linked to each other via “ some dualityform”. Furthermore, in many (probably all) cases even when the utility is nice enough such as logutility, it is more convenient, more efficient, and more easier to solve a dual problem and describe theoptimal deflator than getting the optimal portfolio directly. For this latter fact, we refer the readerto [12], where the authors prove that dealing with the dual problem gives sharp and precise results.When considering the impact of τ on optimal portfolio, we refer the reader to [13] for direct applicationof the current paper.This paper contains four sections including the current one. Section 2 presents the mathematicalmodel and its preliminaries. Section 3 states the explicit parametrization of deflators (that are localmartingales) for ( S τ , G ) in terms of deflators of ( S, F ) and the “survival” processes associated withthe random time. Section 4 addresses the case of general supermartingale deflators, while Section 5illustrates the results on particular models for the triplet ( τ, S, F ). Among these cases, we considerthe jump-diffusion and the discrete-times settings for ( S, F ). The paper contains an appendix wheresome proofs are relegated and some useful technical (new and existing) results are detailed. This section defines the notations, the financial and the mathematical concepts that the paper ad-dresses or uses, the mathematical model that we focus on, and some useful existing results. Throughoutthe paper, we consider the complete probability space (Ω , F , P ). By H we denote an arbitrary fil-tration that satisfies the usual conditions of completeness and right continuity. For any process X ,the H -optional projection and dual H -optional projection of X , when they exist, will be denote by o, H X and X o, H respectively. Similarly, we denote by p, H X and X p, H the H -predictable projectionand dual predictable projection of X when they exist. The set M ( H , Q ) denotes the set of all H -martingales under Q , while A ( H , Q ) denotes the set of all optional processes with integrable variationunder Q . When Q = P , we simply omit the probability for the sake of simple notations. For an H -semimartingale X , by L ( X, H ) we denote the set of H -predictable processes that are X -integrablein the semimartingale sense. For ϕ ∈ L ( X, H ), the resulting integral of ϕ with respect to X is denotedby ϕ • X . For H -local martingale M , we denote by L loc ( M, H ) the set H -predictable processes ϕ thatare X -integrable and the resulting integral ϕ • M is an H -local martingale. If C ( H ) is the set of pro-cesses that are adapted to H ∈ { F , G } , then C loc ( H ) is the set of processes, X , for which there exists2 sequence of H -stopping times, ( T n ) n ≥ , that increases to infinity and X T n belongs to C ( H ), for each n ≥
1. For any H -semimartingale, L , we denote by E ( L ) the Doleans-Dade (stochastic) exponential,it is the unique solution to the stochastic differential equation dX = X − dL, X = 1 , given by E t ( L ) = exp( L t − h L c i t ) Y
Definition 2.1.
Consider the triplet ( X, H , Q ) such that H is a filtration, X is an H -semimartingale,and Q be a probability. Let Z be a process. (a) We call Z an H -local martingale deflator for X under Q (or a local martingale deflator for ( X, Q, H ) ) if Z > and there exists a real-valued and H -predictable process ϕ such that < ϕ ≤ and both processes Z and Z ( ϕ • X ) are H -local martingales under Q . Throughout the paper, the set ofall local martingale deflators for ( X, Q, H ) will be denoted by Z loc ( X, Q, H ) . (b) We call Z an H -deflator for X under Q (or a deflator for ( X, Q, H ) ) if Z > and Z E ( ϕ • X ) isan H -supermartingale under Q , for any ϕ ∈ L ( X, H ) such that ϕ ∆ X ≥ − . The set of all deflatorsfor ( X, Q, H ) will be denoted by D ( X, Q, H ) . When Q = P , for the sake of simplicity, we simple omitthe probability in notations and terminology. The rest of this section is divided into two subsections. The first subsection introduces the mathe-matical model that we are interested in studying, while the second subsection recalls an importantmartingales representation results.
Throughout the paper, we consider (Ω , F , F := ( F t ) t ≥ , P ) a filtered probability space satisfying theusual conditions of right continuity and completeness. Here F is the public flow of information. Onthis stochastic basis, we suppose given an F -semimartingale, S , that represents the discounted priceprocess of risky assets. In addition to this initial market model, we consider a random time τ , thatmight represent the death time of an agent or the default time of a firm, and hence it might not bean F -stopping time. Throughout the paper, we will be using the following associated non-decreasingprocess D and the filtration G := ( G t ) t ≥ given by D := I [[ τ, + ∞ [[ , G t := G t + where G t := F t ∨ σ ( D s , s ≤ t ) . (2)The agent who has access to F , can only get information about τ through the survival probabilities,called Az´ema supermartingales in the literature, G t := o, F ( I [[0 ,τ [[ ) t = P ( τ > t |F t ) and e G t := o, F ( I [[0 ,τ ]] ) t = P ( τ ≥ t |F t ) . (3)The process m := G + D o, F , (4)is a BMO F -martingale. Then thanks to [6, Theorem 3], the process T ( M ) := M τ − e G − I ]]0 ,τ ]] • [ M, m ] + I ]]0 ,τ ]] • (cid:16) X ∆ M I { e G =0
Theorem 2.2. ([8, Theorems 2.19, 2.22, 2.23]): Suppose
G > . Let h be an element of L ( O ( F ) , P ⊗ D ) , and M ( h ) and J be two processes given by M ( h ) := o, F (cid:16) Z ∞ h u dD o, F u (cid:17) , J := (cid:16) M ( h ) − h • D o, F (cid:17) G − . (9) Then the following assertions hold. (a)
The G -martingale H ( h ) t := o, G ( h τ ) t = E [ h τ |G t ] satisfies H ( h ) = H ( h )0 + G − − • T ( M ( h ) ) − J − G − − • T ( m ) + ( h − J ) • N G . (10)(b) For any G -martingale M G , there exists a unique triplet ( M F , ϕ ( o ) , ϕ ( pr ) ) belonging to M ,loc ( F ) ×I oloc (cid:0) N G , G (cid:1) × L loc (cid:16)e Ω , Prog( F ) , P ⊗ D (cid:17) such that E h ϕ ( pr ) τ (cid:12)(cid:12) F τ i I { τ< + ∞} = 0 , P -a.s. and (cid:16) M G (cid:17) τ = M G + G − − • T ( M F ) + ϕ ( o ) • N G + ϕ ( pr ) • D. (11)For the sake of a self-contained paper, we outline the key ideas with some details for the proof of thistheorem in Appendix B. This version, whose proof is less technical than that of [8, Theorems 2.19,2.22, 2.23] due to G >
0, will be used throughout the paper. The following extends slightly Theorem2.2-(b) to the local martingales setting.
Theorem 2.3.
Suppose that
G > . Then for any G -local martingale M G , there exists a uniquetriplet ( M F , ϕ ( o ) , ϕ ( pr ) ) satisfying the following properties: M F ∈ M ,loc ( F ) , ϕ ( o ) ∈ I oloc (cid:0) N G , G (cid:1) , ϕ ( pr ) ∈ L loc (Prog( F ) , P ⊗ D ) , E h ϕ ( pr ) τ (cid:12)(cid:12) F τ i I { τ< + ∞} = 0 , P -a.s. , (12) and (cid:16) M G (cid:17) τ = M G + G − − I ]]0 ,τ ]] • T ( M F ) + ϕ ( o ) • N G + ϕ ( pr ) • D. (13) Proof.
Let M G ∈ M ,loc ( G ), then there exists a sequence of G -stopping times that increases to infinitysuch that ( M G ) T n is a G -martingale. On the one hand, due to G > F -stopping times ( σ n ) n that increases to infinity and T n ∧ τ = σ n ∧ τ for any n ≥
1. On the other hand, by applying Theorem 2.2 to each ( M G ) T n − ( M G ) T n − , we deduce the existence4f a unique triplet ( M F ,n , ϕ ( o,n ) , ϕ ( pr,n ) ) belonging to M ,loc ( F ) × I oloc (cid:0) N G , G (cid:1) × L loc (Prog( F ) , P ⊗ D )and satisfying E h ϕ ( pr,n ) τ (cid:12)(cid:12) F τ i I { τ< + ∞} = 0 , P -a.s. , and ( M G ) τ ∧ T n − ( M G ) τ ∧ T n − = G − − I ]]0 ,τ ]] • T ( M F ,n ) + ϕ ( o,n ) • N G + ϕ ( pr,n ) • D. Then notice that ( M G ) τ − M G = P n ≥ (( M G ) τ ∧ T n − ( M G ) τ ∧ T n − ), and put σ := 0 , ϕ ( o ) := X n ≥ I ]] σ n − ,σ n ]] ϕ ( o,n ) , ϕ ( pr ) := X n ≥ I ]] σ n − ,σ n ]] ϕ ( pr,n ) , and M F := X n ≥ I ]] σ n − ,σ n ]] • M F ,n . This ends the proof of the theorem.We end this section by the following
Lemma 2.4.
Let σ be an H -stopping time. Z is a deflator for ( X σ , H ) if and only if there existsunique pair of processes ( K , K ) such that K = ( K ) σ , E ( K ) is also a deflator for ( X σ , H ) , K isany H -local supermartingale satisfying ( K ) σ ≡ , ∆ K > − , and Z = E ( K + K ) = E ( K ) E ( K ) . The proof of this lemma is straightforward and will be omitted. This lemma shows, in a way oranother, that when dealing with the stopped model ( X σ , H ), there is no loss of generality in focusingon the part up-to- σ of deflators, and assume that the deflator is flat after σ . ( S τ , G ) This subsection focuses on describing completely the set of all local martingale deflators, defined inDefinition 2.1-(a), for ( S τ , G ) in terms of those of ( S, F ). Theorem 3.1.
Suppose
G > , and let K G be a G -local martingale. Then the following assertionsare equivalent. (a) Z G := E (cid:0) K G (cid:1) is a local martingale deflator for ( S τ , G ) (i.e. Z G ∈ Z loc ( S τ , G ) ). (b) There exists (cid:0) K F , ϕ ( o ) , ϕ ( pr ) (cid:1) such that ( K F , ϕ ( o ) ) ∈ M ,loc ( F ) ×I oloc ( N G , G ) , ϕ ( pr ) ∈ L loc (Prog( F ) , P ⊗ D ) and E [ ϕ ( pr ) τ (cid:12)(cid:12) F τ ] I { τ< + ∞} = 0 P -a.s., and the following three conditions hold: (b.1) Z F := E (cid:0) K F (cid:1) is a local martingale deflator for ( S, F ) (i.e. Z F ∈ Z loc ( S, F ) ). (b.2) The following inequalities hold. ϕ ( pr ) > − (cid:2) G − (1 + ∆ K F ) + ϕ ( o ) G (cid:3) / e G, P ⊗ D -a.e , (14) − G − G (1 + ∆ K F ) < ϕ ( o ) < (1 + ∆ K F ) G − ∆ D o, F , P ⊗ D o, F -a.e . (15)(b.3) The following decomposition holds K G = T ( K F ) − G − − • T ( m ) + ϕ ( o ) • N G + ϕ ( pr ) • D. (16) Proof.
The proof will be achieved in two steps. The first step proves that E ( K G ) is a local martingalefor which there exists a G -predictable process ϕ G satisfying 0 < ϕ G ≤ E ( K G )( ϕ G • S τ ) is a G -local martingale if and only if there exist K F ∈ M ,loc ( F ) and a triplet (cid:16) ϕ ( o ) , ϕ ( pr ) , ϕ F (cid:17) ∈ I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D ) × L ( S, F )5uch that (16) holds, E [ ϕ ( pr ) τ (cid:12)(cid:12) F τ ] I { τ< + ∞} = 0 P -a.s., 0 < ϕ F ≤ E ( K F )( ϕ F • S ) is an F -localmartingale . The second step proves that E ( K G ) > K G , ϕ ( o ) , ϕ ( pr ) ), foundin the first step satisfying (16), should fulfill (14)-(15) and 1 + ∆ K F > Step 1.
Suppose that Z G is a local martingale deflator for ( S τ , G ). Then, on the one hand, thanks toTheorem 2.3, there exists ( N F , ϕ ( o ) , ϕ ( pr ) ) that belongs to M ,loc ( F ) ×I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D ), E [ ϕ ( pr ) τ (cid:12)(cid:12) F τ ] I { τ< + ∞} = 0 P -a.s. and K G = K G + G − − I ]]0 ,τ ]] • T ( N F ) + ϕ ( o ) • N G + ϕ ( pr ) • D. (17)On the other hand, thanks to a combination of Definition 2.1 and Lemma A.1-(a), we deduce theexistence of an F -predictable process ϕ such that 0 < ϕ ≤ Z G ( ϕ • S τ ) is G -local martingale orequivalently ϕ • S τ + [ ϕ • S τ , K G ] is a G -local martingale . (18)Thus, using the decomposition S = S + M + A + P ∆ SI {| ∆ S | > } , where M is an F -locally bounded localmartingale and A an F -predictable process with finite variation, (5), and the fact that the processes ϕ • [ A τ , K G ], ϕ ∆ M ϕ ( o ) • N G and ϕ ∆ M ϕ ( pr ) • D are G -local martingales, we derive ϕ • S τ + ϕ • [ K G , S τ ] = ϕ • M τ + ϕ • A τ + X ϕ ∆ S τ I {| ∆ S | > } + ϕ • [ K G , S τ ] , = G -local martingale + ϕ e G I ]]0 ,τ ]] • [ M, m ] + ϕ • A τ + ϕG − e G I ]]0 ,τ ]] • [ N F , M ]+ X ϕ ∆ SI {| ∆ S | > } " (1 + ∆ N F G − e G ) I ]]0 ,τ ]] + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D . Then ϕ • S τ + ϕ • [ K G , S τ ] is a G -local martingale if and only if W := X ϕ ∆ SI {| ∆ S | > } [(1 + ∆ N F G − e G ) I ]]0 ,τ ]] + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D ] , (19)has G -locally integrable variation (i.e. W ∈ A loc ( G )) and (due to Lemma A.2)0 ≡ ϕG − I ]]0 ,τ ]] • h M, m i F + ϕ • A τ + ϕG − I ]]0 ,τ ]] • h N F , M i F + W p, G . (20)In virtue of Lemma A.3, we conclude that W ∈ A loc ( G ) iff both processes W (1) := X ϕ ∆ SI {| ∆ S | > } [(1 + ∆ N F G − e G )] I ]]0 ,τ ]] and W (2) := P ϕ ∆ SI {| ∆ S | > } [ ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D ] belong to A loc ( G ).It is clear that W (2) belongs to A loc ( G ) if and only if it is a G -local martingale, and hence in this casewe get W p, G = ( W (1) ) p, G = G − − I ]]0 ,τ ]] • X ϕ ∆ SI {| ∆ S | > } G − e G + ∆ N F G − ! p, F . As a result, by inserting these remarks in (20), we obtain0 ≡ ϕ • h M, m i F + ϕG − • A + ϕG − • h N F , M i F ++ G − • (cid:18)X ϕ ∆ SI {| ∆ S | > } [1 + ∆ mG − + ∆ N F G − ] (cid:19) p, F ,
6r equivalently0 = ϕ • h M, G − • m + 1 G − • N F i F + ϕ • A + (cid:18)X ϕ ∆ SI {| ∆ S | > } [1 + ∆ mG − + ∆ N F G − ] (cid:19) p, F Thanks to Itˆo’s formula, this is equivalent to E ( K F )( ϕ • S ) is an F -local martingale with K F := G − − • m + G − − • N F , and the first step is completed. Step 2.
Herein, we assume that (16) holds, and prove that E ( K G ) > K F > G − e G (1 + ∆ K F ) − , and we derive ∆ K G = ∆ T ( K F ) − G − − ∆ T ( m ) + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D = (cid:20) Γ − ϕ ( o ) ∆ D o, F e G (cid:21) I ]]0 ,τ [[ + (cid:20) Γ + ϕ ( pr ) + ϕ ( o ) G e G (cid:21) I [[ τ ]] . Therefore, E ( K G ) > K G > , τ [[ ⊂ (cid:26) G − e G (1 + ∆ K F ) − ϕ ( o ) ∆ D o, F e G > (cid:27) , and (21)[[ τ ]] ⊂ (cid:26) G − e G (1 + ∆ K F ) + ϕ ( o ) G e G + ϕ ( pr ) > (cid:27) . (22)Thus, by putting Σ := n G − e G − (1 + ∆ K F ) − ϕ ( o ) e G − ∆ D o, F > o ∩ ]]0 , + ∞ [[, (21) is equivalent to I ]]0 ,τ [[ ≤ I Σ . Hence, by taking the F -optional projection on both sides of this inequality, we get0 < G ≤ I Σ on ]]0 , + ∞ [[. This proves the right inequality in (15). Notice that (22) is equivalent to G − e G − (1 + ∆ K F ) + ϕ ( o ) G e G − + ϕ ( pr ) > , P ⊗ D − a.e., and (14) is proved. Now, we focus on proving that 1 + ∆ K F > E P ⊗ D [ ϕ ( pr ) |O ( F )] = 0, P ⊗ D − a.e. , by taking conditional expectation under P ⊗ D withrespect to O ( F ) on the both sides of the above inequality, we getΣ := G − (1 + ∆ K F ) + ϕ ( o ) G > , P ⊗ D − a.e., (23)or equivalently I [[ τ ]] ≤ I { Σ > } . Remark that (23) is equivalent to the left inequality in (15) since Σis F -optional, and hence the proof of (15) is completed. By taking the F -optional projection in bothsides of I [[ τ ]] ≤ I { Σ > } , we get ∆ D o, F ≤ I { Σ > } . Therefore, we derive { ∆ D o, F > } ⊆ { G − (1 + ∆ K F ) > − ϕ ( o ) G } . (24)On the one hand, due to the right inequality in (15), we deduce that on { ∆ D o, F = 0 } , we have1 + ∆ K F >
0. On the other hand, using (24) and the right inequality in (15) afterwards again, we get { K F ≤ } ∩ { ∆ D o, F > } ⊆ { ϕ ( o ) > , K F ≤ , ∆ D o, F > } = ∅ , or equivalently { ∆ D o, F > } ⊆ { K F > } . Thus, 1+∆ K F > Theorem 3.2.
Suppose
G > . Then Z G ∈ Z loc ( S τ , G ) iff there exists unique (cid:0) Z F , ϕ ( o ) , ϕ ( pr ) (cid:1) ∈Z loc ( S, F ) × I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D ) such that ϕ ( pr ) > − , − e GG < ϕ ( o ) < e G e G − G , P ⊗ D -a.e (25) E [ ϕ ( pr ) τ (cid:12)(cid:12) F τ ] I { τ< + ∞} = 0 P -a.s. (26) and Z G = ( Z F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) . (27) Proof.
Thanks to Theorem 3.1, we conclude that Z G is a local martingale deflator for ( S τ , G ) if andonly if there exists a triplet (cid:0) Z F , ϕ ( o ) , ϕ ( pr ) (cid:1) such that Z F := E ( K F ) belongs to Z loc ( S, F ) , ϕ ( o ) belongsto I oloc ( N G , G ), ϕ ( pr ) belongs to L loc (Prog( F ) , P ⊗ D ) and E [ ϕ ( pr ) τ (cid:12)(cid:12) F τ ] I { τ< + ∞} = 0 P -a.s., and (14),(15) and (16) hold. Thus, we put Y := T ( K F ) − G − − • T ( m ) , X := Y + ϕ ( o ) • N G ϕ ( o ) := e Gϕ ( o ) G − (1 + ∆ K F ) , ϕ ( pr ) := e Gϕ ( pr ) G − (1 + ∆ K F ) + ϕ ( o ) G . (28)Since the pair ( ϕ ( o ) , ϕ ( pr ) ) satisfies (14)-(15), we conclude that the pair ( ϕ ( o ) , ϕ ( pr ) ) satisfies (25)-(26).Furthermore, put Γ := G − e G − (1 + ∆ K F ) − , e Ω := Ω × [0 , + ∞ ) , and calculate1 + ∆ X = " Γ + 1 − ∆ D o, F ϕ ( o ) e G I ]]0 ,τ [[ + (cid:20) Γ + 1 + ϕ ( o ) G e G (cid:21) I [[ τ ]] + I e Ω \ ]]0 ,τ ]] > . Y = G − e G (1 + ∆ K F ) I ]]0 ,τ ]] + I ]] −∞ , ∪ ]] τ, + ∞ [[ > . Thanks to Yor’s formula (i.e. E ( X ) E ( X ) = E ( X + X + [ X , X ])) we derive E ( X + X ) = E ( X ) E ( X −
11 + ∆ X • [ X , X ]) , for any semimartingales X , X with 1 + ∆ X >
0. By applying this formula repeatedly, and using ϕ ( o ) = ϕ ( o ) / (1 + ∆ Y ) and ϕ ( pr ) = ϕ ( pr ) / (1 + ∆ X ) P ⊗ D -a.e. which follow directly from (28), weobtain E ( K G ) = E ( X + ϕ ( pr ) • D ) = E ( X ) E ( ϕ ( pr ) X • D ) = E ( X ) E ( ϕ ( pr ) • D )= E ( Y ) E ( ϕ ( o ) Y • N G ) E ( ϕ ( pr ) • D ) = E ( Y ) E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) . Therefore, the equality (27) follows immediately from combining this equality with E ( Y ) = E ( K F ) τ / E ( G − − • m ) τ . This latter equality is a direct consequence of 1 / E ( G − − • m ) τ = E ( G − − • T ( m )) and E ( K ) τ E ( − G − − • T ( m )) = E ( T ( K ) − G − − • T ( m )) . This ends the proof of the theorem. 8s a direct consequence of Theorem 3.1 (or equivalently Theorem 3.2), we describe below a family of G -local martingales. Corollary 3.3.
For any F -local martingale (respectively an element of Z loc ( S, F ) ) Z F := E ( K F ) , theprocess Z G is given by Z G := E (cid:16) T ( K F ) − G − − • T ( m ) (cid:17) = ( Z F ) τ E ( G − − • m ) τ , (29) is a G -local martingale (respectively an element of Z loc ( S τ , G ) ). ( S τ , G ) This section focuses on explicitly parametrizing the set of all deflators for ( S τ , G ) in terms of deflatorsfor ( S, F ). Throughout the rest of the paper, processes will be compared to each other in the followingsense. Definition 4.1.
Let X and Y be two processes with X = Y . Then X (cid:23) Y if X − Y is an increasing process. We start this section by parametrizing deflators as follows.
Lemma 4.2.
Let X be an H -semimartingale, and Z be a positive H -supermartingale. Then thefollowing assertions are equivalent. (a) Z is a deflator for ( X, H ) . (b) There exists unique ( N, V ) such that N ∈ M ,loc ( H ) , V is nondecreasing and H -predictable, Z := Z E ( N ) E ( − V ) , N = V = 0 , ∆ N > − , ∆ V < − o . (32)For the sake of easy exposition, we postpone the proof of this lemma to Appendix C. In the lemma, by“abuse of notations” for the sake of simplicity, we use Y p, H to denote the predictable with finite varia-tion part in the Doob-Meyer decomposition of Y whenever this process is a special H -semimartingale. Theorem 4.3.
Suppose
G > , and let Z G be a G -semimartingale. Then the following assertions areequivalent. (a) Z G is a deflator for ( S τ , G ) (i.e. Z G ∈ D ( S τ , G ) ). (b) There exists a unique (cid:0) K F , V F , ϕ ( o ) , ϕ ( pr ) (cid:1) such that K F ∈ M ,loc ( F ) , V F is an F -predictable andnondecreasing process, ϕ ( o ) ∈ I oloc ( N G , G ) , ϕ ( pr ) belongs to L loc (Prog( F ) , P ⊗ D ) such that E ( K F ) E ( − V F ) ∈D ( S, F ) , ϕ ( pr ) > − (cid:2) G − (1 + ∆ K F ) + ϕ ( o ) G (cid:3) / e G, P ⊗ D − a.e., (33)9 G − G (1 + ∆ K F ) < ϕ ( o ) < (1 + ∆ K F ) G − ∆ D o, F , P ⊗ D o, F -a.e. . (34) Z G = E ( K G ) E ( − V F ) τ , K G = T ( K F − G − • m ) + ϕ ( o ) • N G + ϕ ( pr ) • D. (35)(c) There exists unique (cid:0) Z F , ϕ ( o ) , ϕ ( pr ) (cid:1) such that Z F ∈ D ( S, F ) , ( ϕ ( o ) , ϕ ( pr ) ) belongs to I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D ) , ϕ ( pr ) > − , P ⊗ D − a.e., − e GG < ϕ ( o ) < e G e G − G , P ⊗ D o, F -a.e., (36) and Z G = ( Z F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) . (37) Proof.
The proof will be achieved in three steps, where we prove the implications (a)= ⇒ (b), (b)= ⇒ (c),and (c)= ⇒ (a) respectively. Step 1.
Herein, we prove (a)= ⇒ (b). To this end, we suppose that Z G is a deflator for ( S τ , G ).Thus, due to Lemma 4.2, we deduce the existence of K G ∈ M ,loc ( G ) and V G a G -predictable andnondecreasing process such that Z G = Z E ( K G ) E ( − V G ) , sup − , ∆ V G < , (1 − ∆ V G ) − • V G (cid:23) ( ϕ • S τ + [ ϕ • S τ , K G ]) p, G , for any bounded ϕ ∈ L ( S τ , G ). Then a direct application of Theorem 2.3 to K G and Lemma A.1to V G , leads to the existence of the triplet ( N F , ϕ ( o ) , ϕ ( pr ) ) that belongs to M loc ( F ) × I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D ) and an F -predictable and nondecreasing process V with finite values such that K G = K G + 1 G − I ]]0 ,τ ]] • T ( N F ) + ϕ ( o ) • N G + ϕ ( pr ) • D, V G = V τ . Consider a bounded ϕ ∈ L ( S, F ), and remark that ϕ • S τ + [ ϕ • S τ , K G ] ∈ A loc ( G ) is equivalent to W := P ( ϕ ∆ S τ + ϕ ∆[ S τ , K G ]) I {| ∆ S | > } ∈ A loc ( G ). As a result, W p, G = lim n −→ + ∞ (cid:16) I { n ≥| ∆ S | > } • W (cid:17) p, G . By combining ϕ • S τ + [ ϕ • S τ , K G ] = ϕ • S τ + I ]]0 ,τ ]] G − • [ T ( N F ) , ϕ • S τ ]+ ϕ ∆ Sϕ ( o ) • N G + ϕ ∆ Sϕ ( pr ) • D, the fact that ϕ ∆ Sϕ ( o ) I {| ∆ S |≤ n } • N G and ϕ ∆ Sϕ ( pr ) I {| ∆ S |≤ n } • D are G -local martingales for any n ≥ T ( N F ) , ϕ • S τ ] = G − e G − I ]]0 ,τ ]] • [ N F , ϕ • S τ ], we deduce that (cid:0) ϕ • S τ + [ ϕ • S τ , K G ] (cid:1) p, G = (cid:0) ϕ • S τ + 1 G − I ]]0 ,τ ]] • [ T ( N F ) , ϕ • S ] (cid:1) p, G = (cid:0) ϕ • S τ + 1 G − e G I ]]0 ,τ ]] • [ N F , ϕ • S ] (cid:1) p, G .
10y inserting in this equation the following decomposition of S , S = S + M + A + X ∆ SI {| ∆ S | > } , where M is a locally bounded F -local martingale and A is an F -predictable process with finite variation,we obtain (cid:0) ϕ • S τ + [ ϕ • S τ , K G ] (cid:1) p, G = ϕ • A τ + I ]]0 ,τ ]] G − • h m, ϕ • M i F + I ]]0 ,τ ]] G − • h N F , ϕ • M i F ++ X ϕ ∆ S (1 + ∆ N F G − e G ) I ]]0 ,τ ]] I {| ∆ S | > } ! p, G = ϕ • A τ + I ]]0 ,τ ]] G − • h m, ϕ • M i F + I ]]0 ,τ ]] G − • h N F , ϕ • M i F ++ G − − I ]]0 ,τ ]] • (cid:18)X ϕ ∆ S ( e G + ∆ N F G − ) I {| ∆ S | > } (cid:19) p, F . As a result, for any bounded ϕ ∈ L ( S, F ), V ( ϕ ) := ϕ • S + [ ϕ • S, G − • m + G − • N F ] has an F -compensator,and 11 − ∆ V F • V F (cid:23) ϕ • A + h G − • m + 1 G − • N F , ϕ • M i F + (cid:16)X ∆ V ( ϕ ) I {| ∆ S | > } (cid:17) p, F = (cid:18) ϕ • S + [ ϕ • S, G − • m + 1 G − • N F ] (cid:19) p, F . On the one hand, this is equivalent to E ( K F ) E ( − V ) E ( ϕ • S ) is an F -supermartingale for any bounded ϕ ∈ L ( S, F ), where K F := G − − • m + G − − • N F . On the other hand, as in the proof of Theorem 3.1 (seestep 2 of that proof), it is clear that E ( K G ) > K G > K F , ϕ ( o ) , ϕ ( pr ) ) satisfies 1 + ∆ K F > Step 2.
This step proves (b)= ⇒ (c). Hence, we suppose that assertion (b) holds. Then there exists aunique (cid:0) K F , V F , ϕ ( o ) , ϕ ( pr ) (cid:1) such that Z G = E (cid:16) T ( K F ) − G − − • T ( m ) + ϕ ( o ) • N G + ϕ ( pr ) • D (cid:17) E ( − V F ) τ . Then by mimicking the analysis (calculations) that starts from (28), we derive Z G = E (cid:0) K F (cid:1) τ E ( − V F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D )= ( Z F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) . Here ϕ ( o ) := e Gϕ ( o ) /G − (1 + ∆ K F ) and ϕ ( pr ) := e Gϕ ( pr ) [ G − (1 + ∆ K F ) + Gϕ ( o ) ] − are the F -optionaland F -progressive processes respectively that satisfy (36) as a direct consequence of the conditions(33)-(34) fulfilled by the pair ( ϕ ( o ) , ϕ ( pr ) ). This ends the proof of (b)= ⇒ (c). Step 3.
Herein, we deal with (c)= ⇒ (a). Thus, we suppose that assertion (c) holds, and deduce theexistence of a triplet (cid:0) Z F , ϕ ( o ) , ϕ ( pr ) (cid:1) that belongs to D ( S, F ) × I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D )satisfying (36) and Z G = ( Z F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) . (38)11hen for any bounded ϕ ∈ L ( S, F ), Z F E ( ϕ • S ) is an F -supermartingale, and hence there exist N ∈M ,loc ( F ) and V is an F -predictable and non decreasing process such that Z F E ( ϕ • S ) = E ( N ) E ( − V ) . Therefore, by combining this with (38), we deduce that, for any bounded ϕ ∈ L ( S, F ), Z G E ( ϕ • S ) τ = E ( N ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) E ( − V ) τ . Thus, thanks to Corollary 3.3 which allows us to conclude that the process ( E ( N ) τ / E ( G − − • m ) τ ) E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) is in fact a G -local martingale, we deduce that Z G E ( ϕ • S ) τ is a G -supermartingale,for any bounded ϕ ∈ L ( S, F ). Then assertion (a) follows immediately from combining this with thefact that, for any bounded ϕ G that belongs to L ( S τ , G ), there exists a bounded ϕ F ∈ L ( S, F ) suchthat ϕ G = ϕ F on ]]0 , τ ]] (see Lemma A.1-(a)). This ends the proof of the theorem. In this section, we illustrate the obtained results on particular cases and/or examples. Precisely,in three subsections, we discuss the case when τ avoids F -stopping times or all F -martingales arecontinuous, the case of jump-diffusion for ( S, F ), and the case of discrete time model for ( S, F ). τ avoids F -stopping times or all F -martingales are continuous As explained in [8], when either τ avoids all F -stopping times or all F -martingales are continuous, the G -local martingales T ( M ) and N G –given by (5) and (6)– coincide with M and N G respectively where M := M τ − G − − I ]]0 ,τ ]] • h M, m i F , N G := D − G − − I ]]0 ,τ ]] • D p, F . (39)It is clear that both M and N G are the G -local martingale parts in the Doob-Meyer decomposition,under G , of M τ and D respectively. Theorem 5.1.
Suppose that
G > , and either τ avoids F -stopping times or all F -martingales arecontinuous. Let K G be a G -local martingale and V G be a nondecreasing and G -predictable. Then thefollowing assertions are equivalent. (a) Z G := E (cid:0) K G (cid:1) E ( − V G ) is a deflator for ( S τ , G ) if and only if there exists a unique (cid:0) K F , V F , ϕ ( p ) (cid:1) such that ( K F , ϕ ( p ) ) ∈ M loc ( F ) × L loc ( N G , G ) , V F is nondecreasing with finite values and F -predictable, E ( K F ) E ( − V F ) ∈ D ( S, F ) , − G − p, F G < ϕ ( p ) < G − G − − p, F G , P ⊗ D p, F -a.e. , (40) V G = ( V F ) τ , and K G = K F − G − − • m + ϕ ( p ) • N G . (41)(b) Z G := E (cid:0) K G (cid:1) E ( − V G ) is a deflator for ( S τ , G ) if and only if there exists a unique pair (cid:0) Z F , ϕ ( p ) (cid:1) ∈D ( S, F ) × L loc ( N G , G ) such that P ⊗ D p, F -a.e, − G − p, F G < ϕ ( p ) < G − G − − p, F G , and Z G = ( Z F ) τ E ( ϕ ( p ) • N G ) E ( G − − • m ) τ . (42)12 roof. (a) The proof of assertion (a) follows from combining Theorem 4.3-(a) with the following threefacts. Under the assumption that either τ avoids F -stopping times or all F -martingales are continuous,the following facts hold:1) For ϕ ( pr ) ∈ L loc (Prog( F ) , P ⊗ D ), there exists ϕ ∈ L loc ( O ( F ) , P ⊗ D ) such that ϕ ( pr ) τ = ϕ τ P -a.son { τ < + ∞} . Therefore, we deduce that P -a.s. on { τ < + ∞} we have ϕ ( pr ) τ = E ( ϕ ( pr ) τ |F τ ) = 0, orequivalently ϕ ( pr ) • D ≡ N G = N , and T ( M ) = M for any F -local martingale M .3) For any ϕ ( o ) ∈ I oloc ( N G , G ), there exists an F -predictable process ϕ that is N G -integrable such that( ϕ ( o ) − ϕ ) • N G ≡
0. This proves assertion (a).(b) The proof of assertion (b) follows immediately from combining assertion (a) and the fact that,under the assumption that either τ avoids F -stopping times or all F -martingales are continuous, wehave [ N G , M ] = [ N G , M ] ≡ F -local martingale M . This fact, indeed, implies that E ( M + ϕ ( p ) • N G ) = E ( M ) E ( ϕ ( p ) • N G ) , for M ∈ M ,loc ( F ) with 1 + ∆ M >
0. This ends the proof of the theorem.Theorem 5.1 states universal results that work for both cases of τ avoids F -stopping times and whenall F -martingales are continuous. The main difference between the two cases lies in the conditionon the parameter ϕ ( p ) . Indeed, for the case when τ avoids F -stopping times, the condition (40) (orequivalently the inequalities in (42)) becomes − < ϕ ( p ) P ⊗ D p, F -a.e. instead. This is due to e G = G which follows from the avoidance property of τ . However, when all F -martingales are continuous, (40)takes the form of − e G/G < ϕ ( p ) < e G/ ( e G − G ) P ⊗ D p, F -a.e., since in this case e G = G − and p, F ( G ) = G . ( S, F ) This subsection focuses on the important case of a jump-diffusion framework for the market model( S, F ). For this model, we consider two situations depending whether τ is left to be an arbitraryrandom time or a particular example. Herein, we suppose that a standard Brownian motion W anda Poisson process N with intensity λ > , F , P ), and thefiltration F is the completed and right continuous filtration generated by W and N . The stock’s priceprocess is supposed to have the following dynamics S t = S E ( X ) t , X t = Z t σ s dW s + Z t ζ s dN F s + Z t µ s ds, N t F := N t − λt, (43)where the processes µ , ζ, and σ are bounded, F -adapted and there exists a constant δ ∈ (0 , + ∞ ) suchthat σ > , ζ > − , σ + | ζ | ≥ δ, P ⊗ dt -a.e. . (44)Since m is an F -martingale, then there exists two F -predictable processes ϕ ( m ) and ψ ( m ) such that R t (( ϕ ( m ) s ) + | ψ ( m ) s | ) ds < + ∞ P -a.s. for any t ≥ G − • m = ϕ ( m ) • W + ( ψ ( m ) − • N F . (45) Theorem 5.2.
Suppose S given by (43), G > , and let Z G := E (cid:0) K G (cid:1) be a positive G -local martingale.Then the following assertions are equivalent. (a) Z G is a local martingale deflator for ( S τ , G ) , There exist ( ψ , ψ ) ∈ L loc ( W, F ) × L loc ( N F , F ) , ϕ ( o ) ∈ I oloc ( N G , G ) and ϕ ( pr ) ) ∈ L loc (Prog( F ) , P ⊗ D) satisfying the following K G = ψ • T ( W ) + ( ψ − • T ( N F ) − G − − • T ( m ) + ϕ ( o ) • N G + ϕ ( pr ) • D.ϕ ( pr ) > − (cid:2) G − ψ + ϕ ( o ) G (cid:3) / ˜ G, and − ψ G − G < ϕ ( o ) < ψ G − ∆ D o, F P ⊗ D -a.e. , and µ + ψ σ + ( ψ − ζλ ≡ , ψ > P ⊗ dt − a.e. (c) There exists unique quadruplet ( ψ , ψ , ϕ ( o ) , ϕ ( pr ) ) that belongs to the set L loc ( W, F ) × L loc ( N F , F ) ×I oloc ( N G , G ) × L loc (Prog( F ) , P ⊗ D) , and satisfies Z G = E ( ψ • W + ( ψ − • N F ) τ E ( G − − • m ) τ E ( ϕ ( o ) • N G ) E ( ϕ ( pr ) • D ) ,ϕ ( pr ) > − , P ⊗ D -a.e. , − G − G < ϕ ( o ) < G − ∆ D o, F P ⊗ D o, F -a.e. , and µ + ψ σ + ( ψ − ζλ ≡ , ψ > P ⊗ dt − a.e. Proof.
The proof follows immediately from Theorems 3.1-3.2 and the fact that for any M ∈ M loc ( F ),there exists a unique ( ψ , ψ ) ∈ L loc ( W, F ) × L loc ( N F , F ) such that M = M + ψ • W + ( ψ − • N F . In the following, we discuss a particular model for τ that was considered in [2, Example 2.12] and [5,Subsection 5.2.2, page 108]. Example 5.3.
Consider the same model for ( S, F ) as in Theorem 5.2, and let τ := ( aT ) ∧ T , where a ∈ (0 , and T and T are the first and the second jump times of the Poisson process N (i.e. N := P + ∞ n =1 I [[ T n , + ∞ [[ ). Since F is generated by ( W, N ) and W is independent of τ , the samecalculations for the three processes ( G, G − , e G ) as in [2, 5] remain valid. Thus, we get e G t = e − βt ( βt + 1) I [[0 ,T [[ ( t ) + e − βt I [[ T ]] ( t ) , G t = e − βt ( βt + 1) I [[0 ,T [[ ( t ) G t − = e G t − = e − βt ( βt + 1) I [[0 ,T ]] ( t ) . However the arguments for the calculations of m and D o, F differ slightly from that of [2, 5]. Let m c be the continuous local martingale part of m , and hence m − m c is a pure jump local martingale withjumps equal to ∆ m = e G − G − = φ ∆ N F where φ t := − βte − βt , β := λ ( a − − . Hence m = m c + φ • N F on the one hand. On the other hand, by writing G t = e − βt ( βt + 1)(1 − H (1) t ) , H (1) := I [[ T , + ∞ [[ , M (1) := H (1) − λ ( t ∧ T ) = ( N F ) T , and by applying Itˆo’s formula to the process G and using G = m + D o, F (see (4)), we deduce that m c ≡ , m = m + φ • N F and D o , F t = Z t0 e − β s dH (1)s + ( β + λ ) β Z t ∧ T se − β s ds . Since in the current case we have { e G = 0 < G − } = ∅ , we derive T ( W ) = W τ , T ( N F ) = ( N F ) τ + βt βt • ( N ) τ , G − • T ( m ) = − βt βt • T ( N F ) ,N G = I { aT
11 + βt ) • T ( N F ) + ϕ ( o ) • N G + ϕ ( pr ) • D,µ + ψ σ + ( ψ − ζλ ≡ , ψ > , − ψ < ϕ ( o ) I ]]0 ,T [[ , P ⊗ dt − a.e. and the following inequalities hold P -a.s. ϕ ( o ) ( T ) < ψ ( T )(1 + βT ) ,ϕ ( pr ) ( aT ∧ T ) > − [ ψ ( aT ) + ϕ ( o ) ( aT )] I { aT
Let M G be a G -martingale. Then there exists a unique pair ( M F , ϕ ) of F -adaptedprocesses such that M F is an F -martingale and M G n ∧ τ = M G + n X k =1 ∆ T k ( M F ) P ( τ ≥ k |F k − ) + n X k =1 ϕ k ∆ N G k . (50) Proof.
The proof follows from combining Theorem 2.3 and Lemma 5.5.Below, we state our main result in this subsection.
Theorem 5.7.
Let Z G be a G -adapted process and e Q be a probability given by e Q := Z T · P and Z n := n Y k =1 e G k G k − I { G k − > } + I { G k − =0 } ! . (51) Then the following assertions are equivalent. (a) Z G is a deflator for ( S τ , G ) (i.e. Z G ∈ D ( S τ , G ) ). (b) There exists a unique pair (cid:16) Z ( e Q, F ) , ϕ (cid:17) such that Z ( e Q, F ) ∈ D ( S, e Q, F ) , ϕ is an F -adapted processsatisfying for all n = 0 , ..., T P -a.s. − P ( τ ≥ n |F n ) P ( τ > n |F n ) < ϕ n < P ( τ ≥ n |F n ) P ( τ = n |F n ) , and Z G = ( Z ( e Q, F ) ) τ Z ( ϕ ) . (52) Here Z ( ϕ ) is given by Z ( ϕ ) t := t Y n =1 (cid:18) ϕ n P ( τ > n |F n ) P ( τ ≥ n |F n ) I { τ = n } − ϕ n P ( τ = n |F n ) P ( τ ≥ n |F n ) I { τ>n } (cid:19) . (53) Proof.
We start this proof by making the following three remarks:1) It is easy to check that (see also [9] for details and related results) the process Z is a martingaleand hence e Q is a well defined probability. Furthermore, the process Z is the discrete-time version of16he process E ( G − − I { G − > } • m ) (which is well defined even in the case where G might vanish) see [21,Subsection 2.3].2) It is clear that X is a supermartingale under e Q if and only if Y := ZX is a supermartingale .3) Thanks to (48), the discrete-time version of E ( ϕ • N G ) coincides with Z ( ϕ ) given in (53), for any F -optional process ϕ .Then by combining these remarks and Theorem 4.3, the proof of the theorem follows immediately. APPENDIX
A Some G -properties versus those in F Lemma A.1.
The following assertions hold. (a)
For any G -predictable process ϕ G , there exists an F -predictable process ϕ F such that ϕ G = ϕ F I ]]0 ,τ ]] .Furthermore, if ϕ G is bounded, then ϕ F is bounded with the same constants. (b) Suppose that
G > . Then for any bounded θ ∈ L ( S τ , G ) , then there exists a bounded ϕ ∈ L ( S, F ) that coincides with θ on [[0 , τ ]] . (c) Suppose
G > , and let V G be a G -predictable and nondecreasing process with finite values and ( V G ) τ = V G . Then there exists a unique nondecreasing with finite values and F -predictable process, V , such that V G = V τ .If furthermore ∆ V G < , then ∆ V < holds also.Proof. Remark that the boundedness condition for ϕ G can be reduced to the condition 0 ≤ ϕ G ≤ Part 1.
Here we prove assertion (b). Consider a bounded θ ∈ L ( S τ , G ). Then θ is a bounded and G -predictable process satisfying θ tr ∆ S τ > −
1. Thus, in virtue of assertion (a), there exists a boundedand F -predictable process ϕ such that θI [[0 ,τ ]] = ϕI [[0 ,τ ]] . Then by inserting this equality in θ tr ∆ S τ > −
1, we deduce that ϕ tr ∆ SI ]]0 ,τ ]] > − , which is equivalent to I ]]0 ,τ ]] ≤ I { ϕ tr ∆ S> − } . By taking the F -optional projection on both sides of thisinequality, we get 0 < G ≤ I { ϕ tr ∆ S> − } on ]]0 , + ∞ [[, or equivalently ϕ tr ∆ S > −
1. Hence ϕ belongs to L ( S, F ), and the proof of assertion (b) is complete. Part 2.
This part proves assertion (c). Consider a G -predictable and nondecreasing process withfinite values V G such that ( V G ) τ = V G . It is clear that there is no loss of generality in assumingthat V G is bounded. Then due to [20, Lemma 4.4 (b)] (see also [2, Lemma B.1]), there exists an F -predictable process V such that V G I [[0 ,τ ]] = V I [[0 ,τ ]] . (54)By writing V G I [[0 ,τ [[ = V G − V G τ I [[ τ, + ∞ [[ –which is obviously a RCLL bounded G -semimartingale– andby taking the F -optional projection on both sides of (54), we get V = o, F ( V G I [[0 ,τ [[ ) /G . Hence V is17 RCLL F -semimartingale that is predictable. As a result, there exists a continuous F -martingale L with L = 0 and an F -predictable process with finite variation B such that V = L + B . Since V G ispredictable with finite variation and V G = V τ = L τ + B τ = (cid:16) L τ − G − − I ]]0 ,τ ]] • h L, m i F (cid:17) + G − − I ]]0 ,τ ]] • h L, m i F + B τ , then we conclude that the G -local martingale L τ − G − − I ]]0 ,τ ]] • h L, m i F is null. This implies that [ L, L ] τ is also a null process since L is continuous, or equivalently L ≡ G >
0. Thisproves that V = B has a finite variation. To prove that V is nondecreasing it is enough to remarkthat ( V G ) p, F is nondecreasing and V = G − − • ( V G ) p, F . This proves the first statement of assertion (c),while the proof of the last statement of assertion (c) follows the same foot steps of part 1). Indeed∆ V G = ∆ V I ]]0 ,τ ]] < I ]]0 ,τ ]] ≤ I { ∆ V < } holds, and this implies that –after takingthe F -predictable projection on both sides of this inequality– 0 < G − ≤ I { ∆ V < } on ]]0 , + ∞ [[. This isequivalent in fact to ∆ V <
1. The fact G − > G > { G − > } and { G > } have the same d´ebut (see [20, Lemme (4.3)]). This ends the proofof the lemma.The following lemma recalls the G -compensator of any F -optional process stopped at τ . Lemma A.2.
Let V ∈ A loc ( F ) , then we have ( V τ ) p, G = I ]]0 ,τ ]] G − − (cid:5) ( e G (cid:5) V ) p, F . For the proof of this lemma and other related results, we refer to [2, 3, 4].
Lemma A.3.
Let ϕ is a real-valued and F -predictable process, N F ∈ M ,loc ( F ) , ϕ ( o ) ∈ I oloc ( N G , G ) ,and ϕ ( pr ) ∈ L loc (Prog( F ) , P ⊗ D ) such that N F G − e G ! I ]]0 ,τ ]] + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D > , < ϕ ≤ . (55) Then the process W := X ϕ ∆ SI {| ∆ S | > } " N F G − e G ! I ]]0 ,τ ]] + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D has a G -locally integrable variation if and only if both processes W (1) := X ϕ ∆ SI {| ∆ S | > } (1 + ∆ N F G − e G ) I ]]0 ,τ ]] and W (2) := X ϕ ∆ SI {| ∆ S | > } [ ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D ] belong to A loc ( G ) .Proof. Due to the first condition in (55), it is clear that W ∈ A loc ( G ) iff W + := X | ϕ ∆ S | I {| ∆ S | > } [(1 + ∆ N F G − e G ) I ]]0 ,τ ]] + ϕ ( o ) ∆ N G + ϕ ( pr ) ∆ D ] ∈ A + loc ( G ) .
18y stopping, there is no loss of generality to assume that E [ W + ∞ ] < + ∞ . Thus, since both P | ϕ ∆ S | I { k ≥| ∆ S | > } ϕ ( o ) ∆ N G = | ϕ ∆ S | I { k ≥| ∆ S | > } ϕ ( o ) • N G and P | ϕ ∆ S | I { k ≥| ∆ S | > } ϕ ( pr ) ∆ D = | ϕ ∆ S | I { k ≥| ∆ S | > } ϕ ( pr ) • D are G -localmartingale, we derive E "X | ϕ ∆ S | I {| ∆ S | > } [(1 + ∆ N F G − e G ) I ]]0 ,τ ]] = lim k −→ + ∞ E "X | ϕ ∆ S | I { < | ∆ S |≤ k } (1 + ∆ N F G − e G ) I ]]0 ,τ ]] = lim k −→ + ∞ E [( I {| ∆ S |≤ k } • W + ) ∞ ] ≤ E [ W + ∞ ] < + ∞ . This proves that W (1) ∈ A + loc ( G ), and hence W (2) = W − W (1) ∈ A + loc ( G ). Thus, the proof of thelemma is complete. B Proof of Theorem 2.2
Proof.
We start this proof by highlighting some useful implications of the assumption
G >
0. Indeed,thanks to [20, Lemme (4.3)], which states that the three sets { G = 0 } , { G − = 0 } and { e G = 0 } havethe same d´ebut, we deduce that the assumption G > G , G − and e G , are positive (strictly) processes,2) G − − is locally bounded, and G − is a well defined RCLL semimartingale.3) The operator T defined in (5) takes the following form T ( M ) = M τ − e G − I ]]0 ,τ ]] • [ M, m ] , for all M ∈ M ,loc ( F ) . (56)The rest of the proof is divided into three steps. The first step discusses some integrability propertiesuseful for both assertions (a) and (b). The second step proves assertion (a) of the theorem, while thethird step proves assertion (b). Step 1.
Consider k ∈ L loc (Prog( F ) , P ⊗ D ) and let h be an F -optional process such that k τ = h τ P -a.s. on ( τ < + ∞ ). In this step, we prove that ( ϕ (0) , ϕ ( pr ) ) belongs to I oloc (cid:0) N G , G (cid:1) × L loc (Prog( F ) , P ⊗ D ),where ϕ ( pr ) := k − h, ϕ ( o ) := h − J ( h ) . and J := Y ( h ) G , Y ( h ) := o, F ( h τ I [[0 ,τ [[ ) (57)Since k ∈ L (Prog( F ) , P ⊗ D ), then h ∈ L ( O ( F ) , P ⊗ D ) and hence ϕ ( pr ) belongs to L (Prog( F ) , P ⊗ D )on the one hand. On the other hand, we derive E h ( | h | G e G − • D ) ∞ i ≤ E (cid:2) | h τ | I { τ< + ∞} (cid:3) < + ∞ , which is equivalent to h ∈ I o (cid:0) N G , G (cid:1) ⊂ I oloc (cid:0) N G , G (cid:1) . Now consider the sequence of F -stopping times T n := inf { t ≥ (cid:12)(cid:12) | J ( h ) t | > n } with the convention inf( ∅ ) = + ∞ . Then ( T n ) increases to infinity and19 J ( h ) | I [[0 ,T n [[ ≤ n , and hence E h ( | J ( h ) | G e G − • D ) T n i ≤ n + E h | Y ( h ) T n | e G − T n I { τ = T n < + ∞} i = n + E h | Y ( h ) T n | e G − T n ∆ D o, F T n I { T n < + ∞} i ≤ n + E h | h τ | I { T n <τ } e G − T n ∆ D o, F T n i ≤ n + E (cid:2) | h τ | I { T n <τ } (cid:3) < + ∞ . This proves that J ( h ) belongs to I oloc (cid:0) N G , G (cid:1) and hence ϕ ( o ) does also. Step 2.
Here, we prove assertion (a). To this end, we remark that the process H := o, F ( h τ ) can bedecomposed as follows H = h • D + J ( h ) I [[0 ,τ [[ = ( h − J ( h ) ) • D + ( J ( h ) ) τ , (58) Y ( h ) := M ( h ) − h • D o, F , M ( h ) := o, F ( Z + ∞ h s dD s ) (59)For full details about these facts, we refer the reader to [8]. Thus, thanks to Itˆo (applied to 1 /G ) and∆ G = ∆ m − ∆ D o, F , we get d (cid:18) G (cid:19) = − G τ − ) dm + 1 G ( G τ − ) d [ m, m ] + G − − ∆ mG ( G − ) dD o, F . By combining this equation, (56) and e G ( G − − ∆ m ) + (∆ m ) = G − , we obtain d (cid:18) G τ (cid:19) = − G − ) d T ( m ) + 1 e GG I ]]0 ,τ ]] dD o, F . (60)Then again Itˆo and (56) combined with (59) and Y = GJ , we derive (for full details about the followingequalities we refer the reader to [8]) d ( J ( h ) ) τ = d (cid:18) Y τ G τ (cid:19) = 1 G τ − dY τ + Y τ − d (cid:18) G τ (cid:19) + d (cid:20) G τ , Y τ (cid:21) = − J ( h ) − G − d T ( m ) + 1 G − d T ( M ( h ) ) + G − J ( h ) − + ∆ M ( h ) − h e G e GG I ]]0 ,τ ]] dD o, F = − J ( h ) − G − d T ( m ) + 1 G − d T ( M ( h ) ) + J ( h ) − h e G I ]]0 ,τ ]] dD o, F . Hence, by inserting this latter equality in (58) and using (6), the representation (10) follows immedi-ately, and the proof of assertion (a) is complete.
Step 3.
This step proves assertion (b). Consider a G -martingale M G . Therefore, see [8] for fulldetails, there exist a unique k ∈ L (Prog( F ) , P ⊗ D) such that M G τ = k τ . Hence, to this process k , wededuce the existence of a unique h ∈ L ( O ( F ) , P ⊗ D ) satisfying E ( k τ (cid:12)(cid:12) F τ ) = h τ in fact we have h := E µ ( k (cid:12)(cid:12) O ( F )) , µ := P ⊗ D. Then we combine these remarks (see full details in the proof of [8, Theorem 2.22]) and write( M G ) τ = o, G ( M G τ ) = o, G ( k τ ) = ( k − h ) • D + o, G ( h τ ) .
20y applying assertion (a) to the G -martingale o, G ( h τ ) and putting ϕ ( pr ) := k − h, ϕ ( o ) := h − J ( h ) := h − M ( h ) − h • D o, F G ,M F := G − • M ( h ) − G − J ( h ) − • m, where M ( h ) is given by (59), (13) follows immediately, and hence assertion (b) is proved. This endsthe proof of the theorem. C Proof of the lemma 4.2
Proof.
The proof of this lemma will be achieved in two steps. The first step proves that there existsa unique pair (
N, V ) satisfying (30) as soon as Z is a deflator for ( X, H ). The second step showsthat for a process Z , for which there exists a pair ( N, V ) satisfying (30), there is equivalence between Z E ( ϕ • X ) is supermartingale and (31), for any bounded ϕ ∈ L ( X, H ). Step 1.
Suppose that Z is a deflator. This implies that Z is a positive supermartingale (since ϕ = 0 ∈ L ( X, H )), and hence X := Z − − • Z is a local supermartingale having the unique Doob-Meyerdecomposition X := K − V , where K ∈ M ,loc ( H ) and V is nondecreasing and predictable with∆ V <
Z > − ∆ V ) − is well defined and is locallybounded. Hence N := 11 − ∆ V • K ∈ M ,loc ( H ) , ∆ N > − E (N) E ( − V) . This ends the first step.
Step 2.
Suppose that there exists a pair (
N, V ) such that Z = Z E ( N ) E ( − V ) and (30) holds. Let ϕ be a bounded element of L ( X, H ). Then by applying Itˆo to Z E ( ϕ • X ) = Z E ( N ) E ( − V ) E ( ϕ • X ), oneget Z E ( ϕ • X ) = Z E ( N ) E ( ϕ • X ) E ( − V ) = Z E (cid:16) N + ϕ • X + ϕ • [ X, N ] (cid:17) E ( − V )= Z E ( Y ( ϕ ) ) E ( − V ) = Z E (cid:16) Y ( ϕ ) − V − [ Y ( ϕ ) , V ] (cid:17) = Z E (cid:16) (1 − ∆ V ) • Y ( ϕ ) − V (cid:17) , where Y ( ϕ ) := N + ϕ • X + ϕ • [ X, N ] . Since Z is positive and ϕ ∈ L ( X, H ), then the process Z E ( ϕ • X )is an H -supermartingale if and only if (1 − ∆ V ) • Y ( ϕ ) − V is a local H -supermartingale, or equivalently Y ( ϕ ) is a special semimartingale (which is equivalent to the first condition of (31)) and its predictablewith finite variation part, ( Y ( ϕ ) ) p, H , satisfies ( Y ( ϕ ) ) p, H (cid:22) (1 − ∆ V ) − • V . This finishes the secondstep, and the proof of the lemma as well. acknowledgements: This research is fully supported financially by the Natural Sciences and Engi-neering Research Council of Canada, through Grant G121210818.The authors would like to thank Ferdoos Alharbi, Safa Alsheyab, Jun Deng, Monique Jeanblanc, YouriKabanov, Martin Schweizer and Michele Vanmalele for several comments and/or suggestions, fruitfuldiscussions on the topic, and/or for providing important and useful related references.
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