Enlargement of Filtration and Insider Trading

Transcription

1 Enlargemen of Filraion and Insider Trading A. Es-Saghouani Under supervision of Dr. P.J.C. Spreij Faculei der Nauurweenschappen, Wiskunde en Informaica, Koreweg-de Vries Insiue for Mahemaics, Planage Muidergrrach 24, 8 TV Amserdam, The Neherlands A hesis submied for candidacy for he degree of Maser of Mahemaics Amserdam, 6 January 26

2 Nederlandse Samenvaing He onderwerp van deze scripie is Vergroing van Filraies en Handelen me Voorkennis. De meese financiële verschijnselen worden besudeerd me behulp van de heorie van maringalen. Een belangrijk onderdeel van deze heorie is de filraie, da is een sijgende rij van σ-algebra s. In een financiële mark correspondeer een σ-algebra me alle publieke informaie die beschikbaar is voor alle handelaren o en me een ijdsip. De bedoeling van deze scripie is een kore inleiding e geven in he vergroen van een filraie en hoe di gerelaeerd is aan he handelen me voorkennis. Wij doen di onder de aanname da de handelaar me voorkennis (ies meer informaie o zijn beschikking heef dan de gewone handelaar, da kan bijvoorbeeld zijn de prijs van een aandeel in de oekoms. In wiskundige ermen is die exra informaie gegeven als een sochasische variabele. Wij kijken naar de filraie gegenereerd door de oorspronkelijke filraie en de σ-algebra gegenereerd door de sochas. We besuderen hoe de objecen van de oorspronkelijke filraie erui zien in de groe filraie. Onder andere besuderen we hoe he probleem van he maximaliseren van de nusfuncie van de wee handelaren kan worden opgelos, en we kijken ook naar he verschil ussen de wee. i

3 Absrac We consider a probabiliy space (Ω, F, P equipped wih wo filraions F = (F and G = (G = F σ(g, where G is a random variable aking values in a Polish space. We give a condiion on G such ha every F-semimaringale remains a G-semimaringale. Then we ransfer maringale represenaion heorems from F o G. We hen use hese heorems o solve he problem of maximizing he expeced logarihmic uiliy for an invesor having he filraion G a his disposal, and rewrie his addiional expeced logarihmic uiliy, wih respec o an invesor hoe has only he filraion F a his disposal, in erms of relaive enropy. A las we give anoher approach for he problem of enlargemen of filraions. ii

4 Conens Nederlandse Samenvaing Absrac Inroducion i ii iv Enlargemen of filraions. Iniial enlargemen of filraions (I.E.F Sochasic exponenial represenaion of he process /p G Examples Maringale represenaion heorems for I.E.F 2. Noaions The maringale preserving probabiliy measure Maringale represenaion heorems for I.E.F Insider rading and uiliy maximizaion 2 3. The ordinary invesor problem Uiliy Maximizaion The model Soluion of he logarihmic uiliy maximizaion problem Insider s addiional expeced logarihmic uiliy Explici calculaions of he insider s expeced logarihmic uiliy Enlargemen of filraions and Girsanov s heorem Preliminaries and noaions Girsanov-ype for change of filraions Conclusion A Some Imporan Theorems and Lemma s 38 A. Appendix Chaper A.2 Appendix Chaper A.3 Appendix Chaper iii

5 Inroducion In he pas decades an exensive mahemaical heory using maringale echniques has been developed for he problems of characerizaion of no arbirage, hedging and pricing of financial derivaives and uiliy maximizaion of invesors in financial markes. One of he imporan feaures of his heory is he assumpion of one common informaion flow (filraion on which he porfolio decisions of all economic agens are based. In his hesis we give a shor inroducion o enlargemen of filraion and insider rading. We do his by considering a financial marke ha is a probabiliy space (Ω, F, P equipped wih a filraion F = (F [,T (public informaion. While he ordinary rader makes is decisions based on he informaion flow (F [,T, he insider possesses from he beginning exra informaion abou he oucome of some random variable G aking values in a Polish space (U, U, foe example, he fuure price of a sock. The insider s informaion flow is herefore described by he enlarged filraion G = (G [,T = (F σ(g [,T. This hesis is based on aricles of Amendinger [, 2, 3, Jacod [7, 9, Pikovsky and Karazas [3 and Ankirchner [4. The problem of he enlargemen of filraions consiss of he following hree imporan issues:. Give condiions on he random variable G such ha every F-semimaringale become a G-semimaringale. 2. If par is saisfied, give he decomposiion of G-semimaringales. 3. If a version of he maringale represenaion holds under he filraion F, find a version of he maringale represenaion heorem wih respec o he enlarged filraion G. For he insider rading, we will be ineresed in is expeced logarihmic uiliy maximizaion and give some example where we obain explici formulae for he uiliy gain. The ouline of his hesis is as follows. In Chaper we give mos of he resuls of Jacod [9 and Amendinger [3. For his we fix a ime horizon T. If he regular condiional disribuions of he random variable G given F, [, T are absoluely coninuous wih respec o he law of G, Jacod [9 proved ha every (P, F-semimaringale remains a (P, G-semimaringale on he inerval [, T, and gave he canonical decomposiion of (P, F-semimaringales in G which involves he condiional densiy process q l, l U. For mos of oher resuls we will assume he equivalence beween he regular condiional probabiliies of G given F, [, T and he uncondiional law of G. Based on his assumpion we give also an exponenial represenaion of heir condiional densiy process p G. In Chaper 2, we give sufficien condiions such ha he exisence of maringale measures, under which he sock price process S is a maringale, in he filraion F implies heir exisence iv

6 Enlargemen of Filraion & Insider Trading v in he enlarged filraion G. Moreover, we show ha he densiy process of an equivalen G- maringale measure ha decouples he σ-algebras F and σ(g is he produc of he densiy process of an equivalen G-maringale measure and he process /p G. And we use hese wo resuls o show he inheriance of he maringale represenaion heorem in he enlarged filraion G. In Chaper 3, we rea he insider s problem of maximizing his expeced logarihmic uiliy. In his case we give he opimal sraegy and also he expression of he uiliy gain. We esablish also a relaionship beween he addiional expeced logarihmic uiliy of he insider and he relaive enropy of he probabiliy measure P wih respec o he probabiliy measure P defined on (Ω, G T by he process /p G. In Chaper 4 we rea anoher aspec of he problem of enlargemen of filraions, sudied by Ankirchner [4. He considered enlargemen of he filraion F by an oher filraion K, his is done by sudying he filraion G = {G := s> (F s K s, }, and he replaces Jacod s condiion by he a condiion inspired from he noion of he decoupling measure. The idea is ha he enlargemen of he filraion can be inerpreed as a change from he decoupling measure o he original measure. Then Girsanov s heorem is used o obain he semimaringale decomposiion relaive o he enlarged filraion G. The noaions in his hesis will be he same for all he chapers.

7 Chaper Enlargemen of filraions. Iniial enlargemen of filraions (I.E.F Le (Ω, F, P be a probabiliy space wih a filraion F = (F [,T saisfying he usual condiions, i.e., he filraion F is righ-coninuous (F = ɛ> F +ɛ and each F conains all (F, P-null ses. T (, is a fixed ime horizon, and we assume ha he σ-algebra F is rivial. Le G be an F-measurable random variable wih values in a Polish space (U, U. Definiion... A coninuous adaped sochasic process X is called a semimaringale if i has a represenaion of he form X = X + M + A. Where M is a coninuous local maringale and A is a coninuous adaped process of locally bounded variaion and M = A =. Definiion..2. Given he noaions above, we call he filraion G defined by : he iniially enlarged filraion of F. G := F σ(g, [, T In his chaper we will assume ha he enlarged filraion G saisfies he usual condiions. Therefore we redefine G as follows : for every, G = ɛ> (F +ɛ σ(g. The following heorem is due o J. Jacod. Before we give he conen of he heorem we inroduce he following hypohesis called l hypohèse (H : every F-semimaringale is a G- semimaringale. Theorem..3. L hypohèse (H is saisfied under he following condiion: (A For every here exiss a posiive measure σ-finie η on (U, U such ha P[G F (ω η ( almos surely in ω, where P[G F (ω sands for a regular version of he condiional law of G wih respec o F. Proof. For a deailed proof of his resul we refer he reader o he proof of Theorem., Jacod [9. Remark..4. For he exisence of he regular condiional probabiliies of G wih respec o each F see Theorem A.. and Corollary A..2 in he Appendix, or Shiryaev [5.

8 Enlargemen of Filraion & Insider Trading 2 Proposiion..5. The condiion (A is equivalen o he condiion (A : There exiss a posiive measure σ-finie η on (U, U such ha P[G F (ω η( for all >, ω Ω. In his case we can ake for η he law of he variable G. Proof. I is clear ha we only need o prove ha (A implies (A, wih η he law of G. Fix > and suppose ha (A is saisfied. Then by Doob s Theorem, see Theorem A..3 in he Appendix, here exiss a F U-measurable posiive funcion: (ω, x q x (ω such ha P[G dx F (ω = η (dxq x (ω. Le b (x = E [ q x and { q x ˆq x (ω (ω = b (x if b (x >, oherwise Because q x = a.s. if b (x =, we have q x = ˆq x b (x a.s., and he measure η (dxb (xˆq x (ω is sill a version of P[G dx F (ω. Hence for every F -measurable posiive funcion g we have g(xη(dx = E [ g(g = E[ g(xp[g dx F (dω = g(xe [ q x η (dx = g(xb (xη (dx, whence η (dxb (x = η(dx, hus P[G dx F (ω = η(dxˆq x (ω. N.B. Doob s derivaion heorem gives join measurabiliy of (ω, x q(ω, x, whereas he Radon-Nikodym Theorem only gives x q(ω, x is measurable for all ω. Now for he res we will assume he following: Assumpion..6. The condiion (A is saisfied: here exiss a σ-finie posiive measure η on (U, U such ha for all [, T, he regular condiion disribuion of G given F is absoluely coninuous wih respec o η for P-almos all ω Ω, i.e, P[G F (ω η( for P-a.a. ω Ω. (. Remark..7. The measure η is no necessary he law of G, because someimes anoher simple measure could be aken, like he Lebesgue measure on U = R d. We inroduce he following noaions : le H := (H [,T, where H {F, G}, be a generic filraion, H := (H [,T, ˆΩ := Ω U, Ĥ := ɛ> (H +ɛ U, Ĥ := (Ĥ [,T and Ĥ := (Ĥ [,T. The fac ha he ime horizon T is included or excluded is of imporance, as we shall see in he secion of examples ha are given laer on. Le K = (K [,T = (K,..., K d [,T be a d-dimensional coninuous local F-maringale wih quadraic variaion K = ( K i, K j aken wih respec o F. i,j=,...,d Definiion..8. We call he opional σ-algebra on ˆΩ [, T, he σ-algebra generaed by he càdlàg Ĥ -adaped sochasic processes, and we denoe i by O(Ĥ. where by we mean he ranspose.

9 Enlargemen of Filraion & Insider Trading 3 Definiion..9. We call he predicable σ-algebra on ˆΩ [, T, he σ-algebra generaed by he lef-coninuous Ĥ -adaped sochasic processes, and we denoe i by P(Ĥ. The following lemma provides a nice version of he condiional densiy process q l resuling from he absolue coninuiy in Equaion (.. Lemma.. (Lemme.8 and corollaire. of Jacod [9. Suppose Assumpion..6 is saisfied. Then:. There exiss a non-negaive O(ˆF -measurable funcion (ω, l, q l (ω which is righconinuous wih lef limis in and such ha: a. for all l U, q l is an F -maringale, he processes q l, q l are sricly posiive on [, T l, and q l = on [T l, T, where T l := inf{ q l = } T ; (.2 b. for all [, T, he measure q l (.η(dl on (U, U is a version of he condiional disribuion P[G dl F. 2. T G = T P-a.s., where T G (ω = T G(ω (ω = T l (ω on {G = l}. Remark... The condiional densiy process q l is he key o he sudy of coninuous local F-maringales in he enlarged filraion G. The following heorem shows ha under Assumpion..6, every coninuous local F-maringale is a G -semimaringale, and explicily gives is canonical decomposiion. Theorem..2 (Théorème 2. of Jacod [9. Suppose Assumpion..6 is saisfied. For i =,..., d, here exiss a P(ˆF -measurable funcion (ω, l, (k(ω l i such ha q l, K i = (k l i q d K l i. (.3 For every such a funcion k i, we have. (kg s i d K i s < P-a.s. for all [, T, where k G = k l on {G = l}, and 2. K i is a G -semimaringale, and he coninuous local G -maringale in is canonical decomposiion is given by: K i := K i (k G s i d K i s, [, T. (.4 Remark..3. If he absolue coninuiy in Assumpion..6 holds for all [, T, hen K is even a local G-maringale. Le now ake a look a he condiional densiy process q l. Since F is rivial, we have P[G dl = P[G A = P[G A F = qη(dl, l A A

10 Enlargemen of Filraion & Insider Trading 4 for all A U. By choosing U smaller if necessary, we can herefore assume ha q l > for all l U, so we obain for P-a.a. ω and all [, T P[G A F (ω = q(ωη(dl l = p l P[G dl, where A p l (ω := ql (ω q l. (.5 From his, we observe ha we can ake p as he process q appearing in Lemma.. and in Theorem..2 by choosing for η he law of G. In he following we will wrie jus p G, bu we mean by his ha p G = p l on {G = l}. By par 2 of Lemma.., he firs ime p G his is P-a.s. equal o T so ha we can consider he process p G A on [, T. If he regular condiional disribuions of G given F are equivalen o he law of G, hen he process p G urns ou o be a posiive G -maringale saring from and hus defines a probabiliy measure P on (Ω, G for all [, T. P coincides wih P on F, and he σ-algebras F and σ(g become independen under P. These properies are shown in he following proposiion due o Amendinger [3, p.267. Proposiion..4. Suppose ha he regular condiional disribuions of G given F are equivalen o he law of G for all [, T, i.e., for all l U, he process (p l [,T is sricly posiive P-a.s. hen:. For [, T, he σ-algebras F and σ(g are independen under he probabiliy measure P (A := dp for A G, (.6 i.e., for A F and B U, A p G P [A {G B} = P[A P[G B = P [A P [G B. (.7 2. p G is a G -maringale. Proof. To prove Equaion (.7, fix A F and B U. By condiioning on F, we obain [ [ [ [ E A {G B} p G = E A E {G B} p G F = E {G B} A p G F (ωp(dω. The definiion of p l (ω yields [ E {G B} p G F (ω = B p l (ωpl (ωp[g dl = P[G B, and so we ge he firs equaliy in Equaion (.7. The second follows by choosing A = Ω or B = U.

11 Enlargemen of Filraion & Insider Trading 5 Now fix s < T and choose A G s of he form A = A s {G B} wih A s F s and B U. Then we obain by Equaion (.7 and by reversing he above argumen ha Now le D = [ E A p G [ = P[A s P[G B = E A P[G B = p l s(ω pl s(ωp[g dlp(dω A s B [ [ [ = E As E {G B} F s = E p G s A p G s { [ [ } A G s : E A = E p G A, we show now ha D is a d-sysem. p G s [. Ω D, indeed, from he firs par of he proof we have ha E 2. Le A, B D such ha A B. Then we have ha hence B\A D. [ E B\A p G [ = E ( B A [ = E B p G s [ = E B\A p G s p G E, [ [ = E B B p G s p G [ = E. Ω p G [ E B p G ( B A p G [ = E Ω. p G s 3. Le A n A wih A n D for all n. Since he process /p G is sricly posiive, hen by he monoone convergence heorem we have ha hus A D. [ E A p G [ = lim E An n p G [ = lim E An n [ = E A p G s, p G s Hence D is a d-sysem ha conains he π-sysem C = { [ A s {G B}; wih A s F s and B U : E As {G B} p G [ = E As {G B} p G s } generaing he σ-algebra F s σ(g, and by Proposiion below, we have G s = F s σ(g. Therefore, G s D G s. Whence his exends o arbirary ses A G s. Hence he process p G is a G -maringale wih =, hence Equaion (.6 defines indeed a probabiliy measure on p G (Ω, G.

12 Enlargemen of Filraion & Insider Trading 6.2 Sochasic exponenial represenaion of he process /p G This secion shows ha under he assumpion of Proposiion..4, he processes p l and /p G can be represened as sochasic exponenials of a paricular form. More precisely, he F - maringale p l is he sochasic exponenial of he sum of a sochasic inegral wih respec o K wih inegrand κ l and an orhogonal local F -maringale, whereas he G -maringale p G can be wrien as a sochasic exponenial of he sum of a sochasic inegral wih respec o κ G wih respec o K and an orhogonal local F -maringale. To do his, we give he following lemma wihou proof. Lemma.2.. Under Assumpion..6, here exiss an R d -valued, P(F U-measurable process (κ l [,T such ha for all l U, d K s κ l s = ( k l (d K ( s s., [, T. (.8 ( k l (dd K (d s s Proof. For he proof of his resul we refer o he reader o Amendinger [3. For furher developmens, we need a week inegrabiliy condiion on κ. Assumpion.2.2. : The process κ from Lemma.2. saisfies T ( κ l s d K s ( κ l s < P-a.s. for all l U. (.9 Remark.2.3. The process κ G is P(G -measurable, indeed we need only o show he measurabiliy of he mapping ( (ω, ; P(G ( (ω,, G(ω; P(F U. For any A P(F and B U, we have ( {(ω, : (ω, A and G(ω B} = A {ω : G(ω B} [, T ( = A {ω : G(ω B} {} ( {ω : G(ω B} (, T, and herefore we have he measurabiliy of he mapping above. By he measurabiliy of κ l we ge he measurabiliy of κ G. And so he sochasic inegral ( κ G d K is well defined under Assumpion.2.2. For each l U, he process κ l is unique up o null ses wih respec o P K, and so he sochasic inegrals ( κ l ( dk and κ G d K do no depend on he choice of κ. Finally, we can now wrie K := ( K,..., K d more compacly as K = K d K κ G.

13 Enlargemen of Filraion & Insider Trading 7 Proposiion Suppose ha he regular condiional disribuions of G given F are equivalen o he law of G for all [, T. Then here exiss a local G -maringale Ñ null a zero which is orhogonal o K from Equaion (.4 (i.e., K (i, Ñ for i =,..., d and such ha ( = E (κ G s d K s + Ñ, [, T. (. p G 2. Fix l U. If p T l > P-a.s., hen here exiss a local F -maringale N l null a zero which is orhogonal o K and such ha p l = E( (κ l s dk s + N l, [, T. (. Proof. See Proposiion 2.9, p. 27, of Amendinger [3. Remark.2.5. If he regular condiional disribuions of G given F are equivalen o he law of G for all [, T, hen he condiion in he second par of Proposiion.2.4 is auomaically saisfied for all l U. The nex corollary gives an explici expression for Ñ in Equaion (. in erms of N G, if p l is coninuous for all l U. As a consequence, we obain hen in paricular ha /p G can be wrien as a sochasic exponenial of a sochasic inegral wih respec o K, if we have in addiion a maringale represenaion heorem for he filraion F. Corollary If p l is coninuous and sricly posiive for all l U, hen ( = E (κ G s d K s N G + N G, [, T. (.2 p G In paricular, Ñ from Equaion (. is given by 2. In paricular, if p l = E ( (κ l dk for all l U, hen p G Ñ = N G + N G, [, T. (.3 ( = E Proof. See Corollary 2., p. 272, of Amendinger [3 (κ G d K, [, T. (.4

14 Enlargemen of Filraion & Insider Trading 8.3 Examples This secion illusraes he preceding resuls by several examples for G. We will give an example where we have equivalence beween he regular condiional probabiliy of G given F and he law of G for [, T, an example where we have absolue coninuiy for [, T and equivalence only for [, T, and an example where we have equivalence for all [, T. Example.3.. Le G be he end poin W T of a one-dimensional F-Brownian moion W on [, T. Then we have G = ɛ> (F +ɛ σ(w T and we have for all < T P [ W T dl F = P [ (WT W + W dl F = P [ (W T W (dl y W=y = 2π(T exp = p l P [ W T dl, ( (l W 2 2(T dl where p l = T ( T exp (l W 2 2(T + l2 2T, l R, is sricly posiive for all < T. Furhermore, applying Iô s formula o (l W2 (T we ge ( p l l W s = E T s dw s. and hence i is an F -maringale by Novikov s condiion. In his example, he condiional law of G given F is even equivalen o he law of G for all [, T. On he oher hand, he condiional law of W T given F T is he poin mass in W T (ω and herefore no absoluely coninuous wih respec o he law of W T. Example.3.2. Le G be a random variable wih values in a counable { se U } such ha P[G = l > for all l U. Then every A σ(g is of he form A = l J G = l for some J U. herefore we have P [ G A F = P [ G = l F = p l P [ G = l = p l P [ G dl l J for all [, T, where p l = P[G = l F /P[G = l, and so he condiional law of G given F is absoluely coninuous wih respec o he law of G for all [, T. Thus we obain by Theorem..2 and he Remark..3 ha every local F-maringale is a G-semimaringale. However, he condiional laws of G given F are equivalen o he law of G on F for < T only if P[G = l F > P-a.s. for all l U. Moreover, here is no equivalence on F T if G is F T -measurable, because in his case P[G = l F T = {G=l} is zero wih posiive probabiliy (unless G is a consan and equal o l. As a special case, consider he siuaion in which G describes wheher he endpoin of a one-dimensional F-Brownian moion lies in some given inerval, i.e., G := {WT [a,b} for some l J A

15 Enlargemen of Filraion & Insider Trading 9 a < b. Then we have p = P[ G = F P[G = and p = P[ G = F, P[G = and for [, T we have P [ G = [ F = P { } = F = P [W W T [a,b T [a, b F = P [W T W + W [a, b F [ = P W T W [a y, b y and a W=y b ( = exp (l W 2 dl, 2π(T 2(T P[G = = P [ G = F = Φ(b/ T Φ(a/ T, where Φ is he sandard normal disribuion funcion. Hence, P [ G. F is absoluely coninuous wih respec o he law of G for [, T and equivalen o he law of G only for all [, T. Example.3.3. Le G = W T + ε, where W T is he endpoin of a one dimensional (P, F- Brownian moion W and ε is a random variable independen of F T such ha ε N (,. Then we have for all [, T P [ G dl F = P [ (WT W + W + ε dl F = P [ (W T W + ε (dl y y=w = ( exp (l W 2 dl 2π(T + 2(T + = p l P [ (W T + ε dl, where p l = T + ( T + exp (l W 2 2(T + + l 2, l R, 2(T + is sricly posiive for all [, T. Furhermore, applying Iô s formula o (l W2 (T + ( p l l W s = E T s + dw s, we ge and hence i is a F-maringale by Novikov s condiion.in his example, he condiional law of G given F is even equivalen o he law of G for all [, T, he end poin included.

16 Chaper 2 Maringale represenaion heorems for I.E.F 2. Noaions Recall ha we are working in a probabiliy space (Ω, F, P equipped wih a filraion F = (F [,T saisfying he usual condiions. T > is a fixed finie ime horizon. We assume ha F is rivial and F s = F T = F for all s T. For an F-measurable random variable G wih values in a Polish space (U, U, we define he iniially enlarged filraion G = (G [,T by G = F σ(g, [, T. Le H = (H [,T {F, G} be a generic filraion and R be a generic probabiliy measure on (Ω, H T. The collecion of uniformly inegrable coninuous (R, H-maringales is denoed by M(R, H. For p [,, H p (R, H denoes he se of he coninuous (R, H-maringales M such ha : M H p (R,H := ( E [ sup M s p p <. s [,T The se of bounded coninuous (R, H-maringales is denoed by H (R, H. For p [,, L p (M, R, H denoes he space of d-dimensional H-predicable processes φ = (φ (,..., φ (d such ha: φ L p (M,R,H := E R[( T φ p u d[m, M u φ 2 u <. For a d-dimensional coninuous (R, H-semimaringale S, L sm (S, R, H denoes he se of d- dimensional H-predicable processes φ ha are inegrable wih respec o S, in he sense ha φ ds = d i= φ i ds i, where for every i =,..., d he sochasic inegral φ i ds i is well defined. To emphasize he dependence of he sochasic inegral on H, we shall hen wrie H- φ ds, where φ L sm (S, R, H. Throughou his chaper we fix a d-dimensional coninuous process S = (S (,..., S (d, and we assume ha here exiss a probabiliy measure Q F P on (Ω, F T such ha each componen of S is in Hloc 2 (QF, F. This assumpion is moivaed because as we shall see in Secion 3., he process S is no always a maringale in he real world, i.e. a (P, F-maringale. Le Z F be he densiy process of Q F wih respec o P.

17 Enlargemen of Filraion & Insider Trading 2.2 The maringale preserving probabiliy measure In his secion we define he maringale preserving probabiliy measure and also shows how i can be used o ransfer properies of sochasic processes and srucures from F o G. Recall ha in Chaper, we have used The following assumpion in some resuls and will be imposed in he remainder of his hesis. Assumpion The regular condiional disribuions of G given F are equivalen o he law of G, i.e.: P[G. F (ω P[G. for all [, T and P-a.a. ω Ω (2. Theorem If he Assumpion (2.2. is saisfied, hen. Le Q G be he measure defined by has he following properies: Q G (A := Z F T A p G T dp for A G T (2.2 i. Q G = Q F on (Ω, F T, and Q G = P on (Ω, σ(g, i.e. for A T F T and B U, Q G [A T {G B} = Q F [A T P[G B = Q G [A T Q G [G B. (2.3 ii. The σ-algebras F T and σ(g are independen under Q G. 2. Z G := ZF p G is a (P, G-maringale. Proof.. To prove Equaion (2.3, le A T F T and B U. By condiioning on F T, we ge E [ Z F [ T AT {G B} = E p G AT E [ Z F T {G B} T p G F T T = E [ Z F T {G B} A T p G F T (ωp(dω. T The definiion of p G T yields Therefore [ E {G B} p G F T (ω = T Z E [ F T {G B} p G F T (ω = T B B p l T (ωpl T (ωp[g dl ZT F(ω p l T (ω pl T (ωp[g dl = ZT F (ωp[g dl B

18 Enlargemen of Filraion & Insider Trading 2 and hence Q G[ A T {G B} [ ZT F = E AT {G B} p G T = E [ ZT F I {G B} A T p G T = A T B = ZT F (ω A T B = Q F [A T P[G B [ = E AT E [ Z F T {G B} p G F T T F T (ωp(dω Z F T (ωp[g dlp(dω P[G dlp(dω where in he las equaliy we used he fac ha Z F is he densiy process of Q wih respec o P. Thus we ge he firs equaliy in Equaion (2.3. The second follows by choosing A T = Ω or B = U. 2. Now fix s T and choose A G s of he form A = A s {G B} wih A s F s and B U. Then we obain by Equaion (2.3, using he fac ha Z F is a (P, F-maringale and by reversing he above argumen ha E [ A Z F p G = Q F [A s P[G B = E [ As Z F s (ωp[g B = A s [ = E B Z F s (ω p l s(ω pl s(ωp[g dlp(dω As E [ {G B} Z F s p G s F s = E [ A Z F s p G s Then arguing as in Proposiion..4, his exends o arbirary ses A G s. Hence he process ZF is a (P, G-maringale wih ZF p G = because Z p G F = = pg defines indeed a probabiliy measure on (Ω, G.. and so Equaion (2.2 The following heorem shows ha he maringale propery is preserved under an iniial enlargemen of filraion and a simulaneous change o he measure Q G. Theorem If he Assumpion 2.2. is saisfied, hen for all p [, H p (loc( Q F, F = H p (loc( Q G, F H p (loc( Q G, G (2.4 and in paricular M (loc ( Q F, F = M (loc ( Q G, F M (loc ( Q G, G. (2.5 Proof. Le M be a (Q F, F-maringale we have hen E QG [M G s = E QG [M F s σ(g = E QG [M F s = E QF [M F s = M s, where in he second equaliy we used independence of σ(g and F T under Q G and in he hird equaliy we used he equaliy of Q G and Q F on (Ω, F T. Therefore M is a (Q G, G-maringale.

19 Enlargemen of Filraion & Insider Trading 3 Since F-sopping imes are also G-sopping imes, any localizing sequence (τ n n for a process M wih respec o (Q F, F will hen also be a localizing sequence for he process M wih respec o (Q G, F and (Q G, G. The inegrabiliy properies in Equaions (2.4 and (2.5 follow from he equaliy of Q G and Q F on (Ω, F T. Remark Any (Q F, F-Brownian moion W is a (Q G, G-Brownian moion. Indeed, Since he quadraic variaion of coninuous maringales can be compued pahwise wihou involving he filraion and since Q G = Q F on (Ω, F T, we obain for all [, T ha W (QG,G = W (QG,F = W (QF,F and herefore W is also a (Q G, G-Brownian moion, by Lévy s characerizaion heorem. Definiion The probabiliy measure Q G defined by Equaion (2.2 on (Ω, G T, is called he maringale preserving probabiliy measure under iniial enlargemen of filraion. This erminology is jusified by Theorem Using he decoupling propery of Q G, he following proposiion shows ha G inheris he righ-coninuiy from F. Proposiion If he Assumpion 2.2. is saisfied, hen G is righ-coninuous. Proof. Define G s+ := ɛ> G s+ɛ for s [, T. Fix [, T and δ (, T. Because of he independence of F +δ and σ(g under Q G, and using Theorem A.2.3, i is enough o show ha for G +δ -measurable random variables Y +δ of he form Y +δ = h(gh +δ, where h is a bounded U-measurable funcion and H +δ is a bounded F +δ -measurable random variable, we have For all ɛ (, δ, we ge E QG [Y +δ G + = E QG [Y +δ G. = E QG [Y +δ G + = h(ge QG [H +δ G + = h(ge QG [ E Q G [H +δ F +ɛ σ(g G + = h(ge QG [ E Q G [H +δ F +ɛ G +, (2.6 since H +δ and F +ɛ are independen of G under Q G. And because of he righ-coninuiy of F, we can always choose righ-coninuous versions of F-maringales. This implies lim ɛ EQG [H +δ F +ɛ = E QG [H +δ F, and since H +δ is bounded, passing in Equaion (2.6 o he limi as ɛ decreases o and applying he dominaed converging heorem, we obain E QG [Y +δ G + = h(ge QG [ E Q G [H +δ F G + = h(ge Q G [H +δ F = h(ge QG [H +δ G = E QG [Y +δ G, because H +δ and F are independen of G under Q G.

20 Enlargemen of Filraion & Insider Trading 4 In paricular, we have for all G + -measurable random variables X ha X = E QG [X G + = E QG [X G Q G a.s. Since Q G P and G conains all (P, F-null ses, X is herefore G -measurable. This complees he proof. The following proposiion shows ha he sochasic inegrals defined under F remain unchanged under an iniial enlargemen ha saisfies Assumpion Proposiion If he Assumpion 2.2. is saisfied. For a d-dimensional (Q F, F- semimaringale Y, he following equaliies hen hold: L sm (Y, Q F, F = L sm (Y, Q G, F (2.7 = {ϑ : ϑ is F-predicable and ϑ L sm (Y, Q G, G} (2.8 and for ϑ L sm (Y, Q F, F he sochasic inegrals F- ϑdy and G- ϑdy have a common version. Proof. Because Q F = Q G on (Ω, F T, we ge hen he firs equaliy. For he second equaliy we have by Theorem 2.2.3, he (Q G, F-semimaringale Y is also a (Q G, G-semimaringale and hus L sm (Y, Q G, F {ϑ : ϑ is F-predicable and ϑ L sm (Y, Q G, G} by Théorème 7 of Jacod [8, see Theorem A.2. in he Appendix. For he oher inclusion, le ϑ L sm (Y, Q G, F, i.e. here exiss a local (Q G, F-maringale M and an F-adaped process A of finie variaion such ha Y = M + A, and such ha ϑ L loc (M, Q G, F and ϑ da exiss. By Theorem 2.2.3, M is a (Q G, F-maringale. Since F G, he process A is G- adaped. Therefore Y = M +A is also a (Q G, G-semimaringale decomposiion. And since M M loc (Q G, F M loc (Q G, G, Corollaire 9.2 of Jacod [7, see corollary A.2.2 in he Appendix, implies ha ϑ L loc (M, Q G, G, and since he ϑ da can be compued pahwise wihou involving he filraions, we ge ha ϑ L sm (Y, Q G, G hus he proof is complee. 2.3 Maringale represenaion heorems for I.E.F In his secion we ransfer maringale represenaion heorems from F o he iniially enlarged filraion G. For hese purpose we suppose hroughou his secion ha he following represenaion propery holds wih respec o S H 2 loc (QF, F: Assumpion For any ψ L (F T, here exiss φ L 2 (S, Q F, F such ha T ψ = E QF [ψ + φ s ds s. Remark By Theorem 3.4 of He, Wang and Yan [6, see Theorem A.2.5 in he Appendix, he assumpion above is equivalen o he represenaion propery of a local (Q F, F-maringale. Tha is for every local (Q F, F-maringale M here exiss a φ L loc (S, QF, F such ha K = K + φ ds

21 Enlargemen of Filraion & Insider Trading 5 Theorem Suppose Assumpions 2.2. and 2.3. are saisfied.. For any M H 2 (Q G, G, here exiss a process ψ L 2 (S, Q G, G such ha M = M + ψ s ds s, [, T. 2. For any M M loc (Q G, G, here exiss a process ψ L (S, Q G, G such ha M = M + ψ s ds s, [, T. Proof. To prove he firs claim i is sufficien o show ha any random variable X L 2 (Q G, G T can be wrien in he form T X = E QG [X G + ψs ds s, for some ψ L 2 (S, Q G, G. Since G T = F T σ(g, Theorem IV of Malliavin [, see Theorem A.2.3 in he Appendix, implies ha he vecor subspace of L (G T defined by V = { X L (G T : X = m I i J i, wih I i L (F T, J i L (σ(g, m N } i= is dense in L 2 (Q G, G T. Thus here exiss a sequence (X n n N = ( m n i= I i,nj i,n n N in V, wih I i,n L (F T and J i,n L (σ(g, such ha X n converges o X in L 2 (Q G. By Assumpion 2.3., and he fac ha Q G = Q F on (Ω, F T (Theorem 2.2.2, hen here exiss a sequence (φ i,n n N L 2 (S, Q G, F such ha I i,n = E QG [I i,n + T (φ i,n s ds s. (2.9 Since S is a local (Q G, F-maringale and hus a local (Q G, G-maringale by Theorem 2.2.3, Proposiion implies ha he value of he sochasic inegral T (φi,n s ds s is no changed when i is considered under G. Because J i,n is bounded and G -measurable since because F is rivial, we have J i,n T G = F σ(g = σ(g The independence of F T and G under Q G yields T (φ i,n s ds s = J i,n (φ i,n s ds s. (2. E QG [I i,n J i,n G = J i,n E QG [I i,n G = J i,n E QG [I i,n (2. By Equaions (2.9, (2. and (2. we hen obain m n ( X n = J i,n E Q G [I i,n + i= T (φ i,n s ds s = E Q G [X n G + T (ψ n s ds s, (2.2

22 Enlargemen of Filraion & Insider Trading 6 where ψ n := m n i= g i,nφ i,n is in L 2 (S, Q G, G due o he boundedness of J i,n and since φ i,n is in L 2 (S, Q G, G. Since X n converges o X in L 2 (Q G, E QG [X n G converges o E QG [X G in L 2 (S, Q G, G and hus Equaion (2.2 yields ha T (ψn s ds s converges in L 2 (Q G as well. Since each componen of S is in Hloc 2 (QG, G, and since he mapping ϑ ϑ ds is an isomery from (L 2 (S, Q G, G,. L 2 (S,Q G,G o (Hloc 2 (QG, G,. H 2 loc (Q G,G, he space of sochasic inegrals { T } ϑ s ds s : ϑ L 2 (S, Q G, G, is closed in L 2 (Q G. This implies he exisence of a process ψ L 2 (S, Q G, G such ha T (ψn s ds s converges o T ψ s ds s in L 2 (Q G. Hence Equaion (2.2 yields ( X = L 2 lim E QG [X n G + n T T (ψs n ds s = E QG [X G + ψs ds s, and hus he firs claim. For he proof of he second par of he heorem, we will proceed in seps. For convenience we will denoe by G he filraion (G s s [, for all [, T and of course we have G T = G.. Le [, T and K H (QG, G, i.e. K H (Q G, G and K =. The filraion G is righ-coninuous by Proposiion Then Theorem.5 of He, Wang and Yan [6 implies ha H 2(QG, G is dense in H (QG, G, hus here exiss a sequence (K n n in H 2(QG, G such ha lim n K n K H (Q G,G =. For all n, he firs par of he heorem yields he exisence of ψ n L 2 (S, Q G, G such ha K n = (ψ n s ds s, [, T. (2.3 Since K n is in L (S := { ϑ ds : ϑ L (S, Q G, G }, and since L (S is closed in H (QG, G by Theorem 4.6 of Jacod [7, we conclude ha K L (S. 2. Le K H (QG, G. By par, on he inerval [, T K is of he form K = where ψ is G -predicable, [, T, and for all [, T, ψ s ds s, [, T, (2.4 E QG[( /2 ψs d S s ψ s <. We now exend Equaion (2.4 o he inerval [, T. Since for all [, T, he filraion G is righ-coninuous, he Burkholder-Davis-Gundy inequaliies imply for [, T E QG[( /2 ψs d S s ψ s = E QG[ K /2 C E QG[ sup s C E QG[ sup K s s T, K s

23 Enlargemen of Filraion & Insider Trading 7 where C is a posiive consan. Hence by monoone convergence we obain E QG[( T /2 ψs d S s ψ s C E QG[ sup K s, s T because K H (QG, G. Hence ψ ds is a (Q G, G-maringale, and Equaion (2.4 implies ha for all [, T K = E QG[ T ψs G ds s. Since K H (QG, G and lim T K K T L =, by maringale convergence we obain K T = T ψ s ds s. 3. Now le K M,loc (Q G, G, i.e. here is a sequence of G-sopping imes (σ n n N such ha for each n N, K σn M (Q G, G. Since for all [, T, G is righ-coninuous, we have hen ha for all n N, τ n := σ n inf{ : K n} T is a G-sopping ime. Hence sup [,T K τn n + K τn, hence K τn H (QG, G. Therefore par 2 yields he exisence of ψ n L (S, Q G, G such ha K τn = (ψ n ds. Wih τ =, we ge ha ψ := n= ψn τn,τ n is in L loc (S, QG, G and ha K = ψ ds. We will make he following assumpion on S which we will need for he prove of a maringale represenaion heorem wih respec o G and he original probabiliy measure P. Assumpion The (P, F-semimaringale S is coninuous and can be wrien as S = M + d M α, (2.5 where M is a d-dimensional coninuous local (P, F-maringale and α is a d-dimensional process in L (M, P, F. Applicaion of Theorem..2 and Lemma.2. o he d-dimensional coninuous local (P, F- maringale M from Assumpion yields he exisence of a P(F U-measurable funcion (ω,, l κ l (ω such ha M := M d M κ G is a d-dimensional coninuous local (P, G-maringale. And we need he following assumpion on he inegrabiliy of κ. Assumpion For all l U he process κ l L loc (M, P, F.

24 Enlargemen of Filraion & Insider Trading 8 Lemma If Assumpions 2.3. and are saisfied, hen for [, T ( = E αs dm s. (2.6 Z F 2. If Assumpions 2.2., 2.3. and are saisfied hen for [, T ( = E (κ G s d M s, (2.7 Z G p G = ZF p G ( = E (α s + κ G s d M s. (2.8 Proof. Since is a sricly posiive (Q F, F-maringale, here exiss a local (Q F, F-maringale Z F L wih L = such ha = E(L. By Assumpion 2.3. and Remark 2.3.2, here exiss a Z F process φ L loc (S, QF, F such ha Z F = E( φ ds. Then by Assumpion 2.3.4, he densiy process Z F can be wrien as { Z F = exp φ dm φ d M α + } φ d M α, (2.9 2 hence Iô s formula implies ha for i =,..., d d(z F S i = Z F ds i + S i dz F + d Z F, S i = Z F dm i + Z F (d M α i + S i dz F + Z F d M i, φ dm = Z F dm i + S i dz F + Z F( d M (α φ i. Since Z F S i, Z F dm i and S i dz F are coninuous local (P, G-maringales, Z F d M (α φ is a coninuous local (P, G-maringale of finie variaion and hus vanishes. And since Z F >, we ge ha d M α = d M φ and so α dm = φ dm. By Equaion (2.9, we ge hen Equaion (2.6. To prove Equaion 2.7, le l U. Since p l is a sricly posiive (P, F-maringale by Lemma.. and Remark.. (we have F insead of F because we have equivalence insead of absolue coninuiy, whence p l /Z F is a sricly posiive (Q F, F-maringale. Because of Assumpion 2.3. here exiss a process φ l L loc (S, QF, F such ha p l Z F = E( (φ l ds.

25 Enlargemen of Filraion & Insider Trading 9 By Assumpion and he firs claim we have p l = pl ( ( Z F ZF = E (φ l ds E = E( (φ l ds α dm = E( (φ l dm + (φ l d M α = E( (φ l α dm. α dm (φ l d M α α dm (φ l d M α On he oher hand, because of Assumpions 2.2., and par 2 of Proposiion.2.4 can be applied o ge p l = E( (κ l dm + N l, where N l is a local (P, F-maringale wih N l = and orhogonal o M. he uniqueness of he sochasic exponenial hus implies ha (φ l α dm = (κ l dm + N l. (2.2 Now aking he covariaion of boh sides of Equaion (2.2 wih respec o N l, we ge by he orhogonaliy of M and N l ha N l =. By Assumpion and Equaion (2.2 he process N l is coninuous. Hence, N l is a coninuous local (P, F-maringale of finie variaion and hus vanishes. Therefore we obain ha p l = E( (κ l dm. Now par 2 of corollary.2.6 yields Equaion (2.7. By Theorem we have ha Z G = Z F /p G is a (P, F-maringale, hence Z G = ZF ( ( p G = E αs dm s E (κ G s d M s ( = E αs dm s (κ G s d M s + αs dm s, (κ G s d M s ( = E αs dm s (κ G s d M s + αs d M s κ G s ( = E αs (dm s + d M s κ G s (κ G s d M s ( = E αs d M s (κ G s d M s ( = E (α s + κ G s d M s. Then he proof is complee. We now show a maringale represenaion heorem for local (P, G-maringales wih respec o he coninuous (P, G-maringale M.

26 Enlargemen of Filraion & Insider Trading 2 Theorem Suppose Assumpions 2.2., 2.3., and are saisfied. For any K M loc (P, G, here exiss hen φ L loc ( M, P, G such ha K = K + φ s d M s, [, T. (2.2 Proof. Since K M loc (P, G, we have ha K M Z G loc (Q G, G, hence Theorem implies ha for all [, T, K Z G = K + φ s ds s for some φ L loc (S, QG, G. Now applying Iô s formula and using he fac ha we ge for [, T S = M + dk = d (Z G K ( K Z G = Z G dzg ( d M (α + κ G and Z G = E = Z G φ ds + ( K + Z G d Z G ( K + (α s + κ G s d M s, + d ZG, K Z G φ s ds s dz G + d Z G, = Z G φ d M + Z G φ d M (α + κ G ( Z G K + φ s ds s (α + κ G d M Z G φ d M (α + κ G ( ( d = Z G φ K + φ s ds s (α + κ G M. φ ds By seing φ ( ( := Z G φ K + φ s ds s (α + κ G, we herefore obain Equaion (2.2. The inegrabiliy propery of φ is a consequence of he inegrabiliy of φ, he coninuiy of Z G and S and he inegrabiliy assumpions on α and κ G.

27 Chaper 3 Insider rading and uiliy maximizaion In his chaper we consider wo ypes of invesors on differen informaion levels rading in a general coninuous-ime securiy marke. The marke is described by he given probabiliy space (Ω, F, P wih he filraion F = (F [,R saisfying he usual condiions and we assume also ha F is rivial. The price process S = {S, [, R} ha models he sock price is assumed o be a posiive coninuous (P, F-semimaringale. While he ordinary invesor, who has as informaion a ime all he F -measurable random variables, makes his decisions based on hese random variables, he insider invesor observes he same process S bu he/she holds as informaion a bigger filraion G hen he ordinary invesor (ha is, F G, [, R. The addiional informaion of he insider could be for example he knowledge a ime = of he oucome of some F-measurable random variable G. For insance, G migh be he price of S a ime = R, or he value of some exernal source of uncerainy, ec. As in he preceding chaper G is an F-measurable random variable wih values in some Polish space (U, U, and hen he insider informaion is modelled by he iniially enlarged filraion G = (G [,R wih G = F σ(g, [, R. We fix a ime T (, R, and we assume ha he financial marke S on he ime inerval [, T is arbirage free and complee for he ordinary invesor in he following sense: here exiss a unique probabiliy measure Q F equivalen o P on ((Ω, F T such ha S is a local (Q F, F- maringale (arbirage free, and any bounded F T -measurable random variable can be wrien as a some of a consan and a sochasic inegral wih respec o S (compleeness, i.e. Assumpion Furhermore we suppose ha he random variable G saisfies Assumpion Remark The inuiive meaning of Assumpion 2.2. is ha a all imes up o ime T he insider has an informaional advanage over he ordinary invesor consising in he knowledge of all he oucomes of G as possible before and a ime T. For he public he oucome of G is revealed only afer ime T. As we have seen in Example.3.3, if G conains a noise erm ha is independen from F R, hen we can choose T = R. For he oher Examples.3. and.3.2 we can choose any T < R. 2. Assumpion 2.2. combined wih he exisence of an equivalen local F-maringale measure 2

28 Enlargemen of Filraion & Insider Trading 22 for S ensures he exisence of an equivalen local G-maringale measure for S, i.e. under he probabiliy measure Q G given in Theorem wih densiy process Z G. The price process S is hen by Theorem a local (Q G, G-maringale, moreover by Theorem any local (Q G, G-maringale can be wrien as a sum of a G -measurable random variable and a sochasic inegral wih respec o S. Whence Assumpion 2.2. is sufficien o place he insider in a complee marke free of arbirage. 3. The ordinary invesor problem In his secion we will give an exposiion of he problem for he ordinary invesor in he Black- Scholes framework. A general model will be reaed in he nex secion. As in he inroducion of his chaper we suppose ha he marke is described by he given probabiliy space (Ω, F, P and we consider a Brownian moion W = {W ; T } defined on (Ω, F, P and we denoe by F = (F [,T he naural filraion generaed by W and he (P, F-ses of measure zero. The price process S = {S ; [, T } ha models he sock price is assumed o be a posiive coninuous (P, F-semimaringale evolving according o he sochasic equaion ( ds = S µd + σdw, [, T, (3. wih S >, and µ is a consan and σ a sricly posiive consan. Besides we denoe by B = {B, [, T } he risk free asse, and we suppose ha i evolves according o he sochasic differenial equaion, for given posiive consan r, db = B rd, [, T and B =. (3.2 Before we coninue, we need some definiions concerning rading sraegies, self-financing sraegies. Definiion 3... A rading sraegy (porfolio is a wo-dimensional predicable, locally bounded process π = {π = (φ, ψ, [, T } wih values in R 2. Remark The condiions on π ensure ha he sochasic inegrals T φ db and T ψ ds are well defined. Where φ denoes he money ha he invesor invess in he riskless asse, and ψ denoes he number of socks held in he porfolio a ime. Definiion Le π be a rading sraegy.. The value of he porfolio π a ime is given by V = V π = φ B + ψ S. The process V π π. is called he value process, or he wealh process, of he rading sraegy 2. The gains process denoed by G π is defined by G π = φ s db s + ψ s ds s.

29 Enlargemen of Filraion & Insider Trading A rading sraegy π is called self-financing if he wealh process V π saisfies V π = V π + G π for all [, T. Now we reurn o our model and we observe ha he Equaions (3. and (3.2 have a unique soluions and by Iô s formula we have ha he soluions are given by B = exp(r, (3.3 S = S exp ((µ 2 σ2 + σw. (3.4 The process modelling he sock price is no a F-maringale under he measure P. Therefore we need anoher measure equivalen o P under which he process S is an F-maringale. We define he process S by S = S /B = e r S, he process S is called he discouned sock process. By Iô s formula and Equaion (3. we have d S = ( ds + S d B B ( = e r S µd + σdw + S ( re r d ( µ r = S σ σe r d + σe r dw d S = σ S ( µ r σ d + dw Under P he process W is a Brownian moion and hence he process S is a local (P, F-maringale if and only if µ r. Bu his will rarely be he case in he real world. On he oher hand, under anoher equivalen measure P he discouned sock process S may well be an F-maringale, if viewed as a process on he filered space (Ω, F, F, P. Because he drif erm in Equaion (3.5 causes he problem, we could firs rewrie he equaion as (3.5 (3.6 d S = σe r S d W, (3.7 for he process W defined by W = W α. Then we need o find a measure under which he process W is a maringale. By Novikov s condiion he process E( αdw for α = µ r σ is a P-maringale, wih mean. Therefore, we can define a probabiliy measure P by d P = E( αdw T dp. By Girsanov s heorem he process W is hen a P-Brownian moion (for he ime parameer resriced o he inerval [, T, and hence a P-maringale. And we have also ha he Brownian moion possesses he maringale represenaion propery. Therefore we are working in an arbirage free complee marke. 3.2 Uiliy Maximizaion The underlying principle for modelling economic behavior of invesors (or economic agens in general is he maximizaion of expeced uiliy, ha is one assumes ha agens have a uiliy funcion U(. and base heir economic decisions on expeced uiliy consideraions.

30 Enlargemen of Filraion & Insider Trading 24 Definiion A funcion U : (, R is called a uiliy funcion, if. i is sricly concave, sricly increasing and coninuously differeniable, and 2. U ( + = lim + U ( = and U ( = lim U ( = ( called Inada Condiions. Example As examples of a uiliy funcion we have. u(x = log x, 2. u(x = x α, < α <. Since we will work a lo wih exponenial processes, we will assume a logarihmic uiliy funcion because i enables us o obain explici formulae The model We reurn now o our seup as inroduced in he beginning of his chaper. We will work wih he one dimensional case. We fix a coninuous local F-maringale M wih M = and a predicable process α wih [ T E α 2 d M <. (3.8 The discouned price process of he sock denoed again by S is assumed o evolve according o he sochasic differenial equaion ds = S ( dm + α d M, [, T, (3.9 wih S >. Using he Iô s formula we ge ( S = S E M + ( αd M = S exp (α s 2 d M s + M (3. By Theorem..2, and Lemma.2. we have ha M is a G-semimaringale, and he local G-maringale M in is canonical G-decomposiion has he form M = M κ G s d M s, [, T, (3. wih κ = (κ l is a P(F U-measurable process. And so he discouned sock price evoluion from he insider s poin of view is ds = S ( d M + (κ G + α d M Using he Iô s formula we ge ( S = S E M + (α + κ G d M ( = S exp (α s + κ G s 2 d M s + M (3.2

31 Enlargemen of Filraion & Insider Trading 25 Definiion Le x > and denoe by H {F, G} a generic filraion.. An H-porfolio process is an R-valued and H-adaped predicable process π = (π [,T such ha T π2 d M < P-a.s. 2. For an H-porfolio process π, he discouned wealh (value process denoed by V (x, π is defined by V (x, π = x and saisfies dv (x, π := π ds := ψ V (x, π ds S for [, T. ( The class of admissible H-porfolio processes up is defined by { A H (x, T := π : π is an H-porfolio process and E [ log V T (x, π } <, (3.4 where log x = max{, log x}. Remark The process ψ describes he proporion of oal wealh a ime invesed is he risky asse S, and Equaion (3.3 is he well known self-financing condiion. For convenience, we will consider he process ψ, bu we will always keep in our mind ha he porfolio is he S process π, and we obain ψ by inroducing he change of variables ψ = π V. And from now on we will denoe he discouned wealh process by V (x, ψ insead of V (x, π. By Iô s formula, for a rading sraegy ψ A H (x, T wih x >, he wealh process is sricly posiive and explicily given by ( ds ( s V (x, ψ = x E ψ s S = x E ψ s dm s + ψ s α s d M s (3.5 s for [, T. From he insider s poin of view his can also be wrien, similar o Equaion (3.2, as V (x, ψ = x E( ψ s d M s + ψ s (κ G s + α s d M s, [, T. (3.6 Definiion (Opimizaion Problems. Le he iniial wealh x >.. The ordinary invesor s opimizaion problem is o solve: max E[ log V T (x, ψ. ψ A F (x,t 2. The insider s opimizaion problem is o solve: max E[ log V T (x, ψ. ψ A G (x,t

32 Enlargemen of Filraion & Insider Trading Soluion of he logarihmic uiliy maximizaion problem Le us firs work ou he expression log V T (x, ψ for ψ A G (x, T and x >, Equaion (3.6 hen gives T log V T (x, ψ = log x + = log x + = log x + 2 T T T T ψ d M + ψ d M + ψ d M + 2 T T ψ (κ G + α d M 2 ψ (κ G T + α 2 ψ d M (κ G + α 2 d M ψ 2 d M (κ G + α ψ 2 d M. (3.7 [ T Now if we had E ψ2 d M <, he local G-maringale ψ d M would be a rue maringale and hence would have expecaion zero. Then ψ = κ G +α, T, would be an opimal sraegy for he insider up o ime T, yielding a maximal expeced logarihmic uiliy up o ime T of log x + T 2 E[ (κ G + α 2 d M. Seing κ G, and of course M = M we ge he opimal sraegy and maximal expeced logarihmic uiliy for he ordinary invesor. Using he connecion beween he maringale densiy processes Z F and Z G and he logarihmic opimizaion problem, he soluion of he opimizaion problems is of he above form. Bu firs we give he following proposiion. Proposiion The processes Z F S and Z F V (x, φ wih φ A F (x, T and x > are local (P, F-maringales on [, T. 2. The processes Z G S and Z G V (x, ψ wih ψ A G (x, T and x > and x > are local (P, F-maringales on [, T. Proof. We will give he prove for he second claim only as he proof of he firs one is jus he same by aking F insead of G and seing κ G. Then using Iô s formula we ge d ( Z G S = S dz G + Z G ds + d Z G, S = Z F ( S α + κ G d M + Z G ( S d M + (α + κ G d M + + Z G S d (α + κ G d M, M + (α + κ G d M = Z G S (α + κ G d M + Z G S d M + + (α + κ G Z F S d M (α + κ G Z F S d M = Z F S ( (α + κ G d M,