ModernStochastics:Theory andApplications0(0000) 0–0 DOI: 6 1 0 2 n a J Generalization of Doob decomposition Theorem. 4 1 ] R Nicholas Gonchara,∗ P aBogolyubov Institute for Theoretical Phisics of NAS, Kyiv, Ukraine . h t mhonchar@i.ua(N.Gonchar) a m [ 1 Abstract Inthepaper,weintroducethenotionofalocalregularsupermartingale v relative to a convex set of equivalent measures and prove for it an optional Doob 4 decomposition in the discrete case. This Theorem is a generalization of the famous 7 Doob decomposition onto the case of supermartingales relative to a convex set of 5 equivalent measures. 3 0 Keywords random process, convex set of equivalent measures, optional Doob . decomposition, regular supermartingale, martingale. 1 0 2010 MSC 60G07, 60G42 6 1 : v 1 Introduction. i X In the paper, we generalize Doob decomposition for supermartingales relative r a to one measure onto the case of supermartingales relative to a convex set of equivalent measures. For supermartingales relative to one measure for contin- uous time Doob’s result was generalized in papers [12, 13]. Atthebeginning,weprovetheauxiliarystatementsgivingsufficientcondi- tions ofthe existence ofmaximalelementin amaximalchain,ofthe existence ofnonzeronon-decreasingprocesssuchthatthe sumofasupermartingaleand this process is again a supermartingale relative to a convex set of equivalent measuresneeded for the main Theorems.In Theorem 2 we give sufficient con- ditionsoftheexistenceoftheoptionalDoobdecompositionforthespecialcase ∗Corresponding author. www.i-journals.org/vmsta Preprint submitted to VTeX / Modern Stochastics: Theory and Applications 2 N. Gonchar as the set of measures is generated by finite set of equivalent measures with bounded as below and above the Radon - Nicodym derivatives. After that, we introduce the notion of a regular supermartingale. Theorem 3 describes regular supermartingales. In Theorem 4 we give the necessary and sufficient conditionsofregularityofsupermartingales.Theorem5describesthestructure ofnon-decreasingprocessforaregularsupermartingale.Thenweintroducethe notionofa localregularsupermartingalerelativeto a convexsetofequivalent measures. At last, we prove Theorem 6 asserting that if the optional decom- positionfor asupermartingaleisvalid,thenit islocalregularone.Essentially, Theorem6and7givethenecessaryandsufficientconditionsoflocalregularity of supermartingale. After that, we prove auxiliary statements nedeed for the description of local regular supermartingales. Theorem 8 gives the necessary and sufficient conditions for a special class of nonnegative supermartingales to be local regular ones. In Theorems 9 and 10 we describe a wide class of local regular supermartingales.Onthe basisoftheseTheoremsweintroduceacertainclass of local regular supermartingales and prove Theorem 11 giving the necessary andsufficientconditionsfornonnegativeuniformlyintegrablesupermartingale to belong to this class. Using the results obtained we give examples of con- structionoflocalregularsupermartingales.At last,we provealsoTheorem12 giving possibility to construct local regular supermartingales. The optional decomposition for supermartingales plays fundamental role for risk assessment on incomplete markets [6], [7], [8], [9]. Considered in the paperproblemisgeneralizationofcorrespondingonethatappearedinmathe- maticalfinanceaboutoptionaldecompositionforsupermartingaleandwhichis relatedwith constructionof superhedge strategy onincomplete financialmar- kets.First,the optionaldecompositionforsupermartingaleswasopenedbyEl Karoui N. and Quenez M. C. [2] for diffusion processes. After that, Kramkov D. O. [11], [5] proved the optional decomposition for nonnegative bounded supermartingales.FolmerH.andKabanovYu.M.[3],[4]provedanalogousre- sult for an arbitrary supermartingale.Recently, BouchardB. and Nutz M. [1] considered a class of discrete models and proved the necessary and sufficient conditions for validity of optional decomposition. Our statement of the prob- lem unlike the above-mentionedone andit is moregeneral:a supermartingale relative to a convex set of equivalent measures is given and it is necessary to findconditionsonthe supermartingaleandthe setofmeasuresunderthatop- tionaldecompositionexists.Generalityofourstatementoftheproblemisthat wedonotrequirethattheconsideredsetofmeasureswasgeneratedbyrandom process that is a local martingale as it is done in the papers [1, 2, 11, 4] and that is important for the proof of the optional decomposition in these papers. 2 Discrete case. We assume that on a measurable space {Ω,F} a filtration F ⊂F ⊂ m m+1 F, m=0,∞,andafamilyofmeasuresM onF aregiven.Further,weassume that F ={∅,Ω}. A random process ψ ={ψ }∞ is said to be adapted one 0 m m=0 A sample document 3 relative to the filtration {F }∞ if ψ is F measurable random value for m m=0 m m all m=0,∞. Definition 1. An adapted random process f = {f }∞ is said to be a m m=0 supermartingale relative to the filtration F , m = 0,∞, and the family of m measures M if EP|f |<∞, m=1,∞, P ∈M, and the inequalities m EP{f |F }≤f , 0≤k ≤m, m=1,∞, P ∈M, (1) m k k are valid. We consider that the filtration F , m = 0,∞, is fixed. Further, for a su- m permartingalef weuseasdenotation{f ,F }∞ anddenotation{f }∞ . m m m=0 m m=0 Bellow in a few theorems, we consider a convex set of equivalent measures M satisfyingconditions:Radon–NicodymderivativeofanymeasureQ ∈M 1 with respect to any measure Q ∈M satisfies inequalities 2 dQ 00)>0, such that the inequality m0 1 m0 f −EP1{f |F }≥ϕ , m0−1 m0 m0−1 m0 is valid, then l f −EQ{f |F }≥ ϕ , Q∈M , m0−1 m0 m0−1 1+L m0 ε¯0 where M ={Q∈M, Q=(1−α)P +αP , 0≤α≤ε¯ , P ∈M}, P ∈M, ε¯0 1 2 0 2 1 L ε¯ = . 0 1+L Proof. Let B ∈F and Q=(1−α)P +αP , P ∈M, 0<α<1. Then m0−1 1 2 2 [f −EQ{f |F }]dQ= m0−1 m0 m0−1 Z B EQ{[f −f ]|F }dQ= m0−1 m0 m0−1 Z B [f −f ]dQ= m0−1 m0 Z B 4 N. Gonchar (1−α) [f −f ]dP + m0−1 m0 1 Z B α [f −f ]dP = m0−1 m0 2 Z B (1−α) [f −EP1{f |F }]dP + m0−1 m0 m0−1 1 Z B α [f −EP2{f |F }]dP ≥ m0−1 m0 m0−1 2 Z B (1−α) [f −EP1{f |F }]dP = m0−1 m0 m0−1 1 Z B dP (1−α) [f −EP1{f |F }] 1dQ≥ m0−1 m0 m0−1 dQ Z B l (1−α)l ϕ dQ≥(1−ε¯ )l ϕ dQ= ϕ dQ. m0 0 m0 1+L m0 Z Z Z B B B Arbitrariness of B ∈F proves the needed inequality. m0−1 Lemma1. Anysupermartingale{f ,F }∞ relativetoafamilyofmeasures m m m=0 M for which there hold equalities EPf = f , m = 1,∞, P ∈M, is a m 0 martingale with respect to this family of measures and the filtration F , m= m 1,∞. Proof. The proof of Lemma 1 see [10]. Remark 1. If the conditions of Lemma 1 are valid, then there hold equalities EP{f |F }=f , 0≤k ≤m, m=1,∞, P ∈M. (3) m k k Let f = {f ,F }∞ be a supermartingale relative to a convex set of m m m=0 equivalent measures M and the filtration F , m = 0,∞. And let G be a set m of adapted non-decreasing processes g ={g }∞ , g =0, such that f +g = m m=0 0 {f +g }∞ is asupermartingaleconcerningthe familyofmeasuresM and m m m=0 the filtration F , m=0,∞. m Introduce a partial ordering (cid:22) in the set of adapted non-decreasing processes G. A sample document 5 Definition2. Wesaythatanadaptednon-decreasingprocessg ={g1 }∞ , 1 m m=0 g1 =0,g ∈G,doesnotexeedanadaptednon-decreasingprocessg ={g2 }∞ , 0 1 2 m m=0 g2 =0, g ∈ G, if P(g2 −g1 ≥ 0) = 1, m = 1,∞. This partial ordering we 0 2 m m denote by g (cid:22)g . 1 2 For every nonnegative adapted non-decreasing process g = {g }∞ ∈ G m m=0 there exists limit lim g which we denote by g . m ∞ m→∞ Lemma2. LetG˜ beamaximalchaininGandforacertainQ∈M supEQg = 1 g∈G˜ αQ < ∞. Then there exists a sequence gs = {gs }∞ ∈ G˜, s=1,2,..., such m m=0 that supEQg =supEQgs, 1 1 g∈G˜ s≥1 where ∞ EQg EQg = m, g ∈G. 1 2m m=0 X Proof. Let 0 < ε < αQ, s = 1,∞, be a sequence of real numbers satisfying s conditions ε >ε , ε →0, as s→∞. Then there exists anelement gs ∈G˜ s s+1 s such that αQ−ε < EQgs ≤ αQ, s = 1,∞. The sequence gs ∈ G˜, s = 1,∞, s 1 satisfies Lemma 2 conditions. Lemma3. Ifasupermartingale{f ,F }∞ relativetoaconvexsetofequiv- m m m=0 alent measures M is such that |f |≤ϕ, m=0,∞, EQϕ0)>0, satisfying conditions m0 1 m0 f −EPi{f |F }≥∆s , i=1,s, (8) m0−1 m0 m0−1 m0 then the set G of adapted non-decreasing processes g ={g }∞ , g =0, for m m=0 0 which {f +g }∞ is a supermartingale relative to the set of measures M m m m=0 contains nonzero element. Proof. For any point P ∈M let us define a set of measures 0 MP0,ε¯0 ={Q∈M, Q=(1−α)P +αP, P ∈M, 0≤α≤ε¯ }, (9) 0 0 L ε¯ = . 0 1+L Prove that the set of measures MP0,ε¯0 contains some ball of a positive radius, thatis,thereexistsarealnumberρ >0suchthatMP0,ε¯0 ⊇C(P ,ρ ), where 0 0 0 C(P ,ρ )={P ∈M, |P −P|<ρ }. 0 0 0 0 Let C(P ,ρ˜) = {P ∈ M, |P −P| < ρ˜} be an open ball in M with the 0 0 centeratthe pointP ofaradius0<ρ˜<1.Consideramapofthe setM into 0 itself given by the law: f(P)=(1−ε¯ )P +ε¯ P, P ∈M. 0 0 0 ′ ′ The mapping f(P) maps an open ball C(P ,δ) = {P ∈ M,|P −P| < δ} 2 2 ′ with the center at the point P of a radius δ > 0 into an open ball with the 2 ′ center at the point (1−ε¯ )P +ε¯ P of the radius ε¯ δ, since |(1−ε¯ )P + 0 0 0 2 0 0 0 ′ ′ ε¯ P −(1−ε¯ )P −ε¯ P|=ε¯ |P −P|<ε¯ δ. Therefore, an image of an open 0 2 0 0 0 0 2 0 set M ⊆M is an open set f(M )⊆M, thus f(P) is an open mapping. Since 0 0 f(P ) = P , then the image of the ball C(P ,ρ˜) = {P ∈ M, |P −P| < ρ˜} 0 0 0 0 is a ball C(P ,ε¯ ρ˜)= {P ∈M, |P −P|< ε¯ ρ˜} and it is contained in f(M). 0 0 0 0 Thus, inclusions MP0,ε¯0 ⊇ f(M)⊇C(P ,ε¯ ρ˜) are valid. Let us put ε¯ ρ˜=ρ . 0 0 0 0 Then we have MP0,ε¯0 ⊇ C(P ,ρ ), where C(P ,ρ ) = {P ∈ M, |P −P| < 0 0 0 0 0 ρ }. Consider an open covering C(P ,ρ ) of the compact set M. Due to 0 0 0 P0∈M compactness of M, there exists a Sfinite subcovering v M = C(Pi,ρ ) (10) 0 0 i=1 [ withthecenteratthepointsPi ∈M, i=1,v,andacoveringbysetsMP0i,ε¯0 ⊇ 0 C(Pi,ρ ), i=1,v, 0 0 v M = MP0i,ε¯0. (11) i=1 [ 8 N. Gonchar Consider the set of measures Pi ∈M, i=1,v. From Lemma 4 conditions, 0 there exist a natural number 1 ≤ m < ∞, and depending on the set of 0 measures Pi ∈ M, i = 1,v, F measurable nonnegative random variable 0 m0−1 ∆v , P1(∆v >0)>0, such that m0 0 m0 fm0−1−EP0i{fm0|Fm0−1}≥∆vm0, i=1,v. (12) Due to Theorem 1, we have l f −EQ{f |F }≥ ∆v =ϕv , Q∈M. (13) m0−1 m0 m0−1 1+L m0 m0 The last inequality imply EQ{f |F }−EQ{f |F }≥EQ{ϕv |F }, Q∈M, s 0 m0−1 m0 1 m0 0)>0, such that fm0−1−EPi{fm0|Fm0−1}≥ϕnm0, i=1,n. (22) If for the maximal element g0 ={g0 }∞ in a certain maximal chain G˜ ⊆G m m=0 the equalities EPi(f +g0 )=f , P ∈M, i=1,n, (23) ∞ ∞ 0 i are valid, where f = lim f , g0 = lim g0 , then there hold equalities ∞ m ∞ m m→∞ m→∞ EP{f +g0 |F }=f +g0, 0≤k ≤m, m=1,∞, P ∈M. (24) m m k k k Proof. The set M is compact one in the introduced metric topology. From the inequalities (22) and the formula n αiEP1{ϕi|Fm0−1}EPi{fm0|Fm0−1} EQ{f |F }= i=1 , Q∈M, (25) m0 m0−1 P n αiEP1{ϕi|Fm0−1} i=1 P where ϕi = ddPP1i, we obtain f −EQ{f |F }≥ϕn , Q∈M. (26) m0−1 m0 m0−1 m0 The inequalities (21) lead to inequalities 1 dQ ≤ ≤nL, P,Q∈M. (27) nL dP Inequalities (26) and (27) imply that conditions of Lemma 4 are satisfied for any set of measuresQ ,...,Q ∈M. Hence, it follows that the set G contains 1 s 10 N. Gonchar nonzero element. Let G˜ ⊆ G be a maximal chain in G satisfying condition of Theorem 2. Denote by g0 ={g0 }∞ , g0 = 0, a maximal element in G˜ ⊆ G. m m=0 0 Theorem 2 and Lemma 3 yield that as {f }∞ and {g0 }∞ are uniformly m m=0 m m=0 integrable relative to each measure from M. There exist therefore limits lim f =f , lim g0 =g0 m ∞ m ∞ m→∞ m→∞ with probability 1. Due to Theorem 2 condition, in this maximal chain EPi(f∞+g∞0 )=f0, Pi ∈M, i=1,n. Since{f +g0 }∞ isasupermartingaleconcerningallmeasuresfromM,we m m m=0 have EPi(fm+gm0 )≤EPi(fk+gk0)≤f0, k

Generalization of Doob decomposition Theorem

Similar Generalization of Doob decomposition Theorem