The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications

Transcription

1 The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications

2 page 2 Seminar on Stochastic Geometry and its applications June 1, 215 Recall - Class of integrands for 1-dimensional Itô Integral Let V = V(S, T ) be the class of functions such that f (t, ω) : [, ) Ω R (i) (t, ω) f (t, ω) is B F - measurable (ii) f (t, ω) is F t - adapted T (iii) E [ f (t, ω) 2 dt] < S

3 page 3 Seminar on Stochastic Geometry and its applications June 1, 215 Extension of V (ii) There exists a filtration H = (H t) t such that a) B t is a martingale wrt (H t ) t b) f (t, ω) is H t - adapted

4 page 4 Seminar on Stochastic Geometry and its applications June 1, 215 Example Let B t(ω) = (B 1 (t, ω),, B n(t, ω)), t T be n-dimensional Brownian motion and define F (n) t = σ (B i (s, ) : 1 i n, s t) Then B k (t, ω) is a martingale wrt F (n) t thus Hence we can choose H t = F (n) and f (s, ω) db k (s, ω) exists for F (n) t - adapted integrands f t

5 page 5 Seminar on Stochastic Geometry and its applications June 1, 215 Definition - multidimensional Itô Integral Let B(t, ω) = (B 1 (t, ω),, B n(t, ω)) be n-dimensional Brownian motion and v = [v ij (t, ω)] be a m n - matrix where each entry v ij (t, ω) satisfies (i), (iii) and (ii) wrt some filtration H = (H t) t Then we define T S v db = T S v 11 v 1n db 1 v m1 v mn db n to be the m 1 - matrix whose i th component is n T j=1 S v ij (s, ω) db j (s, ω)

6 page 6 Seminar on Stochastic Geometry and its applications June 1, 215 Definition - multidimensional Itô processes Let B(t, ω) = (B 1 (t, ω),, B m(t, ω)) denote m-dimensional Brownian motion If the processes u i (t, ω) and v ij (t, ω) satisfy the conditions given in the definition of the 1-dimensional Itô process for each 1 i n, 1 j m then we can form n 1-dimensional Itô processes dx 1 = u 1 dt + v 11 db v 1m db m dx n = u n dt + v n1 db v nm db m

7 page 7 Seminar on Stochastic Geometry and its applications June 1, 215 Or, in matrix notation where X(t) = X 1 (t) X n(t), u = dx(t) = u dt + v db(t) u 1 u n, v = v 11 v 1m v n1 v nm Then X(t) is called an n-dimensional Itô process, db(t) = db 1 (t) db m(t)

8 page 8 Seminar on Stochastic Geometry and its applications June 1, 215 Theorem - The general Itô formula Let X(t) = X() + u(s) ds + v(s) db(s) be an n-dimensional Itô process Let g(t, x) = (g 1 (t, x),, g p(t, x)), p N, be a C 2 map from [, ) R n into R p Then the process Y (t, ω) = g(t, X(t)) is again an Itô process, whose k th component, k = 1,, p, is given by

9 page 9 Seminar on Stochastic Geometry and its applications June 1, 215 Y k (t) =Y k () i,j=1 n i=1 ( g k n (s, X(s)) + t i=1 n 2 g k (s, X(s))v i (s)v j (s) T ds x i x j with v i (s) the i th row of v g k x i (s, X(s))v i (s) db(s) g k x i (s, X(s))u i (s)

10 page 1 Seminar on Stochastic Geometry and its applications June 1, 215 Examples a) Let B(t, ω) = (B 1 (t, ω),, B n(t, ω)) be an n-dimensional Brownian motion, n 2, and consider ( ) 1/2 R(t, ω) = B1(t, 2 ω) + + Bn(t, 2 ω) Then it follows with Itô s formula R(t) = n i=1 B i (s) t R(s) db i(s) + n 1 2R(s) ds

11 page 11 Seminar on Stochastic Geometry and its applications June 1, 215 b) Let B t be an 1-dimensional Brownian motion and Y t = 2 + t + e B t Then Y t = 3 + (1 + e Bs ) ds + e Bs db s b) Let B(t, ω) = (B 1 (t, ω), B 2 (t, ω)) be a 2-dimensional Brownian motion and Y t = B 2 1(t) + B 2 2(t) Then Y t = 2 ds + 2B 1 (s) db 1 (s) + 2B 2 (s) db 2 (s)

12 page 12 Seminar on Stochastic Geometry and its applications June 1, 215 d) With Itô s formula it holds that B 2 s db s = 1 3 B3 t B s ds d) Let B t be an 1-dimensional Brownian motion Define β k = E[Bt k ] ; k =, 1, 2 ; t Use Itô s formula to prove that β k = 1 t k(k 1) β k 2 (s) ds ; k 2 2

13 page 13 Seminar on Stochastic Geometry and its applications June 1, 215 Integration by parts Let X t, Y t be two 1-dimensional Itô processes, ie, Then it holds X t = X + Y t = Y + X ty t =X Y + u X (s) ds + u Y (s) ds + v X (s) db s v Y (s) db s (X su Y (s) + Y su X (s) + v X (s)v Y (s)) ds + X sv Y (s) + Y sv X (s) db s

14 page 14 Seminar on Stochastic Geometry and its applications June 1, 215 Exponential martingales Suppose θ(t, ω) = (θ 1 (t, ω),, θ n(t, ω)) with θ k (t, ω) V(, T ) k = 1,, n, where T Define Z t = exp θ(s, ω) db(s) 1 θ(s, ω) T θ(s, ω) ds 2, t T where B(s) is an n-dimensional Brownian motion Then it holds a) Z t = 1 + b) Z t is a martingale for t T, provided that Z sθ(s, ω) db(s) Z tθ k (t, ω) V(, T ) k = 1,, n

15 page 15 Seminar on Stochastic Geometry and its applications June 1, 215 References Øksendal, B (23) Stochastic Differential Equations: An Introduction with Applications, Sixth Edition, Springer-Verlag, Berlin Karatzas, I, Shreve, SE (1998) Brownian Motion and Stochastic Calculus, Second Edition, Springer-Verlag, New York