Verona Course April Lecture 1. Review of probability

Transcription

1 Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy

2 A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is a σ algebra of subsets of Ω and P a probability measure on Ω (P(Ω) = 1). A σ algebra is a collection U of subsets of Ω with the following properties (i) φ, Ω F. (ii) If A F, then A c F. (iii) If i F, i = 1, 2,..., then i=1 A i, A i F. P is said to be a probability ( measure if ) P(φ) =, P(Ω) = 1, P A i = P(A i ) if A i A j = φ for i j. i=1 i=1 A set A F is called event, points ω Ω are sample points, P(A) is the probability of the event A. A property which is true, except for an event of probability zero, is said to hold almost surely (abbreviated a.s.). i=1

3 Example (Buffon s needle problem) The plane is ruled by parallel lines 2 in apart and a 1-inch long needle is dropped at random on the plane. What is the probability that it hits one of the parallel lines? The first issue is to find some appropriate probability space (Ω, U, P). For this, let { h = distance from the center of needle to nearest line θ = angle ( π 2 ) that the needle makes with the horizontal. These fully determine the position of the needle, up to translations and reflection. Let us next take [ Ω =, π ) [, 1], U = Borel subsets of Ω, }{{ 2 }}{{} values of h values of θ 2 area of B P(B) = for each B U. π

4 Example (continue) We denote by A the event that the needle hits a horizontal line. We can now check that this happens provided h sin θ 1 2. Consequently, { A = (θ, h) Ω h sin θ }, 2 and so P(A) = 2( area of A π = 2 π π sin θ dθ = 1 π.

5 A mapping X : Ω R n is called an n-dimensional random variable if, for each Borelian set B B, we have X 1 (B) F, (1) or, in other words, X is F-measurable. Here B is the collection of all Borel subsets of R n, which is the smallest σ algebra of subsets of R n containing all open sets. If X : Ω R n is a random variable, then F(X) = {X 1 (B); B B}. A random variable τ : Ω [, ) is called a stopping time with respect to F t provided {ω; τ(ω) t} F t, t, that is, the set {ω; τ(ω) t} is F t measurable. A stochastic process is a collection {X(t); t } of random variables. In other words, a stochastic process X assigns to each time t a random variable X(t). In fact, X = X(t, ω), t I R, ω Ω.

6 X is called continuous if t X(t, ω) is continuous with probability 1, i.e., P-a.s. It is called mean square continuous on [, T] if, for each t [, T], lim t t E X(t) X(t ) 2 =. A family (F t ) t F is called a filtration if F s F t for s t, F contains all A F, P(A) =. A stochastic process X=X(t) is said to be (F t ) t adapted if for any Borelian set B B, i.e., X 1 (t)(b) F t ω X(t, ω) is F t -measurable.

7 Integration with respect to the measure P If (Ω, F, P) is a probability space and X = N a i χ Ai, where a i R, χ Ai is the i=1 characteristic function of A i F, A i A j = φ, random variable), we define the integral Ω X dp = N A i = Ω (X is called simple i=1 N a i P(A i ). (2) i=1 If X, then, by definition, X dp = sup Ω Y X, Y simple Ω Ω Ω Y dp. If X : Ω R is a random variable, we define X dp = X + dp X dp. Ω

8 Finally, if X : Ω R n is an n-dimensional random variable, then X = (X 1, X 2,..., X n ), where X i : Ω R, and we define ( ) X dp = X 1 dp,..., X n dp. (3) Ω Ω Ω We call E(X) = X dp (4) the expectation or mean value of X. If X : Ω R n is a random variable, then its distribution function F X : R n [, 1] is defined by Ω F X (x) = P (X x), x R n. (5) If there is a nonnegative integrable function f : R n R such that F X (xd) = F(x 1,..., x n ) = x1 x1 f (y 1,..., y n )dy 1...dy n, then f is called the density function of X. We have for each Borelian set B P (X B) = f (x)dx = df X (x). (6) B B

9 Example If X : Ω R has the density f (x) = 1 2πσ 2 (x m)2 e 2σ 2, x R, we say that the random variable X is Gaussian or has normal distribution with mean m and variance σ 2. X is also called N(m, σ 2 ) random variable. If X : Ω R n has the density 1 f (x) = e (x m) C 1 (x m), (7) (2π) n (det C) 1 2 we say that X has a Gaussian or normal distribution. 1 In this case (n = 1), E(X) = xe (x m) 2 2σ 2 dx = m. 2πσ 2 The random variables X i : Ω R n, i = 1,..., m, are said to be independent if, for all k 2 and all Borel sets B 1,..., B k R n, P(X 1 B 1, X 2 B 2,..., X k B k ) = P(X 1 B 1 )...P(X k B k ). (8) In this case, we also have E(X 1 X m ) = E(X 1 )E(X 2 ) E(X n ). (9)

10 Convergence of random variables ( ) (i) X n X a.s. if P ω Ω; lim X n(ω) = X(ω) = 1. n (ii) X n X in probability if ε >, lim P{ X n X > ε} =. n It turns out that (i) = (ii).

11 Conditional expectation Let Y : Ω R n be a random variable. Then E(X Y) is the random variable defined by X dp = E(X Y)dP, A F(Y). (1) Let (Ω, F A A

12 Martingales Let X = X(t) be a real valued stochastic process. Then F(t) = F(X(s); s t} is the σ algebra generated by the random variables X(s) for s t. More precisely, F(t) = {(X(t)) 1 (B); B B} and is the smallest sub σ algebra of F with respect to which X is measurable. (This is the history of the process X until time t.)

13 Definition Let X : Ω R be a stochastic process such that E(X(t)) <, t. If (i) X(s) = E(X(t) F(s)) P-a.s. for t s >, then X is called a martingale. If (ii) X(s) E(X(t) F(s)) P-a.s. for t s > then X is a submartingale. Theorem (the martingale inequality) If X = X(t) is a martingale with continuous sample paths and 1 < p <, then ( ) p p E max p s t E X(s) p E X(t) p. p 1

14 Brownian motions (Definition) A real valued stochastic process W : Ω R is called Brownian motion or Wiener process if (i) W() =, P-a.s. (ii) W(t) W(s) is Gaussian with mean and variance σ 2 t s. (iii) For all times < t 1 < t 2 < < t n, the random variables W(t 1 ), W(t 2 ), W(t 1 ),..., W(t n ) W(t n 1 ) are independent. In particular, it follows that E[W(t)] =, E[W 2 (t)] = t, t, P[a W(t) b] = 1 b 2πt a e x2 2t dx.

15 An R n -valued stochastic process W(t) = (W 1 (t),..., W n (t)) is an n-dimensional (n D) Wiener process (or Brownian motion) provided (i) For each i, W i is 1 D Wiener process. (ii) The σ-algebras W k = F(W k (t); t ) are independent, k = 1,... It turns out that, if W is an n D Wiener process, then for each Borelian set A R n P(W(t) A) = 1 e x 2 (2πt) n 2t dx. (13) 2 If W is a Wiener process, then P-a.s. the function t W(t) is Hölder continuous with exponent α < 1 2. Moreover, for Ω Ω, t W(t, ω) is nowhere differentiable. A

16 Markov processes If X is a stochastic process, then F(t) = F(X(s); s t), that is, the σ algebra {X 1 (s)(b); B B} is called the history of the process X up to t. The R n valued stochastic process X is called a Markov process if P(X(t) B F(s)) = P(X(t) B X(s)), s t, (14) for all Borelian sets B R n.

17 Stochastic integrals Let W(t) be 1 D Brownian motion on same probability space (Ω, F, P). The σ algebra W(t) = F(W(s) s t) is called the history of W up to time t. Definition A family (F t ) t F is called a filtration with respect to W(t) if (i) F s F t for s t. (ii) W(t) F t for all t. (iii) F t is independent of W + (t) = F{W(s) W(t) s t}. Definition The real valued stochastic process X is called nonanticipating with respect to (F t ) t if, for each t, X(t) is F t -measurable. One says also that X is (F t ) t -adapted. The process X is called progressively measurable if X : (, ) Ω R is measurable and t X(t) is (F t ) t adapted.

18 The process X; [, T] R is called a step process if there is a partition { = t < t 1 < < t m = T} such that X(t) = a k for t k t < t k+1. We define for such a process the Itô stochastic integral T m 1 X dw = a k (W(t k+1 ) W(t k )). (15) k= It turns out that ( T ( T E X dw =, E ) 2 ) T X dw = E X 2 dw. (16)

19 Now, for an arbitrary process X in L 2 (, T) = one defines the Itô integral T { T } X; [, T] R, E X 2 dt < T X dw = lim n X n dw, (17) where the limit is taken in L 2 (Ω) and X n is a family of step processes such that T lim E X X n 2 dt =. (18) n

20 For instance, if t X(t, ω) is continuous, one can choose X n as ( ) k X n (t) = X for k n n t k + 1, k =, 1,..., [n] n and, for general X L 2 (, T), one takes X n (t) = t ne n(s t) X(s)ds. This definition extends to all processes X which are progressively measurable and T I(t) = t X dw is a martingale. X 2 dt <, P-a.s.

21 Stochastic integrals in n-dimension Let W = (W 1,..., W m ) be an m-dimensional Wiener process and let (F t ) t be a filtration such that F t F(W(s) s t) and (F t ) t is independent of F(W(s) W(t) t s < ). Consider an n m stochastic process Then X = (X ij ) n m i,j=1 L 2 n m(, T). T X dw is an R n -valued random variable, whose i-component is m j=1 T X ij dw j, i = 1,..., n.

22 Properties where G 2 = nm i,j=1 ( T E T E X dw = 2) X dw = E T X 2 dt, G ij 2. Moreover, for G L 2 (, T), I(t) = t G(s)dW(s) is a martingale and t I(t) is continuous, P-a.s. (that is, has continuous sample paths).

23 Definition If X = (X 1, X 2,..., X n ) is an R n -valued stochastic process such that X(t) = X(s) + t F(τ)dτ + We say that X has the stochastic differential dx That is, where X = X 1. X n dx i = F i dt +, F = t G dw. (19) dx = F dt + G dw. (2) m G ij dw j, i = 1, 2,..., n, (21) j=1 F 1. F n, G ij = G.

24 Theorem (Itô s formula) Suppose that dx = Fdt + GdW. Let ϕ : [, T] R n R be dϕ(t, X(t)) = ϕ t dt + = ϕ t dt n i,j=1 n i=1 n i=1 2 ϕ X i X j ϕ x i dx i ϕ X i F i dt + n i,j=1 n i=1 m G il G jl dt. l=1 ϕ X i 2 ϕ X i X j m G il G jl dt l=1 m G ij dw j j=1 (22) In compact form, it looks like dϕ(t, X(t)) = ϕ dt + ϕ(t, X(t)) F dt + ϕ(t, X) G dw t + 1 n 2 ϕ (G G) ij dt. 2 X i X j i,j=1 (23)

25 Examples 1 Let dx = Fdt + GdW, F L 1 (, T), G L 2 (, T), W Brownian motion in 1 D. Then, if ϕ C 1 ([, T] R), 2 ϕ X C([, T] R), 2 we have dϕ(t, X(t)) = ϕ t 2 Let X = W, ϕ = ϕ(t, X). Then dϕ(t, W(t)) = ϕ t (t, W(t))dt dx 1 = F 1 dt + G 1 dw dx 2 = F 2 dt + G 2 dw. Then d(x 1 X 2 ) = X 2 dx 1 + X 1 dx 2 + G 1 Gdt. ϕ (t, X(t))dt + (t, X(t))F(t)dt X G2 (t) 2 ϕ (t, X(t)), t (, T). X2 (24) 2 ϕ ϕ (t, W(t))dt+ (t, W(t))dW(t). X2 X

26 Remembering formula (22) dϕ(t, X) = ϕ t dt + n i=1 ϕ X i dx i n i,j=1 where, computing dx i dx j, we use the symbolic rules 2 ϕ X i X j dx i dx j, (25) (dt) 2 =, dtw k =, dw k dw l = δ kl dt, k, l = 1,..., n, m.

27 Exercises 1 Calculate dw m, where W is a Wiener process. ( ) 2 Calculate d e λw(t) λ2 t 2. 3 Calculate d(tw). 4 Calculate t W dw. Hint. Set X(t) = t dw(s) and apply Itô s formula to ϕ(x) = 1 2 x2.