Stochastic Integrals and Itô s formula.

Transcription

1 Chapter 5 Stochastic Itegrals ad Itô s formula. We will call a Itô process a progressively measurable almost surely cotiuous process x(t, ω) with values i R d, defied o some (Ω, F t, P) that is related to progressively measurable bouded fuctios [a(s, ω), b(s, ω)] i the followig maer. exp[ θ, x(t, ω) x(, ω) b(s, ω)ds 1 θ, a(s, ω)θ ds] is a martigale with respect to (Ω, F t, P) for all θ R d. A caoical example is Browia motio that correspods to b(s, ω) ad a(s, ω) 1 or a(s, ω) I i higher dimesios.we will abbreviate it by x( ) I(a, b). Such processes are ot of bouded variatio uless a. I fact they have otrivial quadratic variatio. Lemma 5.1. If x( ) is a oe dimesioal process ad x( ) I(a, b) the lim x( jt 1)T T ) x((j ) = a(s, ω)ds j=1 i probability ad i L 1 (P). Proof. If y(t) = x(t) x() b(s, ω)ds, the y( ) I(a, ) ad the differece betwee x( ) ad y( ) is a cotiuous fuctio of bouded variatio. it is therefore sufficiet to show that lim y( jt 1)T T ) y((j ) = a(s, ω)ds If we deote by j=1 Z j = y( jt ) y((j 1)T 1 ) jt (j 1)T a(s, ω)ds

2 CHAPTER 5. STOCHASTIC INTEGRALS AND ITÔ S FORMULA. the E[Z j ] = ad for i j, E[Z i Z j ] =. It is therefore sufficiet to show E[ Z j ] C(T). This follows easily from the Gaussia boud E[e λ(y(t) y(t1)) ] e Cλ (t t 1 ) provided a(s, ω) C. We see that E[(y(t ) y(t 1 )) 4 ] C(t t 1 ). This meas that itegrals of the form e(s, ω)dx(s, ω) have to be carefully defied. Sice the differece betwee x( ) ad y( ) is of bouded variatio it suffices to cocetrate o e(s, ω)dx(s, ω). We develop these itegrals i several steps, each oe formulated as a lemme. Lemma 5.. Let S be the space of fuctios e(s, ω) that are uiformly bouded piecewise costat progressively measurable fuctios of s. I other words there are itervals [t j 1, t j ) i which e(s, ω) is equal to e(t j 1, ω) which is F tj 1 measurable. We defie for t k 1 t t k ξ(t) = k 1 e(s, ω)dy(s) = e(t j 1, ω)[y(t j ) y(t j 1 )]+e(t k 1, ω)[y(t) y(t k 1 )] j=1 The followig facts are easy to check. 1. ξ(t) is almost surely cotiuous, progressively measurable. Moreover ξ( ) I(e (s, ω)a(s, ω), ).. The space S is liear ad the map e ξ is a liear map. 3. E[ sup ξ(s, ω) ] 4E[ s t e(s, ω) a(s, ω)ds] 4. I particular if e 1, e S, ad for i = 1, the ξ i (t) = E[ sup ξ 1 (s, ω) ξ (s, ω) ] 4E[ s t e i (s, ω)dy(s) Proof. It is easy to see that, because for λ R, E[exp[λ[y(t) y(s)] λ s e 1 (s, ω) e (s, ω) a(s, ω)ds] a(u, ω)du] F s ] = 1

3 3 it follows that if λ is replaced by λ(ω) that is bouded ad F s measurable the E[exp[λ(s, ω)[y(t) y(s)] λ(s, ω) s a(u, ω)du] F s ] = 1 We ca take λ(s, ω) = λe ( s, ω). This proves 1. is obvious ad 3 is just Doob s iequality. 4 is a restatemet of 3 for the differece. Lemma 5.3. Give a bouded progressively measurable fuctio e(s, ω) it ca be approximated by a sequece e S, such that {e } are uiformly bouded ad T lim E[ e (s, ω) e(s, ω) ds] = As a cosequece the sequece ξ (t) = e (s, ω)dy(s) has a limit ξ(t, ω) i the sese lim E[ sup ξ (s) ξ(s) ] = s t It follows that ξ(t, ω) is almost surely cotiuous ad ξ( ) I(e (s, ω)a(s, ω)). Proof. It is eough to prove the approximatio property. Sice Y λ (t) = exp[λξ (t) λ s e (s, ω)a(s, ω)ds] are martigales ad e a has uiform boud C, it follows that sup t T sup E[exp[λξ (t)]] exp[ Cλ T ] providig uiform itegrability. We ote that lim E[ sup ξ (s) ξ m (s) ] =,m s t Now it is easy to show that ξ ( ) has a uiform limit i probability ad pass to the limit. To prove the approximatio property we approximate first e(s, ω) by e h (s, ω) = 1 h s (s h) e(u, ω)du It is a stadard result i real variables that e h ( ) e h ( ) as h ad e h is cotuuos i s. Note that we oly look back ad ot ahead, thus preservig progressive measurability. We ca ow approximate e h (s, ω) by e h ( [s], ω) that are agai progressively measurable, but simple as well, so they are i S. Lemma 5.4. If e(s, ω) is progressively measurable ad satisfies T E[ e (s, ω)a(s, ω)ds] <

4 4 CHAPTER 5. STOCHASTIC INTEGRALS AND ITÔ S FORMULA. we ca defie o [, T], ξ(t) = as a square itegrable martigale ad will be a martigale. ξ(t) e(s, ω)dy(s) e (s, ω)a(s, ω)ds Proof. The proof is elemetary. Just approximate e by trucated fuctios ad pass to the limit. Agai e (s, ω) = e(s, ω)1 { e(s,ω) } (ω) lim E[ sup ξ (s) ξ m (s) ] =,m s t Remark 5.1. If x( ) I(a, b) we ca let y(t) = x(t) b(s, ω)ds ad defie If ξ(t) = e(s, ω)dx(s) = E[ e(s, ω)dy(s) + b (s, ω)e (s, ω)ds] < e(s, ω)b(s, ω)ds the we ca check ξ is well defied. I fact we ca defie for bouded progressively measurable e, c, ξ(t) = e(s, ω)dx(s) + c(s, ω)ds It is easy to check that ξ( ) I(e (s, ω)a(s, ω), e(s, ω)b(s, ω) + c(s, ω)) Recall that if X N[µ, σ ] ad Y = ax + b the Y N[aµ + b, a σ ]. Remark 5.. We ca have x(t) R d ad x( ) I(a, b), where a = a(s, ω) is a symmetric positive semidefiite matrix valued bouded progressively measurable fuctio ad b = b(s, ω) is a R d valued, bouded ad progressively measurable. We ca the defie ξ(t) = e(s, ω) dx(s) + c(s, ω)ds

5 5 where e(s, ω) is a progressively measurable bouded k d matrix ad c is R k valued, bouded ad progressively measurable. The itegral is defied by each compoet. For 1 i k, ξ i (t) = e i,j (s, ω) dx j (s) + c i (s, ω)ds j The oe verifies easily that ξ( ) I(eae, eb + c) Theorem 5.5. Itô s formula. Cosider a smooth fuctio f(t, x) o [, T] R d. Let x(t) with values i R d belog to I(a, b). The almost surely f(t, x(t)) = f(, x()) f s (s, x(s))ds + ( x f)(s, x(s)) dx(s) ai,j (s, ω)(d xi,x j f)(s, x(s))ds Proof. Cosider the d+1 dimesioal process Z(t) = (f(t, x(t)), x(t)). If σ R ad λ d, the if we cosider g(t, x) = σf(t, x) + λ, x we kow that exp[g(t, x(t)) g(, x()) is a martigale. A computatio yields e g [ s e g + L s,ω e g ](s, x(s))ds] e g [ s e g + L s,ω e g ] = s g + L s,ω g + 1 g, a g = σ s f + σl s,ω f + λ, b(s, ω) + 1 (σ f + λ), a(s, ω)(σ f + λ) Implies that Z(t) I(ã, b), where ( ) f, a f (a f) tr ã = (a f) a ad b = ( s f + L s,ω f, b) Now we ca compute that w( ) I(A, B) where w(t) = 1 df(s, x(s)) 1 ( s f)(s, x(s))ds i.j ( s f)(s, x(s)) dx(s) a i,j (s, ω)(d xix j f)(s, x(s))ds

6 6 CHAPTER 5. STOCHASTIC INTEGRALS AND ITÔ S FORMULA. If we ca calculate ad show that A = ad B =, this would imply that w(t) ad that proves the theorem. ( ) ( ) f, a f (a f) tr 1 A = (1, f) = (a f) a f B = s f + L s,ω f b f s f 1 a i,j (s, ω)(d xix j f) = i.j Remark 5.3. If x( ) I(a, b) ad y(t) = σ(s, ω) dx(s) + c(s, ω)ds we saw that y( ) I(ã, b) where ã = σaσ, b = σb + c This is like liear chage of variables of a Gaussia vector. dx N[a dt, b dt] ad σdx + c N[σaσ dt, (σb + c)dt]. We ca develop stochastic itegrals of y( ) ad if dz = σ dy+c dt the dz = σ [σdx+cdt]+c dt = σ σdx+(σ c+c )dt. If σ is a ivertible the dy = σdx+cdt ca be iverted as dx = σ 1 dy σ 1 cdt. Fially oe ca remember Itô s formula by the rules df(t, x(t)) = f t dt + i f xi dx i + 1 f xi,x j dx i dx j If x( ) I(a, b) the dx i dx j = a i,j dt. (dt) = dtdx i =. Because the typical paths have half a derivative (more or less) dx dt. dx i dx j is of the order of dt ad dxdt, (dt) are egligible. i,j