Enlargement of filtration

Transcription

1 Enlargement of filtration Bernardo D Auria bernardo.dauria@uc3m.es web: July 6, 2017 ICMAT / UC3M

2 Enlargement of Filtration

3 Enlargement of Filtration ([1] 5.9) If G is a filtration larger than F, it is not true that an F-martingale is a G-martingale. K Itö [3] was the first to look at problems of enlargement of filtrations. From the seventies, Barlow, Jeulin and Yor started a systematic study of the problem of enlargement of filtrations: Which F-martingale M remains a G-semi-martingale, and if it is the case, what is the semi-martingale decomposition of M in G? 1

4 Immersion of Filtration

5 Immersion of Filtration ([1] 5.9.1) Let F and G be two filtrations such that F G When all F-martingales are G-martingales? That is is equivalent to ask E[D F t ] = E[D G t ], t and D L 1 (F ). Just consider G = F σ(d) with D L 1 (F ) and D not F 0 measurable. E[D G t ] = D is a G-martingale E[D F t ] is an F-martingale However for t = 0, E[D F t ] E[D G t ] = D, that implies that the F-martingale (E[D F t ], t 0) is not a G-martingale. 2

6 Immersion of Filtration ([1] 5.9.1) Definition The filtration F is said to be immersed in G if any square integrable F-martingale is a G-martingale. this is also known as the (H) hypothesis: (H) Every F-square integrable martingale is a G-square integrable martingale. Proposition Hypothesis (H) is equivalent to any of the following properties: (H1) t 0, the σ-fields F and G t are conditionally independent given F t (H2) t 0, G t L 1 (G t), E[G t F ] = E[G t F t] (H3) t 0, F L 1 (F ), E[F G t] = E[F F t] In particular, (H) holds if and only if every F-local martingale is a G-local martingale 3

7 The Brownian Bridge as example of Initial Enlargement

8 The Brownian Bridge example ([1] 5.9.2) Let B = (B t, t 0) be an F B -Brownian motion, and G t = F B t σ(b 1 ). In G the process B is not anymore a martingale, since E[B t G t ] = B 1 B t. Actually as we will see later b is a G-semi-martingale with decomposition t 1 B t = β t + 0 where β is a G-Brownian motion. B 1 B s ds, 1 s 4

9 The Brownian Bridge ([1] 4.3.5) Definition The Brownian Bridge, (b t, 0 t 1) is defined as the conditioned process (B t, t 1 B 1 = 0). The Brownian Bridge is a gaussian process, and each of the processes W, X, Y and Z below are examples of Brownian Bridges W t = B t t B 1, 0 t 1 t db s X t = (1 t) 0 1 s, 0 t 1 ( ) t Y t = (1 t) B, 0 t 1 1 t ( ) 1 t Z t = t B, 0 t 1 t 5

10 The laws on the canonical space Definition We denote by W (1) 0 >1 the law of the Brownian Bridge on the canonical space. Definition We denote by W (T ) x >y the law of the Brownian Bridge between x and y during the time interval [0, T ], on the canonical space. That is the process (x + B t t T B T + t ) T (y x); 0 t T. Theorem ([1] Th ) For every t, W (t) x >y is equivalent to W x on F s for s < t. W x is the Wiener measure such that W x (X 0 = x) = 1. 6

11 Markovian Bridges For general Markov processes with semi-group P t (x, dy) = p t (x, y) dy, P (t) x >y F s = p t s(x s, y) P x Fs p t (x, y) 7

12 n-dimensional Brownian motion For x = y = 0 and s < t ( ) W (t) t n/2 ( 0 >0 Xs 2 ) F s = exp W 0 Fs t s 2(t s) Identifying the density as the exponential martingale E(Z), where Z s = s 0 X u t u dx u, the canonical decomposition of the standard Brownian bridge (under W (t) 0 >0 ) is: X s = β s s 0 X u t u du, s < t, where (β s, s t) is a Brownian motion under W (t) 0 >0. 8

13 SDE for the Brownian bridge The Brownian bridge is solution of the following SDE { db t = bt 1 t dt + dβ t, 0 t < 1 b 0 = 0 9

14 The Brownian Bridge example ([1] 5.9.2) Proposition The decomposition of B in the filtration G is t 1 B t = W t + 0 where W is a G-Brownian motion. B 1 B s ds, 1 s It follows that if M is an F-local martingale such that 1 1 s d M, B s is finite, then 1 0 t 1 M t = M t + 0 where M is an G-local martingale. B 1 B s d M, B s, 1 s 10

15 The Brownian Bridge example ([1] 5.9.2) i Comment ([1] ) The singularity of ξ t = B 1 B t 1 t at t = 1, implies it is not square-integrable between 0 and 1. This prevents Girsanov measure change to transform the (G, P) semi-martingale (G, Q)-martingale. Let ds t = S t (µdt + σ db t ) and enlarge the filtration with S 1 (or equivalently with B 1 ). In the enlarged filtration, the dynamics are ds t = S t ((µ + σξ t )dt + σ dβ t ), and there does not exists an e.m.m. such that the discounted price process (e rt S t, t 1) s a G-martingale. 11

16 The Brownian Bridge example ([1] 5.9.2) ii However for every ɛ ]0, 1], there exists a uniformly integrable G-martingale L defined as dl t = µ r σ ξ t L t dβ t, t 1 ɛ, L 0 = 1, σ such that dq Gt = L t P Gt, the process (e rt S t, t 1 ɛ) is a (G, Q)-martingale. The information of the insider trader is too strong, such that it creates an arbitrage opportunity. 12

17 Bibliography

18 Bibliography M. Jeanblanc, M. Yor and M. Chesney (2009). Mathematical methods for financial markets. Springer, London. D. Revuz and M. Yor (1999). Continuous Martingales and Brownian Motion. Springer Verlag, Berlin. 3 rd ed. K. Itö (1976). Extension of stochastic integrals. Proc. Intern. Symp. on Stochastic Differential Equations,