A Direct Proof of the Bichteler-Dellacherie Theorem and Connections to Arbitrage

Transcription

1 A Direct Proof of the Bichteler-Dellacherie Theorem and Connections to Arbitrage Bezirgen Veliyev, University of Vienna (joint with Mathias Beiglböck and Walter Schachermayer) August 23, 2010 Paris, European Summer School in Financial Mathematics

2 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T.

3 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded.

4 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale.

5 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale.

6 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. S is a good integrator BD S is a semimartingale

7 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. S is a good integrator NFLVR BD DS94 S is a semimartingale

8 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. S is a good integrator weak-nflvr NFLVR BD DS94 S is a semimartingale

9 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. S is a good integrator weak-nflvr NFLVR S is a semimartingale

10 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. S is a good integrator weak-nflvr BD S is a semimartingale

11 Introduction We have a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions and a real-valued, càdlàg, adapted process S = (S t ) 0 t T. For the talk, we assume that S is locally bounded. Theorem (Bichteler 79, 81; Dellacherie 80) S is a good integrator if and only if S is a semimartingale. Theorem (DS 94) If a locally bounded process S satisfies NFLVR for simple integrands, then S is a semimartingale. weak-nflvr NFLVR DS94 S is a semimartingale

12 Good Integrator A simple integrand is a stochastic process of the form H t = n h j 1 ]τj 1,τ j ](t) j=1 where 0 = τ 0 τ 1 τ n = T are stopping times and h j L (Ω, F τj 1, P). Denote by SI the vector space of simple integrands.

13 Good Integrator A simple integrand is a stochastic process of the form H t = n h j 1 ]τj 1,τ j ](t) j=1 where 0 = τ 0 τ 1 τ n = T are stopping times and h j L (Ω, F τj 1, P). Denote by SI the vector space of simple integrands. For each S, we may well-define the integration operator I : SI L 0 (Ω, F, P) n h j 1 ]τj 1,τ j ] j=1 n h j (S τj S τj 1 ) =: (H S) T. j=1

14 Good Integrator A simple integrand is a stochastic process of the form H t = n h j 1 ]τj 1,τ j ](t) j=1 where 0 = τ 0 τ 1 τ n = T are stopping times and h j L (Ω, F τj 1, P). Denote by SI the vector space of simple integrands. For each S, we may well-define the integration operator I : SI L 0 (Ω, F, P) n h j 1 ]τj 1,τ j ] j=1 n h j (S τj S τj 1 ) =: (H S) T. j=1 S is a good integrator if I is continuous i.e. if H n 0, then (H n S) T 0 in probability.

15 Good Integrator and NFLVR S is a good integrator if for every sequence (H n ) n=1 satisfying H n 0, we have (H n S) T 0 in probability.

16 Good Integrator and NFLVR S is a good integrator if for every sequence (H n ) n=1 satisfying H n 0, we have (H n S) T 0 in probability. little investment little outcome

17 Good Integrator and NFLVR S is a good integrator if for every sequence (H n ) n=1 satisfying H n 0, we have (H n S) T 0 in probability. little investment little outcome S satisfies NFLVR if for every sequence (H n ) n=1 satisfying (H n S) 0, we have (H n S) T 0 in probability.

18 Good Integrator and NFLVR S is a good integrator if for every sequence (H n ) n=1 satisfying H n 0, we have (H n S) T 0 in probability. little investment little outcome S satisfies NFLVR if for every sequence (H n ) n=1 satisfying (H n S) 0, we have (H n S) T 0 in probability. vanishing risk little outcome

19 NFLVRLI S satisfies NFLVRLI if for every sequence (H n ) n=1 satisfying H n 0 and (H n S) 0, we have (H n S) T 0 in probability.

20 NFLVRLI S satisfies NFLVRLI if for every sequence (H n ) n=1 satisfying H n 0 and (H n S) 0, we have (H n S) T 0 in probability. little investment and vanishing risk, little outcome.

21 NFLVRLI S satisfies NFLVRLI if for every sequence (H n ) n=1 satisfying H n 0 and (H n S) 0, we have (H n S) T 0 in probability. little investment and vanishing risk, little outcome. S is a good integrator S satisfies NFLVRLI.

22 NFLVRLI S satisfies NFLVRLI if for every sequence (H n ) n=1 satisfying H n 0 and (H n S) 0, we have (H n S) T 0 in probability. little investment and vanishing risk, little outcome. S is a good integrator S satisfies NFLVRLI. S satisfies NFLVR S satisfies NFLVRLI.

23 Theorem (Main Theorem) Let S be locally bounded process. TFAE: S satisfies NFLVRLI. S is a semimartingale.

24 Sketch of Proof Assume S 0 = 0, T = 1.

25 Sketch of Proof Assume S 0 = 0, T = 1. Being a semimartingale is a local property, hence assume S is bounded. WLOG, S 1.

26 Sketch of Proof Assume S 0 = 0, T = 1. Being a semimartingale is a local property, hence assume S is bounded. WLOG, S 1. Consider D n = {0, 1 2 n,..., 2n 1 2 n, 1} and S n sampled on D n.

27 Sketch of Proof Assume S 0 = 0, T = 1. Being a semimartingale is a local property, hence assume S is bounded. WLOG, S 1. Consider D n = {0, 1 2,..., n 2n 1 2, 1} and S n sampled on D n n. Apply discrete Doob-Meyer to obtain S n = M n + A n, where (M n j ) 2n 2 n j=0 is a martingale and (An j ) 2n 2 n j=0 is predictable.

28 Sketch of Proof Continued... Lemma Assume NFLVRLI. For ε > 0, there exist a constant C > 0 and a sequence of { j 2 } n 2n j=1 { }-valued stopping times (ϱ n) n=1 such that P(ϱ n < ) < ε and TV (A n,ϱn ) = 2 n (ϱ n 1) j=1 A n j A n 2 n j 1 2 n C, (1) M n,ϱn 1 2 L 2 (Ω) = Mn ϱ n 1 2 L 2 (Ω) C. (2)

29 Sketch of Proof Continued... Lemma Assume NFLVRLI. For ε > 0, there exist a constant C > 0 and a sequence of { j 2 } n 2n j=1 { }-valued stopping times (ϱ n) n=1 such that P(ϱ n < ) < ε and TV (A n,ϱn ) = 2 n (ϱ n 1) j=1 A n j A n 2 n j 1 2 n C, (1) M n,ϱn 1 2 L 2 (Ω) = Mn ϱ n 1 2 L 2 (Ω) C. (2) Idea: Ht n = 2 2 n j=1 S j n ] j 1 2 n, j (t) 2 n Mn,ϱn ] 1 2 L 2 (Ω) (Hn S) T.

30 Sketch of Proof Continued... Lemma Assume NFLVRLI. For ε > 0, there exist a constant C > 0 and a sequence of { j 2 } n 2n j=1 { }-valued stopping times (ϱ n) n=1 such that P(ϱ n < ) < ε and TV (A n,ϱn ) = 2 n (ϱ n 1) j=1 A n j A n 2 n j 1 2 n C, (1) M n,ϱn 1 2 L 2 (Ω) = Mn ϱ n 1 2 L 2 (Ω) C. (2) Idea: Ht n = 2 2 n j=1 S j n ] j 1 2 n, j (t) 2 n Mn,ϱn ] 1 2 L 2 (Ω) (Hn S) T. Ht n = 2 n j=1 sign( A n j A n ) 2 n j 1 1] j 1 2 n 2 n, j 2 n ] (t) TV (An ) (H n S) T.

31 Sketch of Proof Continued... We want to pass to the limits

32 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and

33 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ].

34 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ]. S = A + M on [0, ϱ]. Since ε was arbitrary, S is a semimartingale.

35 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ]. S = A + M on [0, ϱ]. Since ε was arbitrary, S is a semimartingale. Lemma (Komlos L 2 -version) Let (f n ) n 1 be a sequence of measurable functions on a probability space (Ω, F, P) such that sup n 1 f n 2 <. Then, there exist functions g n conv(f n, f n+1,... ) such that (g n ) n 1 converges almost surely and in. L 2 (Ω).

36 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ]. S = A + M on [0, ϱ]. Since ε was arbitrary, S is a semimartingale. Lemma (Komlos L 2 -version) Let (f n ) n 1 be a sequence of measurable functions on a probability space (Ω, F, P) such that sup n 1 f n 2 <. Then, there exist functions g n conv(f n, f n+1,... ) such that (g n ) n 1 converges almost surely and in. L 2 (Ω). R n T = 1 [0,ϱ n] R T using Komlos.

37 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ]. S = A + M on [0, ϱ]. Since ε was arbitrary, S is a semimartingale. Lemma (Komlos L 2 -version) Let (f n ) n 1 be a sequence of measurable functions on a probability space (Ω, F, P) such that sup n 1 f n 2 <. Then, there exist functions g n conv(f n, f n+1,... ) such that (g n ) n 1 converges almost surely and in. L 2 (Ω). R n T = 1 [0,ϱ n] R T using Komlos. R T is a good enough substitute for ϱ.

38 Sketch of Proof Continued... We want to pass to the limits ϱ n ϱ and A n A and M n M on [0, ϱ]. S = A + M on [0, ϱ]. Since ε was arbitrary, S is a semimartingale. Lemma (Komlos L 2 -version) Let (f n ) n 1 be a sequence of measurable functions on a probability space (Ω, F, P) such that sup n 1 f n 2 <. Then, there exist functions g n conv(f n, f n+1,... ) such that (g n ) n 1 converges almost surely and in. L 2 (Ω). R n T = 1 [0,ϱ n] R T using Komlos. R T is a good enough substitute for ϱ. Using Komlos again, A n A and M n M on [0, ϱ].