3. WIENER PROCESS. STOCHASTIC ITÔ INTEGRAL

Transcription

1 3. WIENER PROCESS. STOCHASTIC ITÔ INTEGRAL 3.1. Wiener process. Main properties. A Wiener process notation W W t t is named in the honor of Prof. Norbert Wiener; other name is the Brownian motion notation B B t t. Wiener process is Gaussian process. As any Gaussian process, Wiener process is completely described by its expectation and correlation functions. We give below a description of main properties of W W t t : 1. W ;. paths traectories of Wiener process are continuous functions of t [, ; 3. expectation EW t ; 4. correlation function EW t W s t s, a b mina, b; 5. for any t 1,..., t n the random vector W t1,..., W tn is Gaussian; 6. For any s, t EWt t E W s ], E W s ] t s ; 7. Increments of Wiener process on non overlapping intervals are independent, i.e. for s 1, t 1 s, t the random variables W t W s, W t1 W s1 are independent; 8. paths of Wiener process are not differentiable functions; 9. martingale property notation W s {W u, u s} EW t W s W s E{W t W s W s } t s. Proofs. 1-5 is nothing but the definition of Wiener process. 6. is implied by 3. and provides the orthogonality of increments for non overlapping intervals, that is for s 1 < s < s 3 < s 4 E W s4 W s3 Ws W s1 s s 1 s s 1. The required independence property for these random variables follows from well known fact: orthogonal Gaussian random variables are independent. To verify the validity of 8., with h > let define h W s+h W s h h h does not exists. 1 and show that

2 Assume that this it exists. Then the it for the Fourier transform here i 1 h Eeiλ h exists and is a continuous function of λ. Hence, since the random variable h is zero mean Gaussian with the variance E W s+h W s 1, we find h h { } Ee iλ h e λ h 1 λ, : Uλ. h λ Since Uλ is discontinuous function the assumed differentiability is not valid. 9. Both follow from the property for the increments of Wiener process to be independent for non overlapping intervals: and EW t W s EW t W s + W s W s W s + EW t W s W s W s E W t W s W s EWt W s t s. There exists the alternative definition of Wiener process based on the martingale property. We formulate this result without of proof. Levy s Theorem: The random process W t t is Wiener process if W, the traectories of W t are continuous and the martingale property hold. 3.. One more property of Wiener process. If ξ is zero mean Gaussian random variable with the variance σ Eξ, then E ξ πσ. Therefore, we have π E W t+1 W t t+1 t and thus the series E W t n +1 W t n diverges with tn +1 t n, where < t n 1 < t n <... t n n t. However the increments of Wiener process obey very important property exposed in Lemma 3.1. Let t < t 1 <... < t n be the subdivision of the interval [, t] with max [t ], n. Then here l.i.m. denotes the it in L sense +1 t l.i.m. n +1 W t ] t.

3 Proof: Note that E +1 W t Consequently, it is sufficient to show only that n The latter holds since and so n E n W t ] t W t ] t ] [t +1 t ] that is E n 1 [W t +1 E E [W t +1 E +1 W t ] [t +1 t ] [t +1 t ]E +1 { 3E +1 3[t +1 t [t +1 t ] max[t +1 t ] t max[t +1 t ], n. 3 W t ] t. W t ] t. W t ] [t +1 t ] W t ] 4 + [t +1 t ] W t ] } W t ] [t +1 t ] ] [t +1 t ] [t +1 t ] 3.3. The Itô Integral. For a pair W t, ft of a Wiener process W t a random process ft, we define the Itô integral If ftdw t.

4 4 Since paths of W t are not differentiable and sums E W t+1 W t diverge, the Itô integral is not classical integral. We give below conditions under with the Itô integral might be defined. A.1. E f t < ; A.. for every fixed time t and any h > the random variables fs, s t and increments W t+h W t are independent. We start with the considerations of the particular case. Assume ft k α k It k t < t k+1, 3.1 where t < t 1 < t,..., < t n,... is deterministic sequences of time values and α k, k, 1,... are random variable such that for fixed k and h > Due the assumption A.1 k Set {α,..., α k } and W tk +h W tk are independent. Eα k [t k+1 t k ] <. If : k α k k+1 W tk ]. 3. The sum in the right side of 3. converges in the mean square sense. In fact, α k k+1 W tk ], k 1 forms the sequence of orthogonal zero mean random variables in the third line of 3.3 k > l: Eα k k+1 W tk ] Eα k E E α k k+1 W tk ] Eα ke Eα k k+1 W tk ]α l l+1 W tell] E k+1 W tk ] k+1 W tk ] Eαkt k+1 t k { α k α l l+1 W tl ]E Particularly, 3.3 provides EI f E α k k+1 W tk ] k k Eα kk+1 W tk ] k+1 W tk ] }. 3.3 k Eα k[t k+1 t k ] Ef t.

5 So, the integral If is a linear function in f, i.e. Ic 1 f 1 + c f c 1 If 1 + c If 3.4 for any constants c 1, c and random processes f 1 t, ft of 3.1 type. To define now the Itô integral for a random process ft satisfying only A.1 and A. we will some additional fact given below without proof. Lemma 3.. Let the random process ft, t be satisfied A.1., A.. Then there exists a sequence f n t, t, n 1 of piece-wise constant random processes 5 f n t k α n kit n k t < t n k+1, where t n k, k, 1,... is a condensing sequence of deterministic time values and for every k the random variables {α 1,... α k } are independent of W t n k +h W t n k, h >, moreover for every n, f n satisfies A.1., A.. and n E ft f n t Proof of existence If. For fixed n, If n is well defined. By linear property of If n we have If n If m If n f m. Hence E If n If m EI f n f m E f n t f m t, n, m. E ft f n t + E ft f m t Consequently, If n, n 1 is the fundamental sequence and so that by the Cauchy criteria this sequence converges the mean square sense to some it, which we denote by If. In other words we get n E If If n. The random variable If is unique in the following sense. If f n t, n 1 is another approximating sequence with a it Ĩf, then E If Ĩf. In fact

6 6 E If Ĩf E If Ifn + If n I f n + I f n Ĩf { 3 E If If n + EI f n f n + E I f } n Ĩg 3 E Ĩfn If If n + E Ĩf + E f n t f n t and the first and second terms in the right side of this inequality tend to zero by the definition while the third E f n t f n t E t f n t + E ft f n t,. The random variable If is named the Itô integral Properties of If. P.1. For f i t, i 1,, satisfying A.1. and A.., and any constants c i, i 1, Ic 1 f 1 + c f c 1 If 1 + c If. P.. EI f Ef t. Proofs: For piece-wise constant processes P.1, P.. are obviously valid. They remain valid under passing to it in the mean square sense see Home work 5.. Remark 1. Instead of A.1 assume: A.1. E T f t <, T >. Then the Itô integral I T f T ftdw t is defined as well by setting I T f If T, with f T t ftit > t. A.1. P T f t < 1. then I T f is well defined as well. Sketch Proof: Set τ n min{t T : t f sds n}, n 1 and put f n t ftiτ n t. Then I T f n is well defined see Problem 4 from Home work. Set I T f I T f 1 + [I T f n+1 I T f n ]. n1

7 1. Home work: Wiener Process and Itô integral 1. Prove the following statement: let ξ, η be Gaussian vector with zero mean and orthogonal components, i.e. Eξ, Eη and Eξη. Show that ξ and η are independent random variables.. Let ξ 1,..., ξ n,... be a sequence of Gaussian random variables. Assume that for any z R and i 1 k Eeizξ k exists. Show that Eξ k k exist. k Eξ k 3. Prove P.1. and P.. 4. With ft, satisfying A.1 and A., to prove that f n t ftiτ n t, where τ n inf{t : t f sds n], satisfies A.1 and A. as well. 7