Optional semimartingale decomposition and no arbitrage condition in enlarged ltration

Transcription

1 Optional semimartingale decomposition and no arbitrage condition in enlarged ltration Anna Aksamit Laboratoire d'analyse & Probabilités, Université d'evry Onzième Colloque Jeunes Probabilistes et Statisticiens Forges-les-Eaux 2014

2 Non-arbitrage up to Random Horizon for Semimartingale Models T. CHOULLI, A. A., J. DENG and M. JEANBLANC, 2013, Optional semimartingale decomposition and NUPBR condition in enlarged ltration A. A., T. CHOULLI, and M. JEANBLANC, 2014, Working paper

3 Problem (Ω, A, H, P) is ltered probability space where ltration H = (H t ) t 0 satises usual conditions. X = (X t ) t 0 is a price process of a risky asset, i.e., an H-semimartingale. θ = (θ t ) t 0 is an H-trading strategy, i.e., an H-predictable process, integrable w.r.t. X in H. By L H (X ) we denote the set of all H-trading strategies. θ X = ( t 0 θ sdx s ) t 0 is a wealth process of H-trading strategy θ. Consider H {F, G} with F G, i.e., for each t 0, F t G t. F represents regular agent and G represents informed agent. Assume that there are no arbitrage opportunities in F. Are there arbitrage opportunities in G?

4 Non-arbitrage condition NUPBR Let X be an H-semimartingale. We say that X satises No Unbounded Prot with Bounded Risk (NUPBR(H)) if for each T <, the set K H T (X ) := { (θ X ) T : θ L H (X ) and θ X 1 } is bounded in probability. The H-semimartingale X satises NUPBR(H) if there exists H-local martingale deator for X, i.e., a strictly positive H-local martingale L such that LX is an H-local martingale.

5 Enlargement of ltration We say that (H ) hypothesis is satised for F G if every F-martingale remains G-semimartingale. In such a case we are interested in G-semimartigale decomposition of F-martingale. Progressive enlargement: Jeulin-Yor's result up to random time. Initial enlargement: Jacod's results on enlargement of ltration and Stricker-Yor's results on calculus with parameter.

6 Random times Let τ be a random time, i.e., a positive random variable. Consider two F-supermartingales associated to τ: The process Z t dened as Z t = P(τ > t F t ) The process Z t dened as Z t = P(τ t F t ) Let A o be a F-dual optional projection of the process A = 1 [τ, [, i.e., for each optional process Y, A o satises E(Y τ 1 {τ< } ) = E( Y s da o s ) [0, [ Denote by m an F-martingale dened as m t = E(A o F t ). Then Z = m A o and Z = m A o m = Z Z and A o = Z Z

7 Progressive enlargement of ltration Progressively enlarged ltration G associated with τ is dened as G t = s>t(f s σ(τ s)). Jeulin-Yor's decomposition For each F-local martingale X, the stopped process X τ is G-semimartingale with semimartingale decomposition where X is G-local martingale. t τ X τ t = X t Z s d X, m F s

8 F-stopping time R The three sets { Z = 0}, {Z = 0} and {Z = 0} have the same début which is an F-stopping time R := inf{t 0 : Z t = 0} We decompose R as R = R R R with F-stopping times: R := R { ZR =0<Z R } R := R { ZR >0} and R := R{ZR =0} Notice that Z τ > 0 and Z τ > 0.

9 Optional semimartingale decomposition in G For H-locally integrable variation process V, we denote by V p,h its H-dual predictable projection, i.e., an H-predictable nite variation process such that for each H-predictable process Y, V p,h satises E( Y s dv s ) = E( Y s dv p,h s ). [0, [ [0, [ Theorem Let X be a F-local martingale. Then X τ t = Xt + t τ 0 ( ) p,f 1 d[x, m] s 1 Z X R s [ R, [ t τ where X is a G-local martingale.

10 Projections in G in terms of projections in F Lemma Let V be an F-adapted process with locally integrable variation. Then, we have (V τ ) p,g = 1 Z 1 [0,τ ] ( Z V ) p,f. The process U := 1 Z 1 [0,τ ] V is locally integrable variation process in G and U p,g = 1 Z 1 [0,τ ] ( 1 { Z>0} V )p,f.

11 G-local martingale deator Dened G-local martingale N = 1 Z 1 [0,τ ] m Then, continuous martingale part and jump process of N are of the form N c = 1 Z 1 [0,τ ] m c 1 N = m 1 [0,τ ] = m ( 1 [0,τ ] p,f 1 Z [ R ] Z ) 1 [0,τ ] Clearly ( N 1 + Z ) 1 [0,τ ] > 1 Z

12 G-local martingale deator Theorem Let L = E ( N). Then, for any F-local martingale X, the process LX τ L is a G-local martingale. (( X R + ) m, X R Z R 1 [ R, [ ) p,f τ Corollary If X is quasi-left continuous and = 0 on { R < }, then L is a G-local X R martingale deator for X τ.

13 Non-arbitrage up to Random Horizon Theorem Let τ be a random time. Then, the following are equivalent: 1. The thin set { Z = 0 & Z > 0} is evanescent. 2. F-stopping time R =. 3. For any F-local martingale X, process X τ L is a G-local martingale. 4. For any process X satisfying NUPBR(F), X τ satises NUPBR(G).

14 Proof Optional decomposition F-martingale X = 1 [ R, [ ( 1 [ R, [) p,f stopped at τ: yields that R = ( ) p,f X τ = 1 [ R, [ τ Takaoka's characterization, localization and equivalent change of measure

15 Initial enlargement of ltration Initially enlarged ltration G associated with random variable ξ is dened as G t = s>t(f s σ(ξ)). Jacod's hypothesis A real-valued random variable ξ satises Jacod's hypothesis if there exists a σ-nite positive measure η such that for every t 0 P(ξ du F t )(ω) η(du) P-a.s. As shown by Jacod, without loss of generality, η can be taken as law of ξ in the above denition.

16 Parameterized processes Stricker-Yor's calculus with parameter Consider a mapping X : (t, ω, u) X u t (ω) with values in R on R + Ω R. Let J be a class of F-optional processes, for example the class of F-(local) martingales or the class of F-locally integrable variation processes. Then, (X u, u R) is called a parametrized J -process if for each u U the process X u belongs to J and if it is measurable with respect to O(F) B(R).

17 Initial enlargement under Jacod's hypothesis For ξ satisfying Jacod's hypothesis, there exists a parameterized positive F-martingale (q u, u R) such that for every t 0, the measure q u t (ω)η(du) is a version of P(ξ du F t )(ω). Semimartingale decomposition Let (X u, u R) be a parameterized F-local martingale. Then t X ξ t = X ξ t + 0 where X ξ is an G-local martingale. 1 q ξ s d X u, q u F s u=ξ

18 F-stopping times R u For each u dene F-stopping time R u = inf{t : q u t = 0}. We have q u > 0 and q u > 0 on [[0, R u [[ and q u = 0 on [[R u, [[. G-stopping time R ξ = a.s. or equivalently q ξ t > 0 and q ξ t > 0 for t 0 P-a.s. Let us decompose F-stopping time R u as R u = R u R u with R u = R u {q u R u >0} and R u = R u {q u R u =0}.

19 Optional semimartingale deomposition in G Theorem Let (X u, u R) be a parameterized F-local martingale. Then X ξ decomposes as G-semimartingale as X ξ t t ξ = X t q ξ s ( p,f d[x ξ, q ξ ] s X ũ 1 R u [ R, [) u=ξ u where X ξ is a G-local martingale.

20 Projections in G in terms of projections in F Lemma Let (V u, u R) be a parameterized F-adapted càdlàg process with locally integrable variation. Then, The G-dual predictable projection of V ξ is ( V ξ ) p,g = 1 q ξ ( q u V u) p,f u=ξ. If V belongs to A + 1 (F), then the process U := V belongs to A + (G). loc q ξ loc The parametrized process (U u, u R) is well dened, its variation is G-locally integrable, and G-dual predictable projection of U ξ is given by (U ξ ) p,g = 1 q ξ ( 1 {qu >0} V u) p,f u=ξ.

21 G-local martingale q ξ Lemma Take the following G-local martingale q ξ := q ξ 1 ( p,f q ξ [qξ ] q ξ 1 [ R, [) u=ξ. u The G-predictable process The G-local martingale 1 q ξ is integrable with respect to q ξ. N := 1 q ξ q ξ has continuous martingale part and jump equal respectively to ( ) N c = 1 q ξ (q ξ ) c 1 ( ) q ξ (q ξ ) c F N = qξ q ξ p,f 1 [ R u ] u=ξ.

22 G-supermartingale 1 q ξ Lemma The process 1 q ξ is G-supermartingale with decomposition 1 q ξ = 1 1 (q ) ξ 2 qξ 1 q ξ ( p,f 1 [ R, [) u=ξ. u Moreover, it can be written as stochastic exponential of the form ( ) 1 q ξ = E 1 ( p,f q ξ q ξ 1 [ R, [) u=ξ. u The process 1 is G-local martingale if and only if R u = P η-a.s. q( ξ ) Then 1 = E 1 q ξ = E ( N). q ξ q ξ

23 G-local martingale deator Theorem Let L = E ( N). Then, for any parametrized F-local martingale (X u, u R), the process LX ξ L is a G-local martingale. (( X ũ R u + qu, X R u q ũ R u ) 1 [ R u, [ ) p,f u=ξ Corollary If X is quasi-left continuous and X ũ = 0 on {R u < } P η-a.s., then L is R u a G-local martingale deator for X ξ in G.

24 NUPBR condition for initial enlargement under Jacod's hypothesis Theorem The following conditions are equivalent: 1. The thin set {q u = 0 < q u } is evanescent η-a.a. 2. The F-stopping time R u = P η-a.s. 3. If (X u, u R) is parameterized F-local martingale, then X ξ satises NUPBR(G). Morover, X ξ martingale deator. q ξ is a G-local martingale, i.e., 1 q ξ is its G-local

25 Proof Optional decomposition under Jacod's hypothesis Parameterized F-martingale (X u, u R) with X u = 1 [ R u, [ ( 1 [ R u, [) p,f at ξ: ( p,f X ξ = 1 [ R, [) u=ξ u yields that R u = P η-a.s.

26 Optional semimartingale decomposition in G Let Q be a probability absolutely continuous with respect to P, and let ζ t = E P ( dq dp F t), S = inf{t > 0 : ζ t = 0} and S = S {ζs >0}. Let X be a P-local martingale. Then, X has decomposition under Q: where X is Q-local martingale. Up to random time τ: Under Jacod's hypothesis: 1 ) p,p X = X + ( X S ζ [X, ζ] 1 [ S, [ X τ 1 ( ) p,f = X + 1 [0,τ ] [X, m] 1 X R [ R, [ Z s τ X ξ = X ξ + 1 ( p,f q ξ [X, qξ ] X ũ 1 R u [ R, [) u=ξ u

27 Thank you!