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1. Stochastic Processes and Filtrations

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1. Stochastic Processes and Filtrations 1. Stochastic Processes and filtrations 1. Stoch. pr., filtrations A stochastic process (Xt )t2T is a collection of random variables on 2. BM as (Ω; F) with values in a measurable space (S; S), i.e., for all t, weak limit 3. Gaussian p. Xt :Ω ! S is F − S-measurable. 4. Stopping times In our case 5. Cond. expectation 1 T = f0; 1;:::; Ng or T = N [ f0g (discrete time) 6. Martingales 2 or T = [0; 1) (continuous time). 7. Discrete stoch. integral The state space will be (S; S) = (R; B(R)) (or (S; S) = (Rd ; B(Rd )) 8. Refl. princ., pass.times for some d ≥ 2). 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Example 1.1 Random walk: 1. Stoch. pr., Let "n, n ≥ 1, be a sequence of iid random variables with filtrations 2. BM as ( 1 weak limit 1 with probability 2 "n = 1 3. Gaussian p. −1 with probability 2 4. Stopping times Define X~0 = 0 and, for k = 1; 2;::: , 5. Cond. expectation k 6. Martingales X X~k = "i 7. Discrete stoch. integral i=1 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 1. Stoch. pr., filtrations 2. BM as weak limit For a fixed ! 2 Ω the function 3. Gaussian p. 4. Stopping t 7! Xt (!) times 5. Cond. expectation is called (sample) path (or realization) associated to ! of the 6. Martingales stochastic process X . 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Let X and Y be two stochastic processes on (Ω; F). 1. Stoch. pr., Definition 1.2 filtrations 2. BM as (i) X and Y have the same finite-dimensional distributions if, for weak limit n all n ≥ 1, for all t1;:::; tn 2 [0; 1) and A 2 B(R ): 3. Gaussian p. 4. Stopping P((X ; X ;:::; X ) 2 A) = P((Y ; Y ;:::; Y ) 2 A): times t1 t2 tn t1 t2 tn 5. Cond. expectation (ii) X and Y are modifications of each other, if, for each t 2 [0; 1), 6. Martingales 7. Discrete P(Xt = Yt ) = 1: stoch. integral 8. Refl. princ., pass.times (iii) X and Y are indistinguishable, if 9. Quad.var., path prop. P(Xt = Yt for every t 2 [0; 1)) = 1: 10. Ito integral 11. Ito's formula Example 1.3 1. Stoch. pr., filtrations 2. BM as weak limit 3. Gaussian p. 4. Stopping times 5. Cond. expectation 6. Martingales 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Definition 1.4 A stochastic process X is called continuous (right continuous), if 1. Stoch. pr., almost all paths are continuous (right continuous), i.e., filtrations 2. BM as P(f! : t 7! Xt (!) continuous (right continuous)g) = 1 weak limit 3. Gaussian p. 4. Stopping times Proposition 1.5 5. Cond. expectation If X and Y are modifications of each other and if both processes are 6. Martingales a.s. right continuous then X and Y are indistinguishable. 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 1. Stoch. pr., Let (Ω; F) be given and fF ; t ≥ 0g be an increasing family of filtrations t σ-algebras (i.e., for s < t, F ⊆ F ), such that F ⊆ F for all t. 2. BM as s t t weak limit That's a 3. Gaussian p. filtration: 4. Stopping times For example: 5. Cond. expectation Definition 1.6 X 6. Martingales The filtration Ft generated by a stochastic process X is given by 7. Discrete stoch. integral X Ft = σfXs ; s ≤ tg: 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 1. Stoch. pr., filtrations 2. BM as weak limit Definition 1.7 3. Gaussian p. A stochastic process is adapted to a filtration (Ft )t≥0 if, for each 4. Stopping times t ≥ 0 5. Cond. Xt : (Ω; Ft ) ! (R; B(R)) expectation 6. Martingales is measurable. (Short: Xt is Ft -measurable.) 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 1. Stoch. pr., filtrations 2. BM as Definition 1.8 ("The usual conditions") weak limit 3. Gaussian p. Let Ft+ = \">0Ft+". We say that a filtration (Ft )t≥0 satisfies the 4. Stopping usual conditions if times 5. Cond. 1 (Ft )t≥0 is right-continuous, that is, Ft+ = Ft , for all t ≥ 0. expectation 2 6. Martingales (Ft )t≥0 is complete. That is, F0 contains all subsets of P-nullsets. 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 2. Brownian motion as a weak limit of random walks Example 2.1 k Scaled random walk: for tk = N , k = 0; 1; 2;::: , we set 1. Stoch. pr., filtrations k 2. BM as ~ N 1 ~ 1 X weak limit X := p Xk = p "i tk N N 3. Gaussian p. i=1 4. Stopping times 5. Cond. expectation 6. Martingales 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Definition 2.2 (Brownian motion) 1. Stoch. pr., filtrations A standard, one-dimensional Brownian motion is a continuous 2. BM as adapted process W on (Ω; F; (Ft )t≥0; P) with W0 = 0 a.s. such weak limit that, for all 0 ≤ s < t, 3. Gaussian p. 1 Wt − Ws is independent of Fs 4. Stopping times 2 Wt − Ws is normally distributed with mean 0 and variance t − s. 5. Cond. expectation 6. Martingales 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula k Recall the scaled random walk: for tk = N , k = 0; 1; 2;::: , we set k 1 1 X X~ N := p X~ = p " 1. Stoch. pr., tk k i filtrations N N i=1 2. BM as weak limit By interpolation we define a continuous process on [0; 1): 3. Gaussian p. 0[Nt] 1 N 1 X 4. Stopping Xt = p @ "i + (Nt − [Nt])"[Nt]+1A (2.1) times N i=1 5. Cond. expectation 6. Martingales 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula 1. Stoch. pr., filtrations 2. BM as weak limit 3. Gaussian p. Theorem 2.3 (Donsker) 4. Stopping times Let (Ω; F; P) be a probability space with a sequence of iid random variables with mean 0 and variance 1. Define (X N ) as in (2.1). 5. Cond. t t≥0 N expectation Then (Xt )t≥0 converges to (Wt )t≥0 weakly. 6. Martingales 7. Discrete stoch. integral 8. Refl. princ., pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Clarification of the statement: 1 Ω~ = C[0; 1) 1. Stoch. pr., ~ ∗ filtrations 2 On Ω = C[0; 1) one can construct a probability measure P (the Wiener measure) such that the canonical process 2. BM as weak limit (Wt )t≥0 is a Brownian motion. Let ! = !(t)t≥0 2 C[0; 1). 3. Gaussian p. Then the canonical process is given as Wt (!) := !(t). 4. Stopping N N ~ times 3 (Xt )t≥0 defines a measure probability P on Ω = C[0; 1). 5. Cond. Indeed, expectation X N : (Ω; F) ! (C[0; 1); B(C[0; 1)) 6. Martingales measurable, that means we consider the random variable 7. Discrete N stoch. integral ! 7! X (!; ·), which maps ! to a continous function in t (i.e. 8. Refl. princ., the path). Let A 2 B(C[0; 1)), then pass.times N N 9. Quad.var., P (A) := P(f! : X (!; · 2 Ag): path prop. 10. Ito integral 4 So, Donsker's theorem says that on N w ∗ 11. Ito's (Ω~; F~) = (C[0; 1); B(C[0; 1))) we have that P −! P . formula 1. Stoch. pr., Steps of the proof: filtrations N 2. BM as 1 Show that the finite-dimensional distributions of X converge weak limit to the finite-dimensional distributions of W . That means, show 3. Gaussian p. that, for all n ≥ 1, and all s1;:::; sn 2 [0; 1) 4. Stopping times (X N ; X N ;:::; X N ) −!w (W ; W ;:::; W ): 5. Cond. s1 s2 sn s1 s2 sn expectation N 6. Martingales 2 Show that (P )N≥1 is a tight familiy of probability measures on 7. Discrete (Ω~; F~). stoch. integral w 8. Refl. princ., 3 (1) and (2) imply PN −! P∗. pass.times 9. Quad.var., path prop. 10. Ito integral 11. Ito's formula Easy partial result of step (1) of the proof: By (CLT), for fixed t 2 [0; 1), for N ! 1, 1. Stoch. pr., filtrations N w Xt −! Wt 2. BM as weak limit Theorem 2.4 (CLT) 3. Gaussian p. 2 4. Stopping Let (ξn)n≥1 be iid, mean=0, variance=σ . Then, for each x 2 R, times 5. Cond. n ! expectation 1 X lim P p ξi ≤ x = Φσ(x); 6. Martingales n!1 n i=1 7. Discrete stoch. integral 2 where Φσ(x) is the distribution function of N(0; σ ). In other words 8. Refl. princ., pass.times n 1 X w 9.
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