An Overview of the Martingale Representation Theorem
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1 An Overview of the Martingale Representation Theorem Nuno Azevedo CEMAPRE - ISEG - UTL nazevedo@iseg.utl.pt September 3, 21 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25
2 Background and notation Definition Denote by {F t } t the filtration generated by the one-dimensional Brownian motion B t and by B the Borel σ-algebra on [, ). Let V = V(S, T ) be the class of functions f : [, ) Ω R such that (i) (t, ω) f (t, ω) is B F-measurable. (ii) f (t, ω) is F t -adapted. [ ] T (iii) E S (f (t, ω))2 dt <. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 2 / 25
3 Background and notation Definition (The Itô integral) Let f V(S, T ). Then the Itô integral of f (from S to T ) is defined by S f (t, ω)db t (ω) = lim n S φ n (t, ω)db t (ω) limit in L 2 (P), (1) where {φ n } is a sequence of elementary functions such that [ ] T E (f (t, ω) φ n (t, ω)) 2 dt as n, (2) S where the limit in (1) exists and does not depend on the choice of {φ n }, as long as (2) holds. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 3 / 25
4 Background and notation Properties of the Itô integral: Itô[ isometry: ( ) ] T 2 E S f (t, ω)db t(ω) = E Linearity [ ] T E S f db t = S [ S f db t is F T -measurable Existence of a continuous version Martingale property Definition (Martingale) ] (f (t, ω))2 dt for all f V(S, T ) Let (Ω, F, P) be a probability space and X = {X t : t } a stochastic process on it. We say that X t is a martingale with respect to the filtration {F t } t if (i) X is adapted to F t. (ii) E [ X t ] < for all t. (iii) E [X s F t ] = X t for all s t. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 4 / 25
5 Background and notation Theorem (Itô formula) Let X t be an Itô process given by and let g(t, x) C 2 ([, ) R). Then is again an Itô process, and dx t = udt + vdb t Y t = g(t, X t ) dy t = g t (t, X t)dt + g x (t, X t)dx t g 2 x 2 (t, X t)(dx t ) 2, where (dx t ) 2 = (dx t ).(dx t ) is computed according to the multiplication table dt.dt = dt.db t = db t.dt =, db t.db t = dt. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 5 / 25
6 Let B(t) = (B 1 (t),..., B n (t)) be a n-dimensional Brownian motion. We know that if v V n then the Itô integral t X t = X + v(s, w)db(s); t is always a martingale w.r.t. filtration F (n) t. In this talk we will prove that the converse is also true: (n) any F t -martingale (w.r.t. P) can be represented as an Itô integral. this result is know as martingale representation theorem. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 6 / 25
7 Theorem (The martingale representation theorem) Let B(t) = (B 1 (t),..., B n (t)) be n-dimensional. Suposse M t is an F (n) t -martingale (w.r.t. P) and that M t L 2 (P) for all t. Then there exists a unique stochastic process g(s, ω) such that g V (n) (, t) for all t and t M t (ω) = E[M ] + g(s, ω)db(s) a.s., for all t. The key result for the proof of the martingale representation theorem is the Itô representation theorem Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 7 / 25
8 Theorem (The Itô representation theorem) Let F L 2 (F (n) T, P). Then there exists a unique stochastic process f (t, ω) V n (, T ) such that F (ω) = E[F ] + f (t, ω)db(t). For the proof of the Itô representation theorem we need to prove some auxiliary lemmas. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 8 / 25
9 Lemma (Doob-Dinkyn Lemma) Let (Ω, Σ) and (S, A) be measurable spaces and f : Ω S be measurable, i.e., f 1 (A) Σ. Then a function g : Ω R is measurable relative to the σ-algebra f 1 (A) [i.e., g 1 (B) f 1 (A)] if and only if there is a measurable function h : S R such that g = h f. ( ) Let g = h f : Ω R be measurable. Then since h 1 (B) A. g 1 (B) = (h f ) 1 (B) = f 1 (h 1 (B)) f 1 (A) Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 9 / 25
10 ( ) Let g be f 1 (A)-measurable, where f 1 (A) is a σ-algebra contained in Σ. We start by checking that it is enough to prove the result for simple functions g = n a i χ Ai, A i f 1 (A). i=1 Recall that for a measurable function g w.r.t. the σ-algebra f 1 (A), there exists a sequence of simple functions g n, measurable w.r.t. f 1 (A), such that as n for each ω Ω. g n (ω) g(ω) Assuming the result holds for simple functions, there is an A-measurable h n : S R, g n = h n f, for each n 1. Define S = {s S : h n (s) h(s), n }. Then: S A and f (Ω) S. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25
11 Let h(s) = h(s) if s S and h(s) = if s S S Then h is A-measurable and g(ω) = h(f (ω)), ω Ω, as required Assume now that g = n i=1 a iχ Ai, A i = f 1 (B i ) f 1 (A) for a B i A. Define h = n i=1 a iχ Bi Then h : S R is A-measurable and simple. Thus and h f = g h(f (ω)) = = n a i χ Bi (f (ω)) = i=1 n a i χ f 1 (B i )(ω) i=1 n a i χ Ai (ω) = g(ω), ω Ω, i=1 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
12 If S = R n and A is the Borel σ-algebra of R n, then there is an h : R n R, Borel measurable, which satisfies the requirements. Corollary Let (Ω, Σ) and (R n, A) be measurable spaces, and f : Ω R n be measurable. Then g : Ω R is f 1 (A)-measurable if and only if there is a Borel measure function h : R n R such that g = h(f 1, f 2,..., f n ) = h f where f = (f 1,..., f n ). Lemma Fix T >. The set of random variables is dense in L 2 (F T, P). {φ(b t1,..., B tn ); t i [, T ], φ C (R n ), n = 1, 2,...} Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
13 Let {t i } i=1 be a dense subset of [, T ]; Let H n be the σ-algebra generated by B t1 ( ),..., B tn ( ). Then clearly Hn H n+1. FT is the smallest σ-algebra containing all the H n. Choose g L 2 (F T, P). then by the martingale convergence theorem, we have that g = E[g F T ] = lim n E[g Hn]. The limit is pointwise a.e. (P) and in L 2 (F T, P). By the Doob-Dynkin Lemma we can write, for each n, E[g H n] = g n(b t1,..., B tn ) for some Borel measurable function g n : R n R. Each such g n (B t1,..., B tn ) can be approximated in L 2 (F T, P) by functions φ n (B t1,..., B tn ), where φ n C (Rn ) and the result follows. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
14 Lemma The linear span of random variables of the type { exp h(t)db t (ω) 1 } h 2 (t)dt ; h L 2 [, T ](deterministic) 2 is dense in L 2 (F T, P). Suppose g{ L 2 (F T, P) is orthogonal (in L 2 (F T, P)) to all functions of the form exp h(t)db } t(ω) 1 T 2 h2 (t)dt. Then in particular G(λ) := exp{λ 1B t1 (ω) λ nb tn (ω)}g(ω)dp(ω) = Ω for all λ = (λ 1,..., λ n) R n and all t 1,..., t n [, T ]. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
15 The function G(λ) is real analytic in λ R n and hence G has an analytic extension to the complex space C n given by G(z) := exp{z 1 B t1 (ω) z n B tn (ω)}g(ω)dp(ω) for all z = (z 1,..., z n ) C n. Ω Since G = on R n and G is analytic, G = on C n. In particular, G(iy1,..., iy n) = for all y = (y 1,..., y n) R n. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
16 But then we get, for φ C (Rn ) φ(b t1,..., B tn )g(ω)dp(ω) where = Ω (2π) n 2 = (2π) n 2 = (2π) n 2 Ω ( ) i(y φ(y)e 1B t ynbtn ) dy g(ω)dp(ω) R n ( ) φ(y) e i(y1bt ynbtn ) g(ω)dp(ω) dy R n is the Fourier transform of φ. Ω R n φ(y)g(iy)dy =, φ(y) = (2π) n 2 R n φ(x)e ix y dx Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
17 We have used the inverse Fourier transform theorem φ(x) = (2π) n 2 φ(y)e ix y dy. Since Ω R n φ(b t1,..., B tn )g(ω)dp(ω) = and from the previous lemma g is orthogonal to a dense subset of L 2 (F T, P), we conclude that g =. Therefore the linear span of the functions must be dense in L 2 (F T, P). { exp h(t)db t (ω) 1 } h 2 (t)dt, 2 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
18 Proof (Itô representation theorem). We consider only the case n = 1 (the proof in the general case is similar). First assume that F is of the form for some h(t) L 2 [, T ]. Define { F (ω) = exp h(t)db t (ω) 1 2 } h 2 (t)dt { t Y t (ω) = exp h(s)db s (ω) 1 t } h 2 (s)ds ; t T. 2 Then by Itô s formula dy t = Y t (h(t)db t 1 2 h2 (t)dt) Y t(h(t)db t ) 2 = Y t h(t)db t Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
19 So that Therefore E[F ] = 1. t Y t = 1 + Y s h(s)db s ; t [, T ]. F = Y T = 1 + Y s h(s)db s F (ω) = E[F ] + f (t, ω)db(t) holds in this case. By linearity F (ω) = E[F ] + f (t, ω)db(t) also holds for linear combinations of functions of the form { exp h(t)db t (ω) 1 2 } h 2 (t)dt. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
20 If F L 2 (F T, P) is arbitrary, we approximate F in L 2 (F T, P) by linear combinations F n of functions of the form Then for each n we have where f n V(, T ). { exp h(t)db t (ω) 1 } h 2 (t)dt. 2 F n (ω) = E[F n ] + f n (s, ω)db s (ω) Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 2 / 25
21 By the Itô isometry E[(F n F m ) 2 ] = E[(E[F n F m ] + = E([F n F m ]) 2 + (f n f m )db) 2 ] E[(f n f m ) 2 ]dt n, m {fn} is a Cauchy sequence in L 2 ([, T ] Ω). and hence converges to some f L 2 ([, T ] Ω). since fn V(, T ) we have f V(, T ). Again using the Itô isometry we see that ( ) F = lim F n = lim E[F n ] + f n db = E[F ] + f db n n Hence the representation F (ω) = E[F ] + F L 2 (F T, P). f (t, ω)db(t) holds for all Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
22 The uniqueness follows from the Itô isometry. Suppose F (ω) = E[F ] + f 1 (t, ω)db t (ω) = E[F ] + f 2 (t, ω)db t (ω) with f 1, f 2 V(, T ). Then = E[( (f 1 (t, ω) f 2 (t, ω))db t (ω)) 2 ] = E[(f 1 (t, ω) f 2 (t, ω)) 2 ]dt. and therefore f1(t, ω) = f 2(t, ω) for a.a. (t, ω) [, T ] Ω. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
23 Proof (Martingale Representation Theorem). n = 1. By the Itô representation theorem applied to T = t and F = M t, we have that for all t there exists a unique f (t) (s, ω) L 2 (F T, P) such that M t(ω) = E[M t] + Now assume t 1 t 2. Then t f (t) (s, ω)db s(ω) = E[M ] + [ t 2 M t1 = E[M t2 F t1 ] = E[M ] + E t f (t) (s, ω)db s(ω). f (t2) (s, ω)db s (ω) F t1 ] t1 = E[M ] + f (t2) (s, ω)db s (ω). (3) Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
24 But we also have t1 M t1 = E[M ] + f (t1) (s, ω)db s (ω). (4) Hence, comparing (3) and (4) we get that [( t 1 ) 2 ] t1 = E (f (t2) f (t1) )db = E[(f (t2) f (t1) )]ds Therefore f (t1) (s, ω) = f (t2) (s, ω) for a.a. (s, ω) [, t 1 ] Ω. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
25 So we can define f (s, ω) for a.a. s [, ) Ω by setting f (s, ω) = f (N) (s, ω), if s [, N]. Then we get t t M t = E[M ] + f (t) (s, ω)db s (ω) = E[M ] + f (s, ω)db s (ω), for all t. Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, / 25
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