MAT4760/MAT9760. ρ X + ρ(x) = ρ(x) ρ(x) = 0 (4.1)

Transcription

1 4.1 Risk Measures FS 153] MAT476/MAT976 Dae A risk measure, denoed by ρ, is a funcion ha quanifies he risk associaed wih some financial posiion X, where X : Ω R. Ω is he se of all possible oucomes, and Xω is he discouned value of he financial posiion in he oucome ω Ω. We define X o be he se of financial posiions. X is a linear space of bounded funcions conaining he consans. The norm we use is he supremum norm: X := sup ω Ω Xω. Definiion 4.1 FS 153] A mapping ρ : X R is called a moneary measure of risk if i saisfies he following condiions for all X, Y X. Monooniciy: X Y = ρx ρy, Xω Y ω, ω Ω Cash Invariance: m R = ρx + m = ρx m, m R. Consequences of cash invariance: ρ X + ρx = ρx ρx = 4.1 ρm = ρ m When we do no have a financial posiion, we do no have any risk. In his case we say he risk measure is normalized. Normalizaion: ρ = Lemma 4.3 FS 154] Any moneary measure of risk ρ is Lipschiz coninuous wr he supremum norm. Lipschiz coninuiy implies coninuiy. ρx ρy X Y Proof X = X ± Y Y + X Y = X Y + X Y By monooniciy and cash invariance: ρx ρ Y + X Y = ρy X Y 1

2 X Y ρy ρx The same reasoning wih Y insead of X. Y = Y ± X X + X Y = Y X + X Y By monooniciy and cash invariance: ρy ρ X + X Y = ρx X Y ρy ρx X Y By combining and we ge: X Y ρy ρx X Y ρy ρx X Y Definiion 4.4 FS 154] A moneary measure of risk ρ : X R is called a convex risk measure if i saisfies he convexiy propery: For some λ 1, ρ λx + 1 λy λρx + 1 λρy Convexiy is a way o implemen a risk reducion for diversifying he invesmen. Lemma FS 154] If ρ is convex and normalized, hen we have he following properies. Proof i By convexiy and normalizaion: ρax aρx, a, 1], X X i ρax aρx, a 1,, X X ii ρax = ρ ax + 1 a aρx + 1 a ρ = aρx }} = ii Using i: ax ρx = ρ a 1 ρax = aρx ρax a 2

3 Definiion 4.5 FS 155] A convex measure of risk ρ is called a coheren risk measure if i saisfies posiive homogeneiy. Forλ, ρλx = λρx Lemma A coheren risk measure is always normalized. Proof Take any m, m 1. ρm = ρ + m = ρ m Using posiive homogeneiy. ρm = ρm 1 = mρ1 = m ρ 1 = mρ m Combining and : ρm = ρm, which gives ρ m = mρ m, mρ ρ = ρ m 1 = }} ρ = Lemma FS 155] For a coheren risk measures ρ, convexiy sub-addiiviy, defined as: ρx + Y ρx + ρy, X, Y X. Proof For a, b > define X = a+b a+b X and Ỹ = Y. Using convexiy and a b posiive homogeneiy: a ρx + Y = ρ a + b X + b a + bỹ a a + b ρ X + b a + b ρỹ = a a + b a + b ρ a X + b a + b a + b ρ Y = ρx + ρy b 3

4 Define X = λx and Ỹ = 1 λy. Using sub-addiiviy and posiive homogeneiy: ρ λx + 1 λy = ρ X + Ỹ ρ X + ρỹ = λρx + 1 λρy Posiive homogeneiy is a resricive assumpion as i requires risk measures o be linear. From he m.r.m moneary risk measure we ge is accepance se. A ρ := X X ρx } The se of all accepable posiions in he sense ha hey do no need any addiional capial. Here are wo resuls ha summarize he relaions beween m.r.m and heir accepance se. Proposiion 4.6 FS 155] Assume ρ is a m.r.m wih accepance se A := A ρ. a A is non-empy, A =, and saisfies he following condiions. inf m R m A } > 4.3 X A, Y X, Y X = Y A 4.4 Moreover, A has he following closure properies. For X A and Y X, S := λ, 1] } λx + 1 λy A is closed in, 1] 4.5 b We can ge ρ jus by knowing A. ρx = inf m R m + X A } 4.6 c ρ is a convex risk measure A is convex. d ρ is posiively homogeneous A is a cone. In paricular, ρ is coheren A is a convex cone. Proof a A = Define z := ρ, hen ρ R. Choose some m R such ha z m. Then, by cash invariance, ρm = ρ m, so m A, and A is nonempy. 4

5 4.3 Consider any consan m A. By cash invariance, ρm = ρ m. If we se z := ρ his means we have z m, which implies - since he lef side is independen of m, z inf m R ρm } z is a number, z R, so z >. 4.4 X A = ρx Y X = ρy ρx Combining hese inequaliies means ρy ρx, hence Y A. 4.5 By assumpion, X A and Y X. For all λ S, λx + 1 λy A = ρ λx + 1 λy. By lemma 4.3, ρ is coninuous. For fixed X, Y f X,Y : λ λx + 1 λy is also a coninuous funcion, which means ha he composie funcion ρ f X,Y is also coninuous, ρ f X,Y λ = ρ λx + 1 λy 1, Now we se S = ρ f X,Y and use he resul ha he couner image of a closed se is sill closed, hus S is closed. b By he defining propery of A: inf m R m + X A } = inf m R ρ m + X } Using cash invariance and he definiion of he infimum: inf m R ρ X m } = ρx c Assume firs ha ρ is convex. Assume X, Y A and λ, 1]. ρ λx + 1 λy λρx + 1 λρy = λx + 1 λy A which means A is a convex se. 5

6 Assume A is convex. We choose arbirary X 1, X 2 X and le λ, 1]. We now consider m 1, m 2 R such ha, by cash invariance, ρm i + X i = ρx i m i for i = 1, 2. By he consrucion of A, boh m 1 + X 1 and m 2 + X 2 are elemens in A. Using ha A is a convex se and he propery for all elemens in A: λm 1 +X 1 +1 λm 2 +X 2 A = ρ λm 1 +X 1 +1 λm 2 +X 2 By cash invariance, ρ λx λx 2 λm λm 2 ] ρ λx λx 2 λm λm 2 = λ inf m 1 m 1 + X 1 A } + 1 λ inf m 2 m 2 + X 2 A } = λρx + 1 λρy = ρ is convex. d Assume ρ is posiively homogeneous. Recall ha X A = ρx and for any λ we have he posiive homogeneiy propery: ρλx = λρx. Since ρx hen λρx, so λρx A for all λ, so A is a cone. Now suppose A is a cone and suppose X X. From par b, m R such ha ρx m, or equivalenly, m+x A. Since A is a cone, we have for λ ha λm + X A. λm + λx A = ρ λm + λx = ρλx λm = ρλx λm ρλx λ inf m R m + X A } = λρx = ρλx λρx We are going o show ha ρ is posiively homogeneous, so now we only need he reverse inequaliy. We consider m R such ha m < ρx. By cash invariance and using λ : m < ρx = < ρx +m = X +m A = λx +m A < ρ λm + X = ρλx λm = λm < ρλx = λ sup m R X + m A } ρλx }} =ρx? If we can show ha he underbraced sup-erm in he proof is equal o ρx he proof will be concluded. We know from above ha: supm } := sup m R X + m A } ρx 6

7 and we know ρx = inf m R ρx m } =: infm} Sraegy: show ha he sup canno be less han ρx, and herefore mus be equal. If supm } < ρx = infm}, hen we can define: ε = infm} supm } 2 >. We also define m := supm } + ε. We noe ha m = supm }+infm} < ρx, 2 and ha ρ X + m = ρx m >, so X + m A, however m > supm }, so we have a conradicion! This means supm } canno be sricly less han ρx which means hey are equal. Reurning o we have λρx ρλx, and wih he firs inequaliy we have shown ha λρx = ρλx. From a m.r.m we can find an accepance se, and conversly, from a se A X we can define a m.r.m ρ A : and we noe ha ρ Aρ = ρ. Proposiion 4.7 FS 156] Assume ha A X saisfies: ρ A := inf m R : m + X A }, i ii iii hen ρ A saisfies: a ρ A is a moneary measure of risk. b A is convex = c A is a cone = X X A = inf m R m A } > X A, Y X, Y X = Y A ρ A is a convex risk measure. ρ A is posiively homogeneous. Noe ha propery b and c combined imply ha ρ A is coheren. d Define A ρa := X X ρ A X } Then A A ρ and if we have he closure propery, i.e for X A, Y X, S := λ, 1] λx + 1 λy A } hen A = A ρ. 7

8 Lemma A If X A, hen ρ A X. Follows from he definiion. B A A ρa. Follows from a. C If i, ii and iii from prop. 4.7 are saisfied, hen X A implies ρ A. In paricular X AρA = ρ A X > C 1 X A ρa \ A = ρ A X = C 2 Proof A, B and C 1 are obvious. Proving C 2 : We know ha, X A ρa \ A = ρ A. Suppose for a conradicion ha ρ A X <, hen m < such ha m + X A. Noe ha m + X < }}}} X A A bu by iii his is absurd! Sricly less is impossible, so we have equaliy. Proof of Proposiion a To prove ha ρ A is a m.r.m, we mus verify ha i is monoone and cash invarian. Monooniciy Le X, Y X and assume X Y. Of course, m+x m+y. Choose m such ha m + X A = iii m + Y A inf m R m + X A } inf m R m + Y A } Cash invariance Le a R. ρ A X ρ A Y ρ A X + a = inf m m + X + a A } = inf m a m + X A } = inf m m + X A } a = ρ A X a 8

9 Verifying wo addiional properies. The risk measure always has an upper bound: ρ A X R, X X. Take X X and Y A X. Since X is he se of bounded funcions, m R such ha X + m > Y. ρ A X m = ρ A X + m ρ A Y Y A = ρ A X m <. Risk measures are no minus infiniy. For X X we can find a R such ha a + X. By monooniciy and cash invariance: ρ A a + X ρ A ρ A X a ρ A ii = inf m m + A} > = ρ A X ρ A + a > b We assume A is convex, and show ha ρ A is convex. We assume X 1, X 2 X and ake λ, 1]. We hen consider m 1, m 2 R such ha m 1 + X 1 A and m 2 + X 2 A. Since we have assumed A is convex: λ m 1 + X λ m2 + X 2 A By definiion of A, and hen cash invariance. ρ A λ m 1 + X λ m2 + X 2 = ρ A λx1 + 1 λx 2 λm1 + 1 λm 2 ] We ge, ρ A λx1 + 1 λx 2 λm λm 2 ρ A λx1 +1 λx 2 λ inf m 1 m 1 + X 1 A } +1 λ inf m 2 m 2 + X 2 A } }}}} ρ A X 1 by 4.6b ρ A X 2 by 4.6b = ρ A λx1 + 1 λx 2 λρ A X λρ A X 2 Wih his we have proved ha ρ A is a convex funcion. 9

10 c We assume A is a cone, and show ha ρ A is posiively homogeneous. We consider some X X and m R : m + X A. Choose a λ >. Since A is a cone, m + X A λm + X A. By definiion of A and cash invarianc: ρ A λm + X = ρa λx λm In general, ρ A λx λm, and in paricular ρ A λx λ inf m m + X A } λ m ρ A λx λρ A X. If we can show he opposie inequaliy as well, we have shown ha ρ A is posiively homogeneous. We consider m R such ha we have he sric inequaliy m < ρ A X. m < ρ A X = < ρ A X m = < ρ A X +m = m+x A For λ >, and using ha A is a cone, we also have λm + X A. < ρ A λx λm = λm < ρa λx = λ sup m R m + X A } ρ A λx From his inequaliy, we can derive he following: sup m R m + X A } inf m R m + X A } We show, as we did in he proof by conradicion of Proposiion 4.6d, ha he lef side can no be sricly less, and herefore we mus have equaliy. This resuls in λρ A X ρ A λx which is he required inequaliy, and we have proved c. d We assume A A ρa := X X ρ A X } and ha X has he closure propery, i.e for every X A, Y X and i, 1] S := λ, 1] λx + 1 λy A } ii where S is closed in, 1], and show ha i and ii imply ha A = A ρ. 1

11 By i, we have A A ρ which implies A c ρ A c. To show ha hese ses are equal we only have o show ha A c A c ρ. We observe ha λ = 1 S since we know ha X A, bu for λ = we do no know, since we don know weher or no Y is in A. Consider a new X A c = X \ A, which we know is bounded since X is bounded. We can find a m R such ha Choose an ε, 1 such ha sup Xω = X < m. ω Ω ε m + 1 ε }} X A A X is no in A by assumpion, bu we don know if m is. ρ A εm + 1 εx = ρa 1 εx εm = εm ρ A 1 εx ρa 1 εx ρa X εx X = ε X < εm Rearranging: ρ A X ρ A 1 εx ε X εm ε = εm X > so ρ A X > which means X A ρ which is equivalen o saying X A c ρ. For an arbirary X Ac we have X A c ρ, so Ac A c ρ, and he proof is compleed. Examples As usual X denoes he linear space of all bounded, measurable funcions on some probabiliy space Ω, F, and M 1 = M 1 Ω, F denoes he class of all probabiliy measures on Ω, F. Ex 4.8 Wors-case Risk measure FS 157] We define he wors-case r.m by ρ max X = inf ω Ω Xω The r.m ρ max is in fac a coheren r.m, which we can see by verifying monooniciy, cash invariance and checking ha A ρmax is a convex cone. 11

12 Monooniciy If X Y, or more precisely, Xω Y ω ω Ω, hen inf Xω inf Y ω = ω Ω ω Ω Y ω = inf ω Ω Xω inf ω Ω ρ max X ρ max Y Cash invariance For some m R we need o show ha ρ max X + m = ρ max X m. } ρ max X + m = inf Xω + m ω Ω = inf Xω m ω Ω = ρ max X m A ρmax is a convex cone A ρmax = X X such ha ρ max X } = X X } From his we see ha A ρmax is convex cone, and by Proposiion 4.7 b and c, i follows ha ρ max is a coheren risk measure. Ex 4.11 Value a Risk V ar FS 158] We assume we have fixed a probabiliy measure P on he measurable space Ω, F. A posiion X X is accepable if PX < λ for some λ, 1. Usually denoe his r.m by V ar λ. The accepance se for V ar λ is: A = X X PX < λ } X does no necessarily have o be less han, could also be e.g a small consan. The corresponding moneary risk measure is V ar λ X = ρ var X = inf m R Pm + X < λ } We verify ha his acually is a m.r.m by verifying monooniciy and cash invariance. 12

13 Monooniciy Assume X Y and derive he inequaliy V ar λ Y V ar λ X. Wriing ou he risk measures. V ar λ X = inf m R Pm + X < λ } V ar λ Y = inf n R Pn + Y < λ } Since Y is larger han X, we need a bigger number o make i negaive, hence n m, so X Y = V ar λ Y V ar λ X. Cash invariance V ar λ X + a = inf m R Pm + a + X < λ } = inf x a R Px + X < λ } m=x a = inf x Px + X < λ } a = V ar λ X a Posiive homogeneiy We assume a >. V ar λ ax = inf m R PaX + m < λ } = inf ax R PX + x < λ } x=m/a = a inf x R PX + x < λ } = a V ar λ X Ex 4.41 Value a Risk 2 FS 158] We consider he case where we inves in wo corporae bonds, each wih payoff r, where r is he reurn on a riskless invesmen and r > r. These bonds are defaulable, i.e hey can defaul and become worhless. We assume he bonds are independenly defauling e.g hey belong o dofferen non-conneced marke secors. We inroduce w > as he size of he iniial invesmen. The discouned ne gain for bond i is given by probabiliies denoed by p i w defaul p i X i = r r w oherwise 1 p i 1 + r 13

14 Now we have enough informaion o find he V ar for he firs bond, which defauls wih probabiliy p 1. We choose some λ, 1 so p 1 λ. V ar λ Y = inf a Pa + Y < λ } a = w = P X 1 < w = λ I is impossible o loose more han he iniial invesmen. So, r r a = w 1 + r = P X 1 < w r r = p 1 λ 1 + r r r V ar λ X 1 = w < = X 1 is accepable. 1 + r Le us see wha happens if we diversify our invesmen, by invesing w/2 in each bond, so we have Z = 1 2 X X 2, and if boh defaul, one defauls or none defaul we ge wih he corresponding probabiliies: Z = w p 1 p 2 w r r r 1 < p 1 1 p 2 p 2 1 p 1 r r w 1 + r 1 p 1 1 p 2 In he second column we used w 2 + w r r = w r 2 and o esablish he las, sric inequaliy, r < r < 2r + 1. r r 1 + r 1 a = w = PZ < w = λ a = w 1 r r = P Z < w r r r r 1 = p 1 p 2 a = w r r 1 + r r r = P Z < w = p 1 p 2 + p 1 1 p 2 + p 2 1 p r = p 1 + p 2 p 1 p 2 14

15 To simplify we assume p 1 = p 2 = p, so p 1 p 2 = p 2 and p 1 +p 2 p 1 p 2 = 2p p 2. Since p λ, we ge p 2 λ 2p p 2, so we ge V ar λ Z = w 1 r r > = V ar λ Z > V ar λ X r so V ar λ is no a convex measure which subsequenly means i punishes diversificaion. This means ha he accepance se, A is no convex, bu ha is no obvious: A = X X PX < λ }. 4.2 Robus Represenaions of Risk Measures FS 161] Linear funcions vs Addiive Se funcions Definiion A.49 FS 426] Le Ω, F be a measurable space. A mapping µ : F R is called a finiely addiive se funcion if i and ii are saisfied. I is called a finiely addiive probabiliy measure if we also have iii: i µ = N ii Pairwise disjoin A 1,...,A N F, implies µ A i = iii µω = 1. i=1 N µa i i=1 We use M 1,f := M 1,f Ω, F o denoe he se of all finiely addiive se funcions µ : F, 1] where µω = 1, so M 1,f is he se of all finiely addiive probabiliy measures. The oal variaion of a finiely addiive se funcion µ is defined, where A 1,..., A N are muually disjoin ses in F and N N µ var := sup A 1,...,A N N µa i Wih ba := baω, F we denoe he collecion of se funcions wih finie oal variaion. We noe ha M 1,f ba, since for some Q M 1,f, A := A 1,...,A N }, Q var sup A i=1 N N QA i = sup Q A i QΩ = 1 < i=1 A 15 i=1

16 Inegraion Theory This is a brief ouline of he inegraion heory wr a measure µ ba. The space X endowed wih he supremum norm, F := sup Fω, F X ω Ω is a Banach space. This is a space wih a nice srucure where we can define a opology, Cauchy sequences converge and we can define a limi. For an elemen F X, we can define he simple or elemenary represenaion: F X = F = N α i ½ Ai i=1 for some consans α i. From his represenaion, we can define he inegral. N Fdµ := α i µa i =: l µ F Ω i=1 We have Fdµ N i=1 α i µa i sup Fω ω N µa i F µ var i=1 For any F X can be approximaed by simple funcions: F n dµ F m dµ = F n F, n. F n F m dµ F n F m µ var m,n som F n dµ } n is Cauchy. By seing µ = Q M 1,f ba, we denoe he inegral by FdQ = E Q F]. Theorem A.5 FS 427] The inegral lf = Fdµ, F X defines a one-o-one correspondence wih he finiely addiive se funcions µ ba, and he linear coninuous funcionals l on X. We have a one-o-one correspondence l µ. 16

17 Proof From µ ba o he definiion of l µ we simply follow he consrucion of he inegral. We define µa = l½ A for A F, where ½ is he indicaor funcion. For pairwise disjoin ses A 1,...,A N, which we abbreviae as A, we ge µ var = sup A N µa i i=1 = l½ Ai We divide he sum ino negaive and posiive erms. N = sup µ + A i + µ A i A i=1 = sup l½ai + l½ Bj A = sup l½ai + ½ Bj l A The norm of l is given by l := sup lx X 1 For linear, coninuous funcions hey are bounded. µ A i = l ½ Ai = l ΣAi = Σl ½Ai = ΣµAi For finie measures, in he sense ha he oal variaion is finie, we have a one-o-one correspondence o linear, coninuous funcionals, The mapping ha links hem is an inegral. Summary The inegral lf = Fdµ for F X defines a one-o-one relaionship beween he spaces X = l X R where l is coninuous and linear } and ba = baω, F = µ : F R signed addiive bounded o. variaion } σ-addiiviy is equivalen o A n ր Ω, ha is, A n A n+1 n A n = Ω. For he measure Q : QA n QA n+1, a monoonic sequence, QA n ր QΩ = 1 for a probabiliy measure. l is coninuous and linear: lx, X, l1 = 1. 17

18 Theorem 4.15 FS 162] - Rep. Theorem for Risk Measures Any convex risk measure ρ : X R admis represenaion as: ρx = sup EQ X] α min Q } Q M 1,f where α min Q = sup X A ρ E Q X], Q M 1,f. If here is any oher α penaly funcion such ha ρx = sup Q M 1,f EQ X] αq } hen αq α min Q, for all Q M 1,f. The penaly funcion isn unique. We can use he max insead of sup. The E Q X] is he linear par, and α min is an elemen in he family of penaly funcions, and i penalizes lineariy. In general: The funcion α : M 1,f R }, such ha inf Q M 1,f αq R. X E Q X] αq is convex, cash invarian and convex, and hese properies are preserved when we ake he sup we call his funcion ρ for now. ρx : X sup EQ X] αq }, ρ = inf αq Q M 1,f Q M 1,f Proof of Theorem 4.15 We show ha ρx boh dominaes and is dominaed by he represenaion in heorem Sep 1 ρx sup EQ X] α min Q } Q M 1,f If we define X := X + ρx for X X, we ge, by cash invariance, ρx = ρ X + ρx = ρx ρx = = X A ρ. 18

19 By he definiion of α min we have ha for all Q M 1,f, α min Q = sup Z A ρ E Q Z] E Q X ] = E Q X ρx] = E Q X] ρx α min Q E Q X] ρx = ρx E Q X] α min Q The lef side of he inequaliy is independen of Q, so we can ake he sup on he righ side, and we have derived he firs inequaliy. ρx sup EQ X] α min Q } Q M 1,f Sep 2 ρx sup EQ X] α min Q } Q M 1,f To derive his inequaliy, we find a measure Q X such ha ρx E QX X] α min Q X }. When his is rue, i is also rue when we ake he sup over all Q, and his will subsequenly resul in he required inequaliy. We need a resul from funcional analysis: Theorem A.54 FS 429] In a opological vecor space E, any wo disjoin convex ses B and C, where a leas one of hem has an inerior poin, can be separaed by a non-zero coninuous linear funcional l on E: lx ly X C, Y B. We define he se X X by X := X X ρx = } Suppose Y X \ X, and define X := Y + ρy If ρx =, hen Y = X ρy?]. Wihou loss of generaliy we can assume ρ =, since if ρ = c which implies, by cash invariance, ρ c =, hen we can se ρx := ρx c for X X. Take X X and define he se B := Y X ρy < }, 19

20 hen we have X B and X C = B c, which are complemenary ses, and one of hem is non-empy. These are he necessary condiions of Theorem A.54, and we have he exisence of a non-zero, coninuous l: In fac, l : X R = lz ly, Z C }} X X, Y B By properies of l we have: 1 } lx, X 2 l1 > lx inf ly := b R. Y B Verifying hese wo properies. 1 Y ly We firs claim ha: By monooniciy, which means we can normalize: lx = lx l1 Y = 1 + λy B, λ > Y = λy = ρλy ρ We can exploi he following relaionship: 1 + λy > λy. ρ1 + λy = ρλy 1 < ρλy ρ = So 1+λY B. Now, for X B wih ρx = we apply he inequaliy for l, and also use ha l is a linear funcional: lx l1 + λy = l1 + λly, λ >. This holds for all λ, so his inequaliy would no be possible if ly <, hence ly. 2 l1 > For some Y X we have < ly = ly + ly Also, Y := sup ω Ω Y ω = sup Y + ω + Y ω } < 1 ω Ω 2

21 From hese equaions, we know ha ly + >, and 1 > Y + + Y Y + = 1 Y + > = l1 Y + Now we can use lineariy of l again: l1 = l1 Y + + Y + = l1 Y + + ly + > = l1 > }}}} > The normalized lx := lx/l1 for X X is in a one-o-one relaionship wih some Q. E bq X] = lx, X X Wih his we can derive he following inequaliy in we are reducing he se, which we will verify α min Q = sup EQ b Y ] Y A ρ sup Y B EQ b Y ] = sup l Y = inf ly = b Y B Y B l1 Now we verify ha we can do he sep indicaed by. This is rue if B A ρ, which is rue if any arbirary elemen Y B is auomaically in A ρ, bu by he defining properies of each se his is rue: Y B = ρy < = ρy = Y A ρ, = B A ρ. To derive he oher inequaliy, Y A ρ, ε > = Y + ε B, since ρy + ε = ρy ε < Now, for all ε > and all Y A ρ, sup E bq Z] E bq Y + ε] = Z B sup E bq Z] sup E bq Y ] = α min Q Z B Y A ρ Combining hese inequaliies, we have α min Q = b l1. E bq X] α min Q = l X b = b lx l1 l1 which is rue for all X X. We have, for some Q M 1,f, ρx E bq X] α min Q 21 = ρx

22 The lef side is independen of Q, so we can ake he sup and keep he inequaliy. ρx sup EQ X] α min Q } Q M 1,f and we have proved he represenaion of ρ from Theorem Now we verify ha α min is he smalles penaly funcion. From our represenaion heorem and for some α: ρx = sup Q M 1,f EQ X αq } E Q X] αq Shifing erms, we ge: αq E Q X] ρx We ge: αq sup EQ X] ρx } X X sup EQ X] ρx } X A ρ since A ρ X sup E Q X] X A ρ since X A ρ ρx = α min Q so αq α min Q. Remarks: From Theorem 4.15 we have: α min Q = sup A ρ E Q X] and ρx E Q X] α min Q, X So, α min Q E Q X] ρx = α min Q sup X A ρ EQ X] ρx } sup X A ρ E Q X] = α min Q = α min Q = sup X A ρ EQ X] ρx } 22

23 Definiion A.59 FS 43] - Fenchel-Lengendre Transform For a funcion f : X R }, we define he Fenchel-Legendre ransform, f : X R ± } by f l := sup X X lx fx }, l X. If f, hen f is a proper convex rue for risk measures and lower semiconinuous funcion as he supremum of affine funcions. lim inf l l f l f l l converges o l in X in he weak sar sense. If f is iself a proper convex funcion, hen f is proper convex and lower semiconinuous. We call f he conjugae funcional. Theorem A.61 FS 431] saes ha if f is lower semiconinuous, hen f = f. Remark FS 164]: In view of his, we have ha he equaion corresponds o he Fenchel-Legendre ransform or conjugae funcion of he convex funcion ρ on he Banach space X. More precisely: α min Q = ρ l Q where ρ : X R + } is defined on he dual X of X by } ρ l = sup lx ρx X X and where l Q X = E Q X] for X X and Q M 1,f. Given a se A, we can find a risk measure: ρ A X = inf m R m + X A } By Theorem 4.15, his r.m has he represenaion: ρ A X = max Q M 1,f EQ X] α min Q } where: α min Q = sup E Q X] sup E Q X]. X A ρa X A 23

24 We wan o have an expression for α min wih A and no A ρa. We know he following inequaliy is rue, bu now we wan o derive he opposie inequaliy and ge he expression wih only A. Y A ρa = ρy ε >, ρy + ε = ρy ε < = sup E Q X] E Q Y + ε] X A sup E Q X] sup E Q Y ]. X A Y A ρa This shows ha we have α min Q = sup X A E Q X]. Y + ε A Corollary 4.18 FS 165] If ρ is a coheren risk measure, hen in Qmax M α min Q = 1,f + in Qmax c Q max is a convex se and i is he larges se where you can find a represenaion We define ρx = sup Q M 1,f EQ X] α min Q } = max Q Q max E Q X]. Q max := Q M 1,f α min Q = } Proof From Proposiion 4.6 we know ha he accepance se A ρ of a coheren r.m is a cone. For λ > we apply posiive homogeneiy: α min Q = sup X A ρ E Q X] = sup λx A ρ E Q λx] = λ sup λx A ρ E Q X] = λ α min Q. This is rue for all Q M 1,f and λ >. Hence α min can only ake values and +. Now we focus our aenion o he convex risk measures ha admi a represenaion in erms of he σ-addiive probabiliy measures. Such measures 24

25 ρ can be represened by a penaly funcion α which is infinie ouside he se M 1 := M 1 Ω, F. ρx = sup EQ X] αq } Q M 1 Lemma 4.2 FS 166] If a convex risk measure admis he represenaion, hen i is a coninuous from above: X n ց X = ρx n ր ρx. b Propery a is equivalen o lower semiconinuiy wih respec o bounded poinwise conergence. If X n } is a bounded sequence in X which converges poinwise o X X, hen ρx lim inf n ρx n. Proof We show ha a, hen a b and finally b a. a X n X is a bounded sequence, so by dominaed convergence, EX n ] EX]. For all Q M 1 ρx = sup Q M 1 lim n E Q X n ] αq } lim inf n = lim inf n ρx n sup EQ X n ] αq } Q M 1 We use he liminf because we don know if we have convergence. b a For X n ց X we have ρx n ρx, since ρx n ρx n+1 ec. This means: lim n ρx n ρx bu as we verified in he firs sep: ρx lim inf n ρx n, hus ρx = lim n ρx n a b X n X is a bounded sequence. If we define Y m := sup n m X n we have 25

26 a decreasing sequence and we know ha Y m ց X. By monooniciy, Y n X n ρy n ρx n, so since Y n ց X: ρx = lim n ρy n lim inf n ρx n Proposiion 4.21 FS 167] Le ρ be a convex risk measure which is coninuous from below: X n ր X = ρx n ց ρx. Le α be some penaly funcion in ρx = max M 1,f EQ X] αq }. Then α is concenraed on he class M 1 of probabiliy measures, i.e αq < = Q is addiive. Proof Resul follows since we know Q M 1,f, which is all we need. Q is σ-addiive A n ր Ω : QA n ր QΩ = 1 Lemma 4.22 FS 167] No proved Le ρ be a convex risk measure wih he usual represenaion: Now consider he se: ρx = max Q M 1,f EQ X] αq }. Λ c := Q M 1,f αq c }, Q M1,f αq < } = c Λ c for c > ρ = inf Q M1,f αq >. Then for any sequence X n } in X where X n 1, he following wo saemens are equivalen. ρλx ρλ, λ 1 26

27 Skech of proof For all λ, inf E Q X n ] 1, c > ρ. Q Λ c ρλ½ An ρλ½ Ω = ρλ. By monooniciy, QA n QA n+1, so inf QA n inf QA n+1, Q Λ c Q Λ c and we have a growing sequence. By monoone convergence, inf QA n ր 1, c = QA n ր 1. Q Λ c Remark: If ρ is convex and coninuous from below, hen, by Proposiion 4.21: ρx = sup EQ X] αq } Q M 1 and hen, by Lemma 4.2, ρ is coninuous from above, which implies ha for all bounded sequences, we can use he dominaed convergence heorem, so X n X means ha ρx n ρx. From now on, we make some assumpions: Ω Separable meric space F = BΩ he Borel σ-algebra on Ω. X linear space of bounded funcions on Ω, F C b Ω X Bounded, coninuous funcions on Ω. Proposiion 4.25 FS 168] If ρ : X R is a convex risk measure on X such ha ρx n ց ρλ for any poinwise convergence X n ր X, where X n C b Ω for all n and λ >, hen here exiss a penaly funcion α : M 1 R } such ha ρx = sup Q M 1 EQ X] αq }, X C b Ω. General represenaion rue for a larger se We can choose α: ρx = max Q M 1 EQ X] α min Q }, X X αq = inf αmin Q }, eq M 1,f E eq Y ] = E Q Y ], Y C b Ω 27

28 The equaliy beween he expecaions is a special kind of equaliy, similar o equaliy under disribuions. Recall: α min Q = sup E Q X] = sup EQ X] ρx } X A ρ X X 4.3 Convex Risk Measures on L FS 171] From now on we assume we have a fixed probabiliy measure P on Ω, F, idenify X as L and consider risk measures ρ such ha ρx = ρy, if X = Y P a.s Lemma 4.3 FS 172] If ρ is a convex risk measure saisfying 4.28, and i has he represenaion ρx = sup Q M 1,f EQ X] αq }, hen αq = + for any Q M 1,f Ω, F which is no absoluely coninuous wr P. Q is abs. con. o P, Q P if PA = = QA =. Proof Assume Q P, hen A F such ha PA = and QA >. Take some X A ρ = X X ρx = }, and consruc a sequence X n := X n ½ A, so X n = X P-a.s, which means ρx n = ρx. ρx n = sup Q M 1,f EQ X n ] αq } E eq X n ] α Q = E eq X] + n } QA α } Q = > α Q EQ e X] + n QA The same argumen applies o he represenaion of ρ via α min α min Q = sup X A ρ EQ X] } E e Q X] + n QA = α Q α min Q E eq X] + n QA n 28

29 Theorem 4.31 FS 172] Suppose ρ : L R is a convex risk measure. Then he following saemens are equivalen. a ρ can be represened by some penaly funcion on M 1 P M 1, where M 1 P = Q Q P } b ρ admis he represenaion ρx = EQ X] α min Q }, X L sup Q M 1,f P where α min M1 P = α min. c ρ is coninuous from above: if X n ց X P-a.s, hen ρx n ր ρx. d The Faou propery. For any bounded sequence X n poinwise convergence: ρx lim inf n ρx n e ρ is lower semiconinuous for he weak opology on L. f The se A ρ L is weak closed. Proof We will prove b a c d e f b. b a. This is obvious. X P-a.s a c. Recall Lemma 4.2 par a. For ρ : X R and ρx given as in Lemma 4.2, we have coninuiy from above. c d. Lemma 4.2 par b. Coninuiy from above is equivalen o ρx lim n ρx n for all X n X where X n } is bounded. P-a.s. d e. We have lower semi-coninuiy for ρ if c R he se C, X L ρx c} := C is closed in he weak sense. For weak convergence we mus specify closedness. Noe ha C is convex, since for X, Y C and some λ, 1] using ha ρ is convex: ρ λx + 1 λy λρx + 1 λρy λc + 1 λc = c 29

30 = λx + 1 λy C The se C is weak closed if r >, he se C r, C r := C Y L Y r } L is closed in L 1. Take a sequence X n C r such ha for all n, X 1 n X. We wan o show ha X C r, which means i is closed. Every X n is bounded, so he limi X is also bounded, and hence X Y L Y r }. Now for L 1 convergence, we have a subsequence ha converges P-a.s.in a poinwise sense: X nk } k X n } n such ha X nk X. Then propery d k ells us: ρx lim inf ρx n k c = X C. k This proves ha C is a closed se. e f. For A ρ : X ρx } jus ake c = in C and we are done. f b. Given a X L, and defining m := sup EQ X] α min Q }. Q M 1 O We wan o show ha m = ρx and do ha by showing ha ρx m and hen ρx m. For he firs inequaliy: ρx = max Q M 1,f EQ X] α min Q } sup E Q X] α min Q } = m. Q M 1 P For he oher inequaliy, we use he represenaion: ρx = inf a R X + a A ρ }. Wih his we wan o show ha ρx m, which by cash invariance amouns o ρx + m which is equivalen o showing ha X + m A ρ. For a conradicion we assume m + X A ρ. Since ρ is a convex measure, we know ha A ρ is a convex se. By our assumpion ha d is rue, we also have ha A ρ is weak closed. 3

31 We define he single poin se B = X +m}, and by our assumpion B A ρ, so B A ρ =. By applying he Hahn-Banach Theorem Thm A.56 FS 429], l : L R which is non-zero, linear and coninuous such ha sup lx + m < inf ly abbreviaed as γ < β. B Y A ρ we also know ha < γ. We always have he represenaion ly = EY Z] for Y L and Z L 1, where Z, P-a.s., and PZ > > 1. Now ake Y and λ >, which means: λy λ = ρλy ρ = ρ λy +ρ = λy +ρ A ρ By, γ < l λy + ρ = λly + lρ = γ λ = ly + lρ λ and when passing o he limi: λ, ly = EY Z]. Now define Q M 1 P, given by dq = Z, which implies dp EZ] Z ] E Q Y ] = E Y EZ] = β = inf Y A ρ ly = inf Y A ρ E Q Y ] EZ] = β EZ] = sup Y A ρ E Q Y ] = α min Q Hence, wih he sric inequaliy a consequence of : E Q X] + m = lx + m EZ] < β EZ] = α minq = m < E Q X] α min Q This is a conradicion, since by definiion: m = sup Q M 1 P EQ X] α min Q }. Our assumpion ha m + X A ρ was false, so ρx m and we have proved ρx = m. 31

32 Enropic Risk Measure For risk measures Q and P we have he relaive enropy: E Q log dq ] ] dq dq = E log if Q P HQ P = dp dp dp + oherwise We have HQ P = if Q = P. For fx = x log x Jensen s inequaliy. For β > we se: αq = 1 β HQ P. Wih his we have risk measure for when we srongly believe ha P is he correc measure. ρx = sup EQ X] 1 Q M 1 P β HQ P} An alernaive represenaion of he expression in, bu one ha isn very imporan o us is HQ P = = sup Z L EQ Z] log Ee Z ] }. If we se Z = βx in his expression, and remove he supremum, we have: HQ P E Q βx] log Ee βx ] 1 β HQ P 1 β E Q βx] 1 β log Ee βx ] 1 β HQ P E Q X] 1 β log Ee βx ] 1 β log Ee βx ] E Q X] 1 β HQ P The lef side is now independen of Q, so we can ake he supremum on he righ. 1 β log Ee βx ] sup EQ X] 1 Q M 1 P β HQ P} = ρx For a paricular choice: d Q dp = e βx Ee βx ] 32

33 we derive from : ] H Q P d Q d Q = E log dp dp ] e βx e βx = E log Ee βx ] Ee βx ] e βx ] = E log e βx log Ee βx ] Ee βx ] e βx = E βx log Ee βx ] ] Ee βx ] and so, E bq X] = E X d Q ] = E X dp ] e βx Ee βx ] EQ b X] 1 β H Q P = ] e βx E X 1β e βx Ee βx ] E βx log Ee βx ] ] = Ee βx ] ] e βx e βx 1 E X E X Ee βx ] Ee βx ] β log Ee βx ] ] = e βx E X Ee βx ] + e βx X Ee βx ] + e βx 1 Ee βx ] β log Ee βx ] ] = e βx 1 ] 1 ] E Ee βx ] β e βx = E bq β log Ee βx ] = 1 β Ee βx ] The Q-expecaion disappears because he inerior is jus a number. Summarising, we have: E bq X] 1 β H Q P = 1 β Ee βx ] Taking he supremum over all Q on he lef side, we ge he reverse inequaliy from, so: ρx = 1 β log Ee βx ] The logarihm is no a linear funcion, so for λ we have ρλx λρx. The measure is in fac normalized, bu i is no coheren since i is no homogeneous. 33

34 For he represenaion: ρx = sup EQ X] α min Q } Q M 1 P we see from equaion ha α min = 1 β HQ P. By definiion X = L : α min Q = sup X A ρ E Q X] = sup X X EQ X] ρx } = sup X L EQ X] 1 β log Ee βx ] } = 1 β HQ P Definiion 4.32 FS 173] A convex risk measure is sensiive or relevan w.r. P if: ρ X > ρ X L such ha X P-a.s., PX > >. Corollary 4.35 FS 175] For a coheren risk measure ρ on L, he following condiions are equivalen. a ρ is coninuous from below: X n ր X P-a.s. = ρx n ց ρx. b There exiss a se Q : Q M 1 P such ha ρx = sup E Q X]. Q Q c There exiss a se Q M 1 P represening ρ such ha he se of densiies: dq } D := Q Q dp is weakly compac in L 1 P. So a coheren risk measure on L can be represened on Q M 1 P if, and only if he condiions of his corollary are me. We can use: Q max := Q M 1 P α min Q = }. 34

35 Recall: M 1,f denoes he se of measures ha sum o 1, and are finiely addiive. M 1 denoes he se of measures ha sum o 1 and are σ-addiive. X denes he se of bounded random variables. If ρ is a convex risk measure ρx = max Q M 1,f EQ X] α min Q } wih α min Q = sup X Aρ E X] for finiely addiive probabiliy measures. ρx = sup EQ X] αq } Q M 1 for σ-addiive probabiliy measures. This implies coninuiy above, which acually is equivalen o he Faou propery: ρx lim inf ρx n, and since n X n } is a bounded sequence, we have P-a.s convergence X n X. I is imporan o know wha kind of convergence you have! Convergence In he box, he probabiliy measure P is fixed. For he -implicaion we use dominaed convergence or monoone convergence. For he implicaion we use uniform inegrabiliy. In general we do no have equivalence beween he convergence ypes. For our applicaions, convergence in measure is convergence in probabiliy measures. Finally, he weakes kind of convergence: convergence in disribuion, is on ha does no depend on he measure space. For a risk measure X = bounded r.v}. ρx R ha is convex P equiv.? we have he represenaion ρx = sup Q M 1 P EQ X] αq } 35

36 where M 1 P = Q prob. measure on F Q P }. For convex risk measures on L we have a represenaion propery, lower semiconinuiy: ρx lim inf ρy Y X for a weak convergence. We will now consider spaces X = L p for p 1, ] and is dual space: X = l X R are linear coninuous }. For X = L p for p 1,, he dual is X = L q where 1 p + 1 q = 1. For X = L 1, he dual is X = L. For X = L, he dual is X bap, which are he bounded, signed, finie, addiive measures ha are absoluely coninuous wr P. In general L 1 bap. Riesz Represenaion Theorem For p 1,. Le l : L p R be linearly coninuous hen l L p, he dual space. Then here exiss a unique g L q L q = L p such ha ellx = gxdp, X L p = EgX] where g is unique P-a.s. All linear funcionals have he form of he expecaions. For p =. Le l : L R be linearly coninuous. Then µ bap such ha lx = Xdµ. we will use he weak opology as a convenion. Expecaions can be wrien as linear funcion and vice versa. Resul The convex? measure ρ : X R admis he represenaion } ρx = sup lx αl l P where α : X R } is convex, and P X is convex. 36

37 Theorem a If ρ : X R is convex and lower semiconinuous, i.e for all c R, C = X : ρx c } is a closed se. Then b The converse is rue. ρx = sup l X lx αl }, X X. Proof Skech Le c R, and C = X L p = X ρx c } Lp. Since ρ is convex, hen so is C. } } C = X L p sup lx αl l X } } = X L p sup EgX] αl g L q L p Take a sequence X n } C for all n and we consider if X n X. If X C, hen C is closed. EgX n ] αl g L q ρx n c, n Passing o he limi: Take he sup and we are done. EgX] αg c g Proposiion If ρ : X R is convex, lower semiconinuous and ρx ρ, X monooniciy, hen where } ρx = sup l X αl, X X X + X + = l X lx >, X } l =. 37

38 Remark: If ρ is convex and lower semiconinuous, hen is equivalen o monoone: ρx ρy, X Y. The converse is also rue. From now on, we will regard X as L p spaces for p 1, ] Proposiion 1 If ρ : X R is convex, lower semiconinuous and X ; ρx ρ, hen } ρx = sup l X αl, X X. l P P = X + = l X lx, X }. The converse is also rue. X + is he dual. The l s are linear funcionals which are relaed o he expecaion. l X = EY X] = E Q X]. See Riesz represenaion heorem, and he Radon-Nikodym heorem. 2 If ρ : X R is convex, lower semi-coninuous and ρ =, hen ρx = sup l X = sup E Y X] l X Y L q where The converse is also rue. X = l X R linear and con. }. Revision: We wrie L p L q, which is idenifiable equaliy. From he definiion of he dual space, we have L q and is dual is: L p = L q when q is he conjugae of p. For l L q, we have l : L p R. For he random variable Y : Ω R, where EY ] <, we have lx = EY X]. By a represenaion heorem, here is a one-o-one correspondene: l = EY ]. Proof of Proposiion 1 Since ρ is convex and lsc lower semiconinuous we have he general represenaion We resric his o finie α s. ρx = sup X X l X αl }, X X. ρx = sup X X,αl< l X αl } = 38

39 From X, ρx ρ, we ge, for all l X and using ha αl < : ρx ρ = l X αl ρ = l X αl + ρ < X = l X X Mus be negaive for his o be rue for all X. = lx = l X + = } = sup l X αl l X + From he shape of he represenaion, we know ha ρ : X R is convex and lsc. I remains o show ha ρx ρ. 2 ρx = sup l X + X = lx = l X = } l X αl sup l X + ρx = } αl = inf αl =: ρ. l X + } sup l X αl = supl X} l X,αl< Used = ρ = sup αl}. Recall ha ρ is always represened by he penaly funcion. The equaliy is verified by showing and. Since we are working wih he sup, his is quie obvious. ρx = sup l X l X,αl= If ρ is lsc, convex and posiive homogeneous, we ge and ρ =. If ρ is lsc, convex, ρ = and sub addiive, we ge and posiive homogeneiy. If ρ is lsc and sub-linear i.e sub-adiive and posiive homogeneous, hen we ge, ρ = and convexiy. For he properies convex, posiive homogeneous and sub-addiiviy, wo of hese combined wih ρ = imply he hird one. Consider ρx ρy, X Y and ρx ρ, X We immediaely have, bu for we need convexiy and lsc. 39

40 6 Backwards Sochasic Differenial Equaions Ph 139] Preliminaries/Revision A complee probabiliy space Ω, F, P, is a space where A F, where PA =, and C A hen C F and an obvious observaion is ha PC =. An probabiliy space is P-augmened if all he P-null ses belong o F For a general space Ω, F, P, if A F and PA =, hen A F bu C A C F. An example: Ω = a, b, c, d}, we can have F =, a, b}, c, d}, Ω }, hen a, b} F bu a F. Augmened and compleeness are no he same conceps. For a measure space Ω, F, P, wih he filraion F and he index se T e.g T =, T], we have he process X } T, where we wrie X : Ω T R d or X : T r.v. : Ω R}. Imporan properies he process can have: 1 Measurabiliy: A BR d, X 1 A F B T. 2 Adapedness: T, X is F -measurable. How randomness is moving wih ime. 3 Progressive Measurabiliy: A R d and T, ω, s Ω, ] Xω, s A } = X 1 A Ω, ] F B, ] Anoher way o see his is: X,] : Ω, ] R d. Noe: for R, τ R, τ, he opology τ is X 1 O τ. The σ-algebra is generaed by he opology, and is bigger han he opology. BR = στ}. Propery 3 implies 1,2, bu 1,2 does no imply 3! However if X saisfies 1,2 hen here exiss a Y, which is a modificaion of X, ha sasifies 3. If X is 2 and is eiher righ or lef coninuous, hen i saisfies 3. Predicabiliy used in inegraion P = A s, ] A Fs, s, : s } For ½ A s,] ω, u = ½ A ω ½ s,] u, hen A F s. For simple funcions φu = N e i ω½ i 1, i u i=1 4

41 If e i ω F i 1 -measurable, e i ω is lef coninuous. Random Time We define he concep of random ime as a random variable τ : Ω, ]. We call random ime a sopping ime if:, τ } F he sopping depends on he filraion. We call random ime for opional ime if:, τ < } F An opional ime implies i is a sopping ime. A sopping ime is opional if he filraion F is righ coninuous. We say ha F is righ coninuous if F = F + := u> F u. This is in general rue for he filraion, since he informaion does no make jumps. If τ and σ are sopping imes, hen minτ, σ and supτ, σ are sopping imes, and i is lef as an exercise o verify ha τ + σ is a sopping ime. Local Maringales - erminology and noaion from Karazas & Shreve. Definiion: A local maringale, denoed wih M for a finie or infinie ime horizon T is such ha τ n } n of F-sopping imes, τ n ր P-a.s as n such ha n M < τ M τn := n is an F-maringale. M τn τ n A local maringale produces an infinie amoun of local maringales. From he definiion of M, i follows ha i is 1 adaped, 2 M L 1 P for all and 3 EM F s ] = M s. Maringales or Local Maringales? 41

42 Uniform Inegrabiliy K&S, Pham and Øksendal appendix For Y } T, lim E Y ½ Y >c}] = Is E Y ½ Y c}? sup c For M,, T] we can wrie M = EM T F ]. For M wih, we can wrie M s = EM F s ] for any fuure, bu in general we can no wrie M s = EM F s ], since M may no exis. If M exiss and lim M = M, we can wrie i. If he maringale converges P-a.s and in L 1 P, i has righ coninuous rajecories wih lef side limis, and we have uniform inegrabiliy. This isn rue for he Girsanov ransform Z? Z Z as, bu Z EZ F ]. - If M is a local maringale and hen M is also a maringale. Esup M s ] <,, s - M is a local maringale, M L 1 P and M is non-negaive, hn M is a super maringale. - If M is a coninuous local maringale, M = and M has finie oal variaion: n Mi M i 1 <, sup 1,..., n i=1 hen M. In Io calculus we have he imporan equaliy db 2 = d, which is a resul from he quadraic variaion. The noaion for local maringales is d < B >. Burkholder-Davis-Gundy BDG For all p >, here exiss consans c p, C p > such ha for all coninuous local maringales M, and for all sopping imes τ < P-a.s., we have he inequaliies: p] c p E < M 2 > τ E sup M s ] p C p E s τ p] < M > τ This inequaliy gives a sufficien condiion o deermine if M is a maringale. 42 2

43 Doob Assume M for, T] or, is a non-negaive submaringale. Then τ sopping imes, τ < P-a.s we have P sup τ E sup M ] p τ M λ EM τ] λ, λ > p pe Mτ p ], p > 1. p 1 Gives us an upper bound on he supremum, which we do no know. Simple inegrals Elemenary inegrals 1 φ n = i e i ½ i 1, i ] where e i is F i 1 -measurable and bounded. The se of such funcions are denoed by V, T] in Øksendals book. From he sep funcion we define he simple inegral: I φ n = e i B i. i 2 The se I φ n } n are all in L 2 P. 3 φ V, T], here exiss φ n such ha φ n φ in L 2 P d. 4 I n φ n } is a Cauchy sequence. Banach spaces L 2 are complee, so he limi exiss. A common assumpion: ] E φ 2 d < bu for many applicaions his is oo srong, for insance when we work wih local maringales. A way o relax his assumpion is: P φ 2 d < = 1, in Øksendal s noaion his is he assumpion for he se W, T]. Under his weaker assumpion we loose he Io isomery. Insead we have ψ n φ, a P- a.s. poinwise convergence, and bound i wih a sopping ime. Iψ n Iφ, and I φ = φ s db s. If φ V, T] we have he sandard Io consrucion. If φ W, T] his is a local maringale and we don know if i is inegrable. 43

44 On a probabiliy space Ω, F, P, we have he Brownian moion W for, T], T < and he filraion F = F W which is he filraion generaed by Brownian moion, F = F T }. For a maringale M L 2 P for all, we have, by he maringale represenaion heorem: M = M + φ s dw s for some φ V, T] and EM ] = M. By he maringale propery we also have: M = EM T F ],. Now consider he equaion dm = d φ dw M T = ξ We can find he soluion o his wih he maringale represenaion heorem MRT. M = Eξ] + M = Eξ F ] φ s dw s If we exchange in he equaion wih fm, φ called he driver, we have a BSDE. Soluions o BSDE s consis of a process M and he funcion φ. We solve BSDE s wih he MRT and sochasic differeniaion. For ξ L 2 P being F -measurable, φ = D ξ a non-anicipaing derivaive. If ξ D 1,2 L 2 P, hen φ = D ξ = ED ξ F ].? They are called Backwards because we know he erminal poin M T. 44

45 BSDE s Ph 139] We have he Brownian moion W, he probabiliy space Ω, F, P wih he filraion generaed by Brownian moion F = F W. For he heory of BSDE s, we always work wih a finie, fixed ime horizon T =, T], T <. Two imporan spaces are: S 2, T, he se of real valued, progressively measurable processes Y such ha E sup Y 2] <, T and H 2, T, he se of R d -values, progressively measurable processes Z such ha ] E Z 2 d <. For each BSDE we are given a pair ξ, g, where ξ is he erminal condiion and he funcion g is called he driver or he generaor. Assumpions we make: A ξ L 2 F T B g : Ω, T] R R d such ha: g,, y, z g, y, z is progressively measurable for all y,z g,, H 2, T g saisfies a uniform Lipschiz coninuiy in y, z: There exiss a consan C g such ha g, y1, z 1 g, y 2, z 2 y1 Cg y 2 + z1 z 2 P-a.s., d-a.e for y 1, y 2 R and z 1, z 2 R d. A BSDE is an equaion on he form: dy = g, Y, Z d Z dw Y T = ξ Definiion Ph 14] A soluion o a BSDE is a pair Y, Z S 2, T H 2, T saisfying: Y = ξ + gs, Y s, Z s ds Z s dw s, T. 45

46 Theorem Exisence & Uniqueness Ph 14] Given a BSDE wih he pair ξ, g saisfying he assumpions above, here exiss a unique soluion Y, Z. Proof We have he funcion Φ : S 2, T H 2, T S 2, T H 2, T, for U, V ΦU, V. We define he maringale M := E ξ + gs, U s, V s ds ] F which, under he assumpions in A,B, is square inegrable: E M 2 ] < for all. By he MRT we have M = M + Z s dw s, Z H 2, T. I follows from he MRT ha Z exiss and is unique. Nex we define Y. Similarly, Y := E ξ + = E ξ + = E ξ + = E ξ + = M = M + gs, U s, V s ds F ] gs, U s, V s ds ± gs, U s, V s ds gs, U s, V s ds F ] gs, U s, V s ds Z s dw s gs, U s, V s ds F ] gs, U s, V s ds F ] gs, U s, V s ds gs, U s, V s ds Y T = ξ = M T = M + gs, U s, V s ds Z s dw s gs, U s, V s ds 46

47 Now we coninue wih : = M + Z s dw s gs, U s, V s ds ± = M + Z s dw s gs, U s, V s ds } } = ξ Z s dw s + = =Y T =ξ gs, U s, V s ds Z s dw s ± Z s dw s + gs, U s, V s ds gs, U s, V s ds which is he general form of he soluion o a BSDE. The nex sep is o show ha Y S 2, T. Use ha a + b + c 2 5 a 2 + b 2 + c 2. E sup Y 2 ] 5Eξ 2 ]+5E T E sup T ] gs, U s, V s 2 ds +5E sup T Z s dw s 2] We wan o deermine if his is finie. The wo firs erms are finie, as he firs erm is a consan in L 2 P and he second erm is a coninuous funcion on a finie inerval. The hird erm is unknown, bu we can find an upper bound by using Doob s inequaliy. sup T Z s dw s 2] 2 2 E 2 ] ] Z s dw s = 4E Z s 2 ds <. 2 1 The square of a maringale is a sub-maringale. The hird erm is also finie, so Y S 2, T. The final sep is o verify ha we have a conracion. For he pairs U, V and U, V, we define U = U U, V = V V and g = g, U, V g, U, V. For some consan β > ha we deermine laer, we apply Io s formula produc formula wih some subsiuions o e βs Y 2 s, for s T: d e βs Y 2 s = βe βs Y 2 s ds 2eβs Y s g s ds + 2Y s e βs Z s dw s + e βs Z 2 s ds e βt Y 2 T Y 2 = e βs βy s 2Y s g s + Z 2 T s ds + 2 e βs Y s Z s dw s EY 2 ]+E e βs βy 2 ] s +Z2 s ds = E ] 2e βs Y s g s ds +E 2 ] e βs Y s Z s dw s } } = The las erm should be, which is he case if Y s Z s is a maringale, which we will show separaely. For now we simply assume ha is. For he remaining 47

48 erms we work wih he righ hand side of he equaliy. Since g is consrained since i is Lipschiz, we ge: ] ] E 2e βs Y s g s ds 2C g E e βs Y s Us + V s ds Using ha ab a2 2 + b2 2, for a = 2 2 C g Y s and b = U s + V s / 2: ] 4Cg 2 E e βs Y 2 s ds E e βs ] U s + V s ds Now we choose β = 1 + 4Cg 2, and obain: EY 2 T ] + E e βs ] Y s + Z s ds 1 2 E e βs U 2 s + V 2 ] s ds This shows ha Φ is a sric conracion on he Banach space S 2, T H 2, T endowed wih he norm: Y, Z β = E e βs ] Ys 2 + Zs 12 2 ds. For a conracion in a complee space, we immediaely ge uniqueness and exisence. To conclude he proof, we have o verify ha he erm E 2 e βs Y s Z s dw s ] is zero, which we do by applying he Burkholder-Davis-Gundy BDG inequaliy, and showing ha i is finie: E < ] eβs Y s Z s dw s > < for all. We sudy he quadraic variaion by he Doob-Meyer decomposiion, and show ha he quadraic variaion is finie. We use ha Y S 2, T which, incidenally is he reason why we require ha assumpion, and use he produc o sum inequaliy ab a 2 + b 2 /2. E e 2βs Y 2 s Z2 s ds E sup T e β 2 Y Z 2 s ds eβt 2 E sup Y 2 T }} S 2,T + ] Z 2 s ds < }} H 2,T This is finie, and his is a sufficien condiion for a local maringale o be a maringale, and hence he inegral is and he proof is concluded. 48

49 Linear BSDE We coninue using he spaces S 2, T and H 2, T. In he Peng aricle, we generally work wih adaped processes Y, while we work wih progressively measurable spaces in Pham, so Pham is more rigorous han Peng. As we saw in he exisence and uniqueness resul, we know ha he BSDE is well defined when he pair ξ, g is given final condiion and a driver. We recall ha we have cerain assumpions on he driver g, denoed by B. Someimes we wrie he Lipschiz condiion: g, y 1, z 1 g, y 2, z 2 C g y1 y 2 + z 1 z 2 where C g = maxν, µ. g, y 1, z 1 g, y 2, z 2 ν y 1 y 2 + µ z 1 z 2 Digression Consider he BSDE given by: dy = g, Y, Z d Z dw Y T = ξ where ξ L 2 Ω and ξ is F T -measurable. Wha happens if ξ is F measurable for some < T? This means ha a some poin we reach he final condiion, which we already know. An exercise: If ξ is F -measurable, prove ha he soluion applies o he inerval, T]. The soluion is a couple Y, Z S 2, T H 2, T. Remember ha he soluion of a BSDE is on he form: Y = E ξ + gu, Y u, Z u duf ]. Since he inegral goes from o T, his is no a maringale. We ge a proper maringale no jus a local maringale if we define M = E ξ + gu, Y u, Z u duf ]. Due o he assumpions we made on g lised in A, g is inegrable from o T. From g you can ge properies on Y. 49

50 Linear BSDEs are BSDEs where we have he ime dependen coefficiens A, B, C, and have he form: dy = A Y + B Z + C d Z dw Y T = ξ Boh A and B are bounded, progressively measurable processes, and C H 2, T. Proposiion Ph 142] The unique soluion Y, Z o he linear BSDE is given by Γ Y = E Γ T ξ + Γ u C u du F ],, T] wih he corresponding linear SDE, also called he adjoin or dual SDE. dγ = Γ A d + Γ B dw Γ = 1 By solving he SDE, you also solve he linear BSDE. Proof By Io s produc formula: dγ Y = Y dγ + Γ dy + dγ dy = Γ Y A d + B dw Γ A Y + Z B + C d + Γ Z dw + Γ Z B d = Γ C d + Γ Y B + Z dw = Γ Y = Γ Y Γ u C u du + Γ u Yu B u + Z u dwu Now, is his a local maringale? Rewrie he equaion: Γ Y + Γ u C u du = Γ Y + Γ u Yu B u + Z u dwu and recall ha Γ = 1. Now we apply he BDG-inequaliy for p = 1, and ge an esimae of he sup. Look a he quadraic variaion hink of Io isomery. See definiio of quadraic variaion which we can use for sochasic inegrals: ] E < Γ u Yu B u + Z u dwu > = E 2du Γ 2 u Yu B u + Z u 5