1. Stochastic Processes and filtrations

Transcription

1 1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S is F S-measurable. 1 T = {0, 1,..., N} or T = N {0} (discrete time) 2 or T = [0, ) (continuous time). The state space will be (S, S) = (R, B(R)) (or (S, S) = (R d, B(R d )) for some d 2).

2 Example 1.1 Random walk: Let ε n, n 1, be a sequence of iid random variables with { 1 with probability 1 ε n = 2 1 with probability 1 2 Define X 0 = 0 and, for k = 1, 2,..., X k = k i=1 ε i

3 For a fixed ω Ω the function t X t (ω) is called (sample) path (or realization) associated to ω of the stochastic process X.

4 Let X and Y be two stochastic processes on (Ω, F). 1. Stoch. pr., Definition 1.2 (i) X and Y have the same finite-dimensional distributions if, for all n 1, for all t 1,..., t n [0, ) and A B(R n ): P((X t1, X t2,..., X tn ) A) = P((Y t1, Y t2,..., Y tn ) A). (ii) X and Y are modifications of each other, if, for each t [0, ), P(X t = Y t ) = 1. (iii) X and Y are indistinguishable, if P(X t = Y t for every t [0, )) = 1.

5 Example Stoch. pr.,

6 Definition 1.4 A stochastic process X is called continuous (right continuous), if almost all paths are continuous (right continuous), i.e.,. P({ω : t X t (ω) continuous (right continuous)}) = 1 Proposition 1.5 If X and Y are modifications of each other and if both processes are a.s. right continuous then X and Y are indistinguishable.

7 Let (Ω, F) be given and {F t, t 0} be an increasing family of σ-algebras (i.e., for s < t, F s F t ), such that F t F for all t. That s a filtration. For example: Definition 1.6 The filtration F X t generated by a stochastic process X is given by F X t = σ{x s, s t}.

8 Definition 1.7 A stochastic process is adapted to a filtration (F t ) t 0 if, for each t 0 X t : (Ω, F t ) (R, B(R)) is measurable. (Short: X t is F t -measurable.)

9 Definition 1.8 ( The usual conditions ) Let F t + = ε>0 F t+ε. We say that a filtration (F t ) t 0 satisfies the usual conditions if 1 (F t ) t 0 is right-continuous, that is, F t + = F t, for all t 0. 2 (F t ) t 0 is complete. That is, F 0 contains all subsets of P-nullsets.

10 2. Brownian motion as a of random walks 1. Stoch. pr., Example 2.1 Scaled random walk: for t k = k N, k = 0, 1, 2,..., we set X N t k := 1 N Xk = 1 N k i=1 ε i

11 Definition 2.2 (Brownian motion) A standard, one-dimensional Brownian motion is a continuous adapted process W on (Ω, F, (F t ) t 0, P) with W 0 = 0 a.s. such that, for all 0 s < t, 1 W t W s is independent of F s 2 W t W s is normally distributed with mean 0 and variance t s.

12 Recall the scaled random walk: for t k = k N, k = 0, 1, 2,..., we set X t N k := 1 Xk = 1 k ε i N N By interpolation we define a continuous process on [0, ): Xt N = 1 [Nt] ε i + (Nt [Nt])ε [Nt]+1 (2.1) N i=1 i=1

13 Theorem 2.3 (Donsker) Let (Ω, F, P) be a probability space with a sequence of iid random variables with mean 0 and variance 1. Define (X N t ) t 0 as in (2.1). Then (X N t ) t 0 converges to (W t ) t 0 weakly.

14 Clarification of the statement: 1 Ω = C[0, ) 2 On Ω = C[0, ) one can construct a probability measure P (the Wiener measure) such that the canonical process (W t ) t 0 is a Brownian motion. Let ω = ω(t) t 0 C[0, ). Then the canonical process is given as W t (ω) := ω(t). 3 (X N t ) t 0 defines a measure probability P N on Ω = C[0, ). Indeed, X N : (Ω, F) (C[0, ), B(C[0, )) measurable, that means we consider the random variable ω X N (ω, ), which maps ω to a continous function in t (i.e. the path). Let A B(C[0, )), then P N (A) := P({ω : X N (ω, A}). 4 So, Donsker s theorem says that on ( Ω, F) = (C[0, ), B(C[0, ))) we have that P N w P.

15 Steps of the proof: 1 Show that the finite-dimensional distributions of X N converge to the finite-dimensional distributions of W. That means, show that, for all n 1, and all s 1,..., s n [0, ) (X N s 1, X N s 2,..., X N s n ) w (W s1, W s2,..., W sn ). 2 Show that (P N ) N 1 is a tight familiy of probability measures on ( Ω, F). 3 (1) and (2) imply P N w P.

16 Easy partial result of step (1) of the proof: 1. Stoch. pr., By (CLT), for fixed t [0, ), for N, Theorem 2.4 (CLT) X N t w W t Let (ξ n ) n 1 be iid, mean=0, variance=σ 2. Then, for each x R, ( ) lim P 1 n ξ i x = Φ σ (x), n n i=1 where Φ σ (x) is the distribution function of N(0, σ 2 ). In other words where Z N(0, σ 2 ). 1 n n w ξ i Z, i=1

17 3. Brownian motion as a Gaussian process 1. Stoch. pr., Definition 3.1 A stochastic process (X t ) t 0 is called Gaussian process, if for each finite familiy of time points {t 1,..., t n } the vector (X t1,..., X tn ) has a multivariate normal distribution. A Gaussian process is called centered if E[X t ] = 0 for all t. The covariance function is given by γ(s, t) = Cov(X s, X t ). Theorem 3.2 A Brownian motion is a centered Gaussian process with γ(s, t) = s t (= min(s, t)). Conversely, a centered Gaussian process with continuous paths and covariance function γ(s, t) = s t is a Brownian motion.

18 Given is a filtered probability space (Ω, F, (F t ) t 0, P). 1. Stoch. pr., A random time T is a measurable function Definition 4.1 T : (Ω, F) ([0, ], B([0, ]). If X is a stochastic process and T a random time, we define the function X T on {T < } by: X T (ω) = X T (ω) (ω)

19 Definition 4.2 A random time T is a stopping time for (F t ) t 0 if, for all t 0, Proposition 4.3 {T t} = {ω : T (ω) t} F t. 1 If T (ω) = t a.s. for some constant t 0, then T is a stopping time. 2 Every stopping time T satisfies that, for all t, {T < t} F t. (4.1) 3 If the filtration is right continuous and a random time T satisfies (4.1) for all t, then T is a stopping time.

20 Lemma 4.4 Let S and T be stopping for (F t ) t 0. Then S T, S T and S + T are stopping for (F t ) t 0. Definition 4.5 Let T be a stopping time for (F t ) t 0. The σ-field F T of events determined prior to the stopping time T is given by Remark: F T = {A F : A {T t} F t, for all t 0}. 1 F T is a σ-field. 2 T is F T -measurable. 3 If T (ω) = t a.s., then F T = F t.

21 Example 4.6 Let (W t ) t 0 be a standard Brownian motion on (Ω, F, (F t ) t 0, P). Let T a be the first passage time of the level a R, i.e., T a (ω) = inf{t > 0 : W t (ω) = a}. (4.2) Then T a is a stopping time for (F t ) t 0.

22 Remark: Let S and T be stopping for (F t ) t Stoch. pr., 1 For A F S we have that A {S T } F T. 2 If, moreover, S T then F S F T. Definition 4.7 A stochastic process (X t ) t 0 is called progressively measurable for (F t ) t 0 if, for all t and all A B(R), the set that means {(s, ω) : 0 s t, ω Ω, X s (ω) A} B([0, t]) F t, (s, ω) X s (ω) is B([0, t]) F t B(R) measurable.

23 Example: If (X t ) t 0 is adapted and right-continuous then it is progressively measurable. Proposition 4.8 Let X be progressively measurable on (Ω, F, (F t ) t 0 ) and let T be a stopping time for (F t ) t 0. Then 1 X T defined on {T < } is an F T -measurable random variable and 2 the stopped process (X T t ) t 0 is progressively measurable.

24 5. Conditional expected value 1. Stoch. pr., Definition 5.1 Let (Ω, F, P) be a probability space and X : Ω R be an F B(R)-measurable random variable such that E[ X ] = X dp <. Let G F be a (smaller) σ-algebra. Then there exists a random variable Y with the following properties: 1 Y is G B(R)-measurable. 2 Y is integrable, i.e., E[ Y ] <. 3 For all A G: E[Y 1I A ] = E[X 1I A ]. The G-measurable random variable Y is called conditional expected value of X given G. We write Y = E[X G] a.s.

25 Remark: 1 E[X G] is unique with respect to equality a.s. 2 Suppose that, additionally, E[X 2 ] <. Then E[X G] is the orthogonal projection of X L 2 (Ω, F) onto the subspace L 2 (Ω, G). 3 If G = {, Ω} then E[X G] = E[X ] a.s.

26 Definition 5.2 Let Z be a function on Ω. Then E[X Z] := E[X σ(z)]. Example 5.3

27 Example 5.4 X : (Ω, F) (R, B(R)). Suppose Ω = k=1 Ω k such that Ω i Ω j = for i j. Let G := σ{ω k, k 1}. E[X G] =

28 Example 5.5 Suppose (X, Z) has a bivariate density f (x, z).

29 Theorem 5.6 (A list of properties of conditional ) In the following let X, Y, X n, n 1, be integrable random variables on (Ω, F, P). The following holds (a) If X is G-measurable, then E[X G] = X a.s. (b) Linearity: E[aX + by G] = ae[x G] + be[y G] a.s. (c) Positivity: If X 0 a.s. then E[X G] 0 a.s. (d) Monotone convergence (MON): If 0 X n and X n X a.s. then E[X n G] E[X G] a.s..

30 (e) Fatou: If 0 X n, for all n, then 1. Stoch. pr., E[lim inf n X n G] lim inf E[X n G]. n (f) Dominated convergence (DOM): If X n Y, for all n 1 and Y L 1 (i.e. integrable) and lim X n = X a.s. then lim n E[X n G] = E[X G] a.s. (g) Jensen s inequality: Let ϕ : R R be a convex function such that ϕ(x ) is integrable. Then ϕ (E[X G]) E[ϕ(X ) G]

31 (h) Tower Property: If G 1 G 2 F then E [E[X G 2 ] G 1 ] = E[X G 1 ] a.s. (i) Taking out what is known: Let Y be G-measurable and let E[ XY ] <. Then E[XY G] = YE[X G] (j) Role of independence: If X is independent of G, then E[X G] = E[X ]

32 and the discrete time stochastic integral 1. Stoch. pr., Definition 6.1 A stochastic process (X t ) t 0 on (Ω, F, (F t ) t 0, P) is a martingale if X is adapted and E[ X t ] <, for all t 0, and, for s < t, E[X t F s ] = X s a.s. (6.1) If the = in (6.1) is replaced by ( ) the process is called supermartingale (submartingale). Remark: 1 (6.1) is equivalent to E[X t X s F s ] = 0 a.s. 2 Suppose (X t ) n=0 is a stochastic process in discrete time adapted to (F n ) n=0 then the martingale property reads as: for all n 1 E[X n F n 1 ] = X n 1 a.s.

33 Example 6.2 (Random walk: sum of independent r.v.) Let ε k, k 1, be independent with E[ε k ] = 0. Let F n = σ{ε 1,..., ε n }, F 0 = {, Ω}. Define X 0 = 0 and, for n 1, X n = n ε k. k=1 Then (X n ) n 0 is a martingale for (F n ) n 0. Example 6.3 (Product of independent r.v.) Let ε k, k 1, be non-negative and independent with E[ε k ] = 1. Let F n = σ{ε 1,..., ε n }, F 0 = {, Ω}. Define X 0 = 1 and, for n 1, X n = n ε k. k=1 Then (X n ) n 0 is a martingale for (F n ) n 0.

34 Example 6.4 Let X be a r.v. on (Ω, F, P) such that E[ X ] <. Let F t ) t 0 be any filtration with F t F, for each t 0. Define X t = E[X F t ]. Then (X t ) t 0 is a martingale for (F t ) t 0.

35 7. Stochastic integral in discrete time Given is a discrete filtered probability space (Ω, F, (F n ) n 0, P). 1. Stoch. pr., Definition 7.1 A discrete stochastic process (H n ) n=1 is called predictable if H n : (Ω, F n 1 ) (R, B(R)) is measurable. (Short: H n is F n 1 -measurable.)

36 Definition 7.2 Let (X n ) n=0 be an adapted stochastic process and (H n) n=1 be a predictable stochastic process on (Ω, F, (F n ) n 0, P). Then the stochastic integral in discrete time at time n 1 is given by (H X ) n = n H k (X k X k 1 ). k=1 We define (H X ) 0 = 0. The stochastic integral process is given by ((H X ) n ) n=0.

37 Theorem 7.3 Let H be bounded and predictable and X be a martingale on (Ω, F, (F n ) n 0, P). Then the stochastic integral process ((H X ) n ) n=0 is a martingale.

38 Theorem 7.4 (Doob s Optional Stopping Theorem) Let (M t ) t 0 be a martingale in continuous time on (Ω, F, (F t ) t 0, P). Then, for a bounded stopping time T it holds that E[M T ] = M 0. More generally, if S T are bounded stopping, then E[M T F S ] = M S a.s.

39 8. Reflection principle, passage, running maximum/minimum 1. Stoch. pr., Definition 8.1 (Markov property) Brownian motion satisfies the Markov property, that means, for all t 0 and s > 0: P(W t+s y F t ) = P(W t+s y W t ) a.s.

40 Reflection principle: Recall that T a = inf{t > 0 : W t = a}. 1. Stoch. pr., P(T a < t) = P(T a < t) = 2P(W t > a) (8.1)

41 In all the following (W t ) t 0 is a standard Brownian motion on (Ω, F, (F t ) t 0, P). Theorem 8.2 (Strong Markov property) The BM (W t ) t 0 satisfies the strong Markov property: for each stopping time τ with P(τ < ) = 1 it holds that Ŵ t = W τ+t W τ,, t 0 is a standard Brownian motion independent of F τ. We will see later that P(T a < ) = 1 for all a R.

42 Theorem 8.3 (Reflection principle) Let a 0 and T a be the passage time for a. { W t W t = 2W Ta W t = 2a W t for t T a for t T a Then ( W t ) t 0 is again a standard Brownian motion.

43 Lemma 8.4 P(sup t 0 W t = + and inf t 0 W t = ) = 1

44 Theorem 8.5 (Recurrence) P(T a < ) = 1 for all a R.

45 Theorem 8.6 (Distribution of passage time) Let a 0. The density of T a is given by f Ta (t) = a 2π t 3 2 e a2 2t, which is the density of an invers-gamma-distribution with parameters 1 a2 2 and 2. In particular, we have that E[T a] = +.

46 Running maximum and minimum of Brownian motion: 1. Stoch. pr., Let M t = max 0 s t W s and m t = min 0 s t W s. Theorem For each a > 0: P(M t a) = P(T a t) = P(T a < t) = 2P(W t > a). 2 For each a < 0: P(m t a) = 2P(W t < a).

47 Theorem 8.8 (Joint distribution) Let y x 0. Then P(W t x, M t y) = P(W t 2y x). This implies that the joint distribution of (W t, M t ) has the following density: for y 0, x y 2 (2y x) f W,M (x, y) = e (2y x)2 2t. π t 3 2

48 Let a < 0 < b and let τ be the first time to leave an interval (a, b) (exit time), i.e., τ = inf{t > 0 : W t / (a, b)}. We have that τ = min(t a, T b ) = T a T b.

49 Theorem 8.9 (Exit time) Let a < 0 < b and τ = T a T b as above. Then P(τ < ) = 1 and E[τ] = ab. Moreover P(W τ = b) = P(W τ = a) = a a + b = a a + b b a + b = b a + b

50 9. Quadratic variation and path properties of BM 1. Stoch. pr., Definition 9.1 A stochastic process has finite quadratic variation if there exists an a.s. finite stochastic process ([X, X ] t ) t 0 such that for all t and partitions π n = {t0 n,..., tn m n } with 0 = t0 n < tn 1 < < tn m n = t and δ n = max i=1,...,mn (ti n ti 1 n ) 0 we have that, for n, V 2 t (π n ) := m n i=1 (X t n i X t n i 1 ) 2 [X, X ] t in probability. (9.1)

51 Theorem 9.2 Let (W t ) t 0 be a Brownian motion. Then Vt 2 (π n ) t in L 2, that means lim E[(V t 2 (π n ) t) 2 ] = 0. n Therefore it follows that V 2 t (π n ) t in probability. Hence [W, W ] t = t a.s.

52 Corollary 9.3 (W t ) t 0 has a.s. paths of infinite variation on each interval.

53 Corollary 9.4 For almost all ω the path t W t (ω) is not monotone in each interval.

54 Remark 9.5 Almost all paths of (W t ) t 0 are not differentiable on each interval.

55 10. Stochastic integral by Ito 1. Stoch. pr., For the whole chapter let (W t ) t 0 be a standard Brownian motion on (Ω, F, (F t ) t 0, P). Everything will be based on this filtered probability space. Recall: Riemann integral: b a f (t)dt = lim δn 0 n f (ξi n )(ti n ti 1) n i=1 where a = t n 0 < tn 1 < < tn n = b, t n i 1 ξn i t n i and δ n = max i=1,...,n (t n i t n i 1 ).

56 Riemann-Stieltjes-integral: Let f be a continuous function and g be a function of bounded variation. Then the following limit exists and is called R-S-integral of f with respect to g: b a f (t)dg(t) := lim δn 0 n f (ti 1)(g(t n i n ) g(ti 1)), n i=1 where a = t n 0 < tn 1 < < tn n = b and δ n = max i=1,...,n (t n i t n i 1 ).

57 Theorem 10.1 Fix [0, t] and let, for each n, 0 = t n 0 < tn 1 < < tn n = t. If g is a function such that lim δ n 0 n f (ti 1)(g(t n i n ) g(ti 1)) n i=1 exists for each continuous function f : [0, t] R, then g is of bounded variation on [0, t]. Remark 10.2 No pathwise R-S-integral for Brownian motion!!!

58 Ito integral with simple integrands: Definition 10.3 A stochastic process (X t ) 0 t T is called simple predictable integrand if n 1 X t = ξ 0 1I [t0,t 1](t) + ξ i 1I (ti,t i+1 ](t), where 0 = t 0 < t 1 < < t n = T and ξ 0,..., ξ n 1 are random variables such that ξ i is F ti -measurable and E[ξi 2 ] <, for i = 1,..., n 1. For each t T the Ito integral at time t for a simple integrand X is given by: t 0 i=1 n 1 X s dw s =: (X W ) t = ξ i (W ti+1 t W ti t). i=0

59 Remark 10.4 If t < T is such that t k t < t k+1 t n = T, this means k 1 (X W ) t = ξ i (W ti+1 W ti ) + ξ k (W t W tk ). i=0 For T this means n 1 (X W ) T = ξ i (W ti+1 W ti ) i=0

60 Theorem 10.5 (Properties of Ito integral for simple integrands) 1 Linearity: X, Y simple, α, β constants, then T 0 T T (αx t + βy t ) dw t = α X t dw t + β Y t dw t Let 0 a < b. Then T 0 1I (a,b](t)dw t = W b W a and T 0 1I (a,b](t)x t dw t = b a X tdw t. 3 Zero mean: E[ T 0 X tdw t ] = 0 [ ( ) ] T 2 4 Isometry: E 0 X tdw t = T 0 E[X t 2 ]dt

61 (More) general integrands: 1. Stoch. pr., Lemma 10.6 Let (X t ) 0 t T be a bounded and progressively measurable stochastic process. Then there exists a sequence of simple predictable processes (Xt m ) 0 t T, m 1, such that [ ] T lim E m 0 X m t X t 2 dt = 0. (10.1)

62 Lemma 10.7 Let (X t ) 0 t T be a progressively measurable stochastic process such that [ ] T E Xt 2 dt <. (10.2) 0 Then there exists a sequence of simple predictable processes (Xt m ) 0 t T, m 1, such that (10.1) holds, i.e., [ ] T lim E m 0 X m t X t 2 dt = 0.

63 Theorem 10.8 (Ito integral for general integrands with (10.2)) Let (X t ) 0 t T be a progressively measurable stochastic process such that (10.2) holds, i.e., E[ T 0 X 2 t dt] <. Then the stochastic integral T 0 X tdw t is defined and has the following properties: 1 Linearity: X, Y as above, then T 0 T T (αx t + βy t ) dw t = α X t dw t + β Y t dw t Let 0 a < b. Then T 0 1I (a,b](t)dw t = W b W a and T 0 1I (a,b](t)x t dw t = b a X tdw t. 3 Zero mean: E[ T 0 X tdw t ] = 0 [ ( ) ] T 2 4 Isometry: E 0 X tdw t = T 0 E[X t 2 ]dt

64 Theorem 10.9 (Ito integral for even more general integrands) Let (X t ) 0 t T be a progressively measurable stochastic process such that ( ) T P Xt 2 dt < = 1. (10.3) 0 Then the stochastic integral T 0 X tdw t is defined and has satisfies (1) and (2) of Theorem Remark Attention: in this case (3) Zero-Mean and (4) Isometry are not satisfied in general!!! If, additionally the stronger condition (10.2) holds, then we are in the case of Theorem 10.8 and Zero-Mean and Isometry hold.

65 Corollary Let X be continuous and adapted. Then T 0 X tdw t exists. In particular, for any continuous function f : R R the stochastic integral T 0 f (W t)dw t exists. Remark The Ito integral is not monotone.

66 Ito integral process: Let X be progressively measurable such that (10.3) holds, i.e., T P( Xs 2 ds < ) = 1. 0 Then t 0 X sdw s is defined for all t T. This gives a stochastic process (I t ) 0 t T with I t = t 0 X s dw s. There exists a modification with continuous paths: we always take this modification.

67 Definition A martingale (M t ) 0 t T is called quadratic integrable on [0, T ] if Theorem sup E[Mt 2 ] <. t [0,T ] Let X be progressively measurable such that (10.2) holds, i.e., E[ T 0 X s 2 ds] <. Then (I t ) 0 t T, where I t = t 0 X sdw s, is a quadratic integrable martingale.

68 Corollary Let f : R R be a continuous and bounded function. Then (I t ) 0 t T, where I t = t 0 f(w s)dw s, is a quadratic integrable martingale.

69 Quadratic variation and covariation of Ito integrals: Theorem Let X be progressively measurable such that (10.3 holds. The quadratic variation of t 0 X sdw s satisfies [ t t ] t X s dw s, X s dw s = Xs 2 ds a.s for all 0 t T.

70 Covariation: Let I t = t 0 X sdw s and Ĩ t = t 0 X s dw s, for 0 t T, and progressively measurable processes X, X such that (10.3) holds. The covariation of I and Ĩ is given by Corollary [I, Ĩ] t := 1 2 It holds that [I, Ĩ] t = t 0 X s X s ds a.s. ( ) [I + Ĩ, I + Ĩ] t [I, I] t [Ĩ, Ĩ] t.

71 Remark Analogously to the quadratic variation the covariation of two stochastic processes Y and X can be defined as follows: m n [Y, X ] t = lim (Y t n n i Y t n i 1 )(X t n i X t n i 1 ), i=1 in probability where 0 = t0 n < tn 1 <... tn m n = t and max i=1,...,mn (ti n ti 1 n ) 0 for n.

72 Integration by parts: Theorem (Integration by parts) Let X, Y be continuous and adapted processes such that (10.3) holds for both processes. Then the following holds, for each 0 t T, t t X t Y t = X 0 Y 0 + X s dy s + Y s dx s + [X, Y ] t. 0 0 Notation: d(x t Y t ) = X t dy t + Y t dx t + d[x, Y ] t

73 : change of variables 1. Stoch. pr.,

74 Theorem 11.1 Let (W t ) t 0 be a Brownian motion and let f : R R be a twice continuously differentiable function (f C 2 (R)). Then, for each t, f (W t ) f (W 0 ) = t 0 f (W s )dw s t 0 f (W s )ds. (11.1) In differential notation (11.1) reads as follows: df (W t ) = f (W t )dw t f (W t )dt