1. Stochastic Processes and filtrations
1. Stoch. pr., filtrations A stochastic process (Xt )t∈T 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 = {0, 1,..., N} or T = N ∪ {0} (discrete time) 6. Martingales 2 or T = [0, ∞) (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 εn, n ≥ 1, be a sequence of iid random variables with filtrations
2. BM as ( 1 weak limit 1 with probability 2 εn = 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 = εi 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 ω ∈ Ω the function 3. Gaussian p.
4. Stopping t 7→ Xt (ω) times
5. Cond. expectation is called (sample) path (or realization) associated to ω 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 ∈ [0, ∞) and A ∈ B(R ): 3. Gaussian p. 4. Stopping P((X , X ,..., X ) ∈ A) = P((Y , Y ,..., Y ) ∈ A). times t1 t2 tn t1 t2 tn
5. Cond. expectation (ii) X and Y are modifications of each other, if, for each t ∈ [0, ∞), 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 ∈ [0, ∞)) = 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({ω : t 7→ Xt (ω) continuous (right continuous)}) = 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 {F , t ≥ 0} 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 = σ{Xs , s ≤ t}. 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+ = ∩ε>0Ft+ε. 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 := √ Xk = √ εi 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 := √ X˜ = √ ε 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, ∞):
3. Gaussian p. [Nt] N 1 X 4. Stopping Xt = √ εi + (Nt − [Nt])ε[Nt]+1 (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. Stoch. pr., ˜ ∗ filtrations 2 On Ω = C[0, ∞) 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 ω = ω(t)t≥0 ∈ C[0, ∞). 3. Gaussian p. Then the canonical process is given as Wt (ω) := ω(t). 4. Stopping N N ˜ times 3 (Xt )t≥0 defines a measure probability P on Ω = C[0, ∞).
5. Cond. Indeed, expectation X N : (Ω, F) → (C[0, ∞), B(C[0, ∞)) 6. Martingales measurable, that means we consider the random variable 7. Discrete N stoch. integral ω 7→ X (ω, ·), which maps ω to a continous function in t (i.e.
8. Refl. princ., the path). Let A ∈ B(C[0, ∞)), then pass.times N N 9. Quad.var., P (A) := P({ω : X (ω, · ∈ A}). path prop.
10. Ito integral 4 So, Donsker’s theorem says that on N w ∗ 11. Ito’s (Ω˜, F˜) = (C[0, ∞), B(C[0, ∞))) 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 ∈ [0, ∞) 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 ∈ [0, ∞), for N → ∞, 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 ∈ R, times
5. Cond. n ! expectation 1 X lim P √ ξi ≤ x = Φσ(x), 6. Martingales n→∞ 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. Quad.var., √ ξ −→ Z, path prop. n i i=1 10. Ito integral 11. Ito’s where Z ∼ N(0, σ2). formula 3. Brownian motion as a Gaussian process
1. Stoch. pr., Definition 3.1 filtrations A stochastic process (X ) is called Gaussian process, if for each 2. BM as t t≥0 weak limit finite familiy of time points {t1,..., tn} the vector (Xt1 ,..., Xtn ) has 3. Gaussian p. a multivariate normal distribution. 4. Stopping A Gaussian process is called centered if E[Xt ] = 0 for all t. times The covariance function is given by γ(s, t) = Cov(Xs , Xt ). 5. Cond. expectation
6. Martingales Theorem 3.2 7. Discrete A Brownian motion is a centered Gaussian process with stoch. integral γ(s, t) = s ∧ t (= min(s, t)). Conversely, a centered Gaussian process 8. Refl. princ., pass.times with continuous paths and covariance function γ(s, t) = s ∧ t is a
9. Quad.var., Brownian motion. path prop.
10. Ito integral
11. Ito’s formula 4. Stopping times
Given is a filtered probability space (Ω, F, (Ft )t≥0, P).
1. Stoch. pr., A random time T is a measurable function filtrations T : (Ω, F) → ([0, ∞], B([0, ∞]). 2. BM as weak limit
3. Gaussian p. Definition 4.1
4. Stopping If X is a stochastic process and T a random time, we define the times function XT on {T < ∞} by: 5. Cond. expectation XT (ω) = XT (ω)(ω) 6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Definition 4.2
A random time T is a stopping time for (Ft )t≥0 if, for all t ≥ 0, 1. Stoch. pr., filtrations {T ≤ t} = {ω : T (ω) ≤ t} ∈ Ft . 2. BM as weak limit
3. Gaussian p.
4. Stopping Proposition 4.3 times 5. Cond. 1 If T (ω) = t a.s. for some constant t ≥ 0, then T is a stopping expectation time. 6. Martingales 7. Discrete 2 Every stopping time T satisfies that, for all t, stoch. integral 8. Refl. princ., {T < t} ∈ F . (4.1) pass.times t
9. Quad.var., path prop. 3 If the filtration is right continuous and a random time T 10. Ito integral satisfies (4.1) for all t, then T is a stopping time. 11. Ito’s formula Lemma 4.4
Let S and T be stopping times for (Ft )t≥0. Then S ∧ T,S ∨ T and 1. Stoch. pr., filtrations S + T are stopping times for (Ft )t≥0. 2. BM as weak limit Definition 4.5 3. Gaussian p. Let T be a stopping time for (F ) . The σ-field F of events 4. Stopping t t≥0 T times determined prior to the stopping time T is given by 5. Cond. expectation FT = {A ∈ F : A ∩ {T ≤ t} ∈ Ft , for all t ≥ 0}. 6. Martingales
7. Discrete stoch. integral Remark: 8. Refl. princ., pass.times 1 FT is a σ-field. 9. Quad.var., path prop. 2 T is FT -measurable.
10. Ito integral 3 If T (ω) = t a.s., then FT = Ft . 11. Ito’s formula Example 4.6 1. Stoch. pr., filtrations Let (Wt )t≥0 be a standard Brownian motion on (Ω, F, (Ft )t≥0, P). 2. BM as Let Ta be the first passage time of the level a ∈ R, i.e., weak limit
3. Gaussian p. Ta(ω) = inf{t > 0 : Wt (ω) = a}. (4.2) 4. Stopping times Then Ta is a stopping time for (Ft )t≥0. 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 Remark: Let S and T be stopping times for (Ft )t≥0.
1 For A ∈ FS we have that A ∩ {S ≤ T } ∈ FT . 1. Stoch. pr., filtrations 2 If, moreover, S ≤ T then FS ⊆ FT . 2. BM as weak limit
3. Gaussian p.
4. Stopping times Definition 4.7 5. Cond. expectation A stochastic process (Xt )t≥0 is called progressively measurable for 6. Martingales (Ft )t≥0 if, for all t and all A ∈ B(R), the set 7. Discrete stoch. integral {(s, ω) : 0 ≤ s ≤ t, ω ∈ Ω, Xs (ω) ∈ A} ∈ B([0, t]) ⊗ Ft , 8. Refl. princ., pass.times that means 9. Quad.var., path prop. (s, ω) 7→ Xs (ω) is B([0, t]) ⊗ Ft − B(R) − measurable. 10. Ito integral
11. Ito’s formula Example: If (Xt )t≥0 is adapted and right-continuous then it is 1. Stoch. pr., filtrations progressively measurable.
2. BM as weak limit
3. Gaussian p. 4. Stopping Proposition 4.8 times
5. Cond. Let X be progressively measurable on (Ω, F, (Ft )t≥0) and let T be a expectation stopping time for (Ft )t≥0. Then 6. Martingales 1 7. Discrete XT defined on {T < ∞} is an FT -measurable random variable stoch. integral and 8. Refl. princ., pass.times 2 the stopped process (XT ∧t )t≥0 is progressively measurable. 9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 5. Conditional expected value
Definition 5.1 1. Stoch. pr., filtrations Let (Ω, F, P) be a probability space and X :Ω → R be an 2. BM as F − B( )-measurable random variable such that weak limit R E[|X |] = R |X |dP < ∞. Let G ⊆ F be a (smaller) σ-algebra. 3. Gaussian p. Then there exists a random variable Y with the following properties: 4. Stopping times 1 Y is G − B(R)-measurable. 5. Cond. expectation 2 Y is integrable, i.e., E[|Y |] < ∞. 6. Martingales
7. Discrete 3 For all A ∈ G: stoch. integral E[Y 1IA] = E[X 1IA]. 8. Refl. princ., pass.times The G-measurable random variable Y is called conditional expected 9. Quad.var., value of X given G. We write Y = E[X |G] a.s. path prop.
10. Ito integral
11. Ito’s formula Remark:
1. Stoch. pr., filtrations 1 E[X |G] is unique with respect to equality a.s.
2. BM as weak limit
3. Gaussian p. 2 4. Stopping 2 Suppose that, additionally, E[X ] < ∞. Then E[X |G] is the times orthogonal projection of X ∈ L2(Ω, F) onto the subspace 5. Cond. L2(Ω, G). expectation
6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop. 3 If G = {∅, Ω} then E[X |G] = E[X ] a.s.
10. Ito integral
11. Ito’s formula Definition 5.2 Let Z be a function on Ω. Then E[X |Z] := E[X |σ(Z)]. 1. Stoch. pr., filtrations 2. BM as Example 5.3 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 Example 5.4 S∞ 1. Stoch. pr., X : (Ω, F) → (R, B(R)). Suppose Ω = k=1 Ωk such that filtrations Ωi ∩ Ωj = ∅ for i 6= j. Let G := σ{Ωk , k ≥ 1}. 2. BM as weak limit E[X |G] = 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 Example 5.5 Suppose (X , Z) has a bivariate density f (x, z). 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 Theorem 5.6 (A list of properties of conditional expectation) In the following let X , Y , X , n ≥ 1, be integrable random variables 1. Stoch. pr., n filtrations on (Ω, F, P). The following holds 2. BM as weak limit (a) If X is G-measurable, then E[X |G] = X a.s.
3. Gaussian p.
4. Stopping times (b) Linearity: E[aX + bY |G] = aE[X |G] + bE[Y |G] a.s. 5. Cond. expectation
6. Martingales (c) Positivity: If X ≥ 0 a.s. then E[X |G] ≥ 0 a.s. 7. Discrete stoch. integral
8. Refl. princ., pass.times (d) Monotone convergence (MON): If 0 ≤ Xn and Xn ↑ X a.s. then 9. Quad.var., path prop. E[Xn|G] ↑ E[X |G] a.s.. 10. Ito integral
11. Ito’s formula (e) Fatou: If 0 ≤ Xn, for all n, then
1. Stoch. pr., filtrations E[lim inf Xn|G] ≤ lim inf E[Xn|G]. n n 2. BM as weak limit
3. Gaussian p.
4. Stopping (f) Dominated convergence (DOM): If |Xn| ≤ Y , for all n ≥ 1 and 1 times Y ∈ L (i.e. integrable) and lim Xn = X a.s. then 5. Cond. expectation lim E[Xn|G] = E[X |G] a.s. 6. Martingales n
7. Discrete stoch. integral 8. Refl. princ., (g) Jensen’s inequality: Let ϕ : → be a convex function such pass.times R R that ϕ(X ) is integrable. Then 9. Quad.var., path prop.
10. Ito integral ϕ (E[X |G]) ≤ E[ϕ(X )|G]
11. Ito’s formula 1. Stoch. pr., filtrations (h) Tower Property: If G1 ⊆ G2 ⊆ F then
2. BM as weak limit E [E[X |G2]|G1] = E[X |G1] a.s.
3. Gaussian p.
4. Stopping (i) Taking out what is known: Let Y be G-measurable and let times E[|XY |] < ∞. Then 5. Cond. expectation E[XY |G] = YE[X |G] 6. Martingales
7. Discrete stoch. integral (j) Role of independence: If X is independent of G, then
8. Refl. princ., pass.times E[X |G] = E[X ] 9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 6. Martingales and the discrete time stochastic integral
Definition 6.1
1. Stoch. pr., A stochastic process (Xt )t≥0 on (Ω, F, (Ft )t≥0, P) is a martingale if filtrations X is adapted and E[|Xt |] < ∞, for all t ≥ 0, and, for s < t, 2. BM as weak limit E[Xt |Fs ] = Xs a.s. (6.1) 3. Gaussian p. 4. Stopping If the = in (6.1) is replaced by ≤ (≥) the process is called times supermartingale (submartingale). 5. Cond. expectation Remark: 6. Martingales
7. Discrete 1 (6.1) is equivalent to E[Xt − Xs |Fs ] = 0 a.s. stoch. integral ∞ 8. Refl. princ., 2 Suppose (Xt )n=0 is a stochastic process in discrete time adapted pass.times ∞ to (Fn)n=0 then the martingale property reads as: for all n ≥ 1 9. Quad.var., path prop. E[Xn|Fn−1] = Xn−1 a.s. 10. Ito integral
11. Ito’s formula Example 6.2 (Random walk: sum of independent r.v.)
Let εk , k ≥ 1, be independent with E[εk ] = 0. Let 1. Stoch. pr., F = σ{ε , . . . , ε }, F = {∅, Ω}. Define X = 0 and, for n ≥ 1, filtrations n 1 n 0 0 2. BM as n weak limit X Xn = εk . 3. Gaussian p. k=1 4. Stopping times Then (Xn)n≥0 is a martingale for (Fn)n≥0. 5. Cond. expectation
6. Martingales Example 6.3 (Product of independent r.v.)
7. Discrete Let εk , k ≥ 1, be non-negative and independent with E[εk ] = 1. Let stoch. integral Fn = σ{ε1, . . . , εn}, F0 = {∅, Ω}. Define X0 = 1 and, for n ≥ 1, 8. Refl. princ., pass.times n 9. Quad.var., Y path prop. Xn = εk .
10. Ito integral k=1
11. Ito’s formula Then (Xn)n≥0 is a martingale for (Fn)n≥0. 1. Stoch. pr., filtrations
2. BM as weak limit Example 6.4
3. Gaussian p. Let X be a r.v. on (Ω, F, P) such that E[|X |] < ∞. Let Ft )t≥0 be 4. Stopping any filtration with Ft ⊆ F, for each t ≥ 0. Define times
5. Cond. expectation Xt = E[X |Ft ].
6. Martingales Then (Xt )t≥0 is a martingale for (Ft )t≥0. 7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 7. Stochastic integral in discrete time
Given is a discrete filtered probability space (Ω, F, (Fn)n≥0, P). Definition 7.1 1. Stoch. pr., filtrations ∞ A discrete stochastic process (Hn)n=1 is called predictable if 2. BM as weak limit Hn : (Ω, Fn−1) → (R, B(R)) 3. Gaussian p. 4. Stopping is measurable. (Short: H is F -measurable.) times n n−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 Definition 7.2 2. BM as ∞ ∞ weak limit Let (Xn)n=0 be an adapted stochastic process and (Hn)n=1 be a
3. Gaussian p. predictable stochastic process on (Ω, F, (Fn)n≥0, P). Then the
4. Stopping stochastic integral in discrete time at time n ≥ 1 is given by times n 5. Cond. X expectation (H · X )n = Hk (Xk − Xk−1). 6. Martingales k=1
7. Discrete stoch. integral We define (H · X )0 = 0. The stochastic integral process is given by ∞ 8. Refl. princ., ((H · X )n)n=0. pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Theorem 7.3 1. Stoch. pr., filtrations Let H be bounded and predictable and X be a martingale on 2. BM as (Ω, F, (Fn)n≥0, P). Then the stochastic integral process weak limit ∞ ((H · X )n)n=0 is a martingale. 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 1. Stoch. pr., filtrations Theorem 7.4 (Doob’s Optional Stopping Theorem) 2. BM as weak limit Let (Mt )t≥0 be a martingale in continuous time on 3. Gaussian p. (Ω, F, (Ft )t≥0, P). Then, for a bounded stopping time T it holds 4. Stopping that times E[MT ] = M0. 5. Cond. expectation More generally, if S ≤ T are bounded stopping times, then 6. Martingales 7. Discrete E[M |F ] = M a.s. stoch. integral T S S
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 8. Reflection principle, passage times, running maximum/minimum
1. Stoch. pr., filtrations
2. BM as weak limit Definition 8.1 (Markov property) 3. Gaussian p.
4. Stopping Brownian motion satisfies the Markov property, that means, for all times t ≥ 0 and s > 0: 5. Cond. expectation P(Wt+s ≤ y|Ft ) = P(Wt+s ≤ y|Wt ) a.s. 6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Reflection principle: Recall that Ta = inf{t > 0 : Wt = a}.
1. Stoch. pr., filtrations
2. BM as weak limit
3. Gaussian p.
4. Stopping times
5. Cond. expectation
6. Martingales P(Ta < t) =
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop. P(Ta < t) = 2P(Wt > a) (8.1) 10. Ito integral
11. Ito’s formula In all the following (Wt )t≥0 is a standard Brownian motion on 1. Stoch. pr., (Ω, F, (F ) , P). filtrations t t≥0
2. BM as weak limit Theorem 8.2 (Strong Markov property)
3. Gaussian p. The BM (Wt )t≥0 satisfies the strong Markov property: for each 4. Stopping stopping time τ with P(τ < ∞) = 1 it holds that times 5. Cond. ˆ expectation Wt = Wτ+t − Wτ ,, t ≥ 0
6. Martingales is a standard Brownian motion independent of Fτ . 7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop. We will see later that P(Ta < ∞) = 1 for all a ∈ R.
10. Ito integral
11. Ito’s formula Theorem 8.3 (Reflection principle)
Let a 6= 0 and Ta be the passage time for a. 1. Stoch. pr., filtrations ( 2. BM as ˜ Wt for t ≤ Ta weak limit Wt = 2WTa − Wt = 2a − Wt for t ≥ Ta 3. Gaussian p. 4. Stopping ˜ times Then (Wt )t≥0 is again a standard Brownian motion.
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 Lemma 8.4
P(sup Wt = +∞ and inft≥0 Wt = −∞) = 1 1. Stoch. pr., t≥0 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 Theorem 8.5 (Recurrence) 1. Stoch. pr., filtrations P(Ta < ∞) = 1 for all a ∈ R. 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 1. Stoch. pr., Theorem 8.6 (Distribution of passage time) filtrations Let a 6= 0. The density of Ta is given by 2. BM as weak limit |a| 3 a2 3. Gaussian p. − 2 − 2t fTa (t) = √ t e , 4. Stopping 2π times
5. Cond. which is the density of an invers-gamma-distribution with parameters expectation 1 a2 2 and 2 . In particular, we have that E[Ta] = +∞. 6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Running maximum and minimum of Brownian motion:
1. Stoch. pr., filtrations Let Mt = max0≤s≤t Ws and mt = min0≤s≤t Ws . 2. BM as weak limit Theorem 8.7 3. Gaussian p. 4. Stopping 1 For each a > 0: times P(Mt ≥ a) = P(Ta ≤ t) = P(Ta < t) = 2P(Wt > a). 5. Cond. expectation 2 For each a < 0:P(mt ≤ a) = 2P(Wt < a). 6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Theorem 8.8 (Joint distribution)
1. Stoch. pr., Let y ≥ x ≥ 0. Then filtrations
2. BM as weak limit P(Wt ≤ x, Mt ≥ y) = P(Wt ≥ 2y − x).
3. Gaussian p. This implies that the joint distribution of (Wt , Mt ) has the following 4. Stopping times density: for y ≥ 0, x ≤ y 5. Cond. r expectation 2 (2y − x) −(2y−x)2 f (x, y) = e 2t . 6. Martingales W ,M 3 π t 2 7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Let a < 0 < b and let τ be the first time to leave an interval (a, b)
1. Stoch. pr., (exit time), i.e., filtrations 2. BM as τ = inf{t > 0 : Wt ∈/ (a, b)}. 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. We have that τ = min(Ta, Tb) = Ta ∧ Tb. 10. Ito integral
11. Ito’s formula Theorem 8.9 (Exit time)
1. Stoch. pr., Let a < 0 < b and τ = Ta ∧ Tb as above. Then P(τ < ∞) = 1 and filtrations E[τ] = |ab|. Moreover 2. BM as weak limit |a| −a P(W = b) = = 3. Gaussian p. τ |a| + b −a + b 4. Stopping times b b P(Wτ = a) = = 5. Cond. |a| + b −a + b expectation
6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 9. Quadratic variation and path properties of BM
1. Stoch. pr., filtrations
2. BM as weak limit
3. Gaussian p.
4. Stopping times Definition 9.1 5. Cond. expectation A stochastic process has finite quadratic variation if there exists an 6. Martingales a.s. finite stochastic process ([X , X ]t )t≥0 such that for all t and 7. Discrete partitions π = {tn,..., tn } with 0 = tn < tn < ··· < tn = t and n 0 mn 0 1 mn stoch. integral n n δn = maxi=1,...,mn (ti − ti−1) → 0 we have that, for n → ∞, 8. Refl. princ., pass.times mn 9. Quad.var., 2 X 2 V (π ) := (X n − X n ) → [X , X ] in probability. (9.1) path prop. t n ti ti−1 t 10. Ito integral i=1
11. Ito’s formula Theorem 9.2 2 2 Let (Wt )t≥0 be a Brownian motion. Then Vt (πn) → t in L , that 1. Stoch. pr., means filtrations 2 2 lim E[(Vt (πn) − t) ] = 0. 2. BM as n→∞ weak limit 2 Therefore it follows that V (πn) → t in probability. Hence 3. Gaussian p. t [W , W ]t = t a.s. 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 1. Stoch. pr., Corollary 9.3 filtrations 2. BM as (Wt )t≥0 has a.s. paths of infinite variation on each interval. 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 1. Stoch. pr., Corollary 9.4 filtrations For almost all ω the path t 7→ W (ω) is not monotone in each 2. BM as t weak limit interval. 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 Remark 9.5
Almost all paths of (Wt )t≥0 are not differentiable on each interval. 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 10. Stochastic integral by Ito
For the whole chapter let (Wt )t≥0 be a standard Brownian motion on (Ω, F, (Ft )t≥0, P). Everything will be based on this filtered
1. Stoch. pr., probability space. filtrations
2. BM as weak limit
3. Gaussian p. Recall:
4. Stopping times Riemann integral: 5. Cond. Z b n expectation X n n n f (t)dt = lim f (ξi )(ti − ti−1) 6. Martingales δn→0 a i=1 7. Discrete n n n n n n stoch. integral where a = t0 < t1 < ··· < tn = b, ti−1 ≤ ξi ≤ ti and n n 8. Refl. princ., δn = maxi=1,...,n(ti − ti−1). pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 1. Stoch. pr., Riemann-Stieltjes-integral: Let f be a continuous function and g be filtrations a function of bounded variation. Then the following limit exists and 2. BM as is called R-S-integral of f with respect to g: weak limit 3. Gaussian p. Z b n X n n n 4. Stopping f (t)dg(t) := lim f (ti−1)(g(ti ) − g(ti−1)), δn→0 times a i=1 5. Cond. expectation n n n n n where a = t0 < t1 < ··· < tn = b and δn = maxi=1,...,n(ti − ti−1). 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., Theorem 10.1 filtrations n n n Fix [0, t] and let, for each n, 0 = t0 < t1 < ··· < tn = t. If g is a 2. BM as weak limit function such that
3. Gaussian p. n X 4. Stopping n n n lim f (ti−1)(g(ti ) − g(ti−1)) times δn→0 i=1 5. Cond. expectation exists for each continuous function f : [0, t] → R, then g is of 6. Martingales bounded variation on [0, t]. 7. Discrete stoch. integral
8. Refl. princ., Remark 10.2 pass.times No pathwise R-S-integral for Brownian motion!!! 9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Ito integral with simple integrands: Definition 10.3 1. Stoch. pr., filtrations A stochastic process (Xt )0≤t≤T is called simple predictable
2. BM as integrand if weak limit n−1 3. Gaussian p. X X = ξ 1I (t) + ξ 1I (t), 4. Stopping t 0 [t0,t1] i (ti ,ti+1] times i=1
5. Cond. expectation where 0 = t0 < t1 < ··· < tn = T and ξ0, . . . , ξn−1 are random 2 6. Martingales variables such that ξi is Fti -measurable and E[ξi ] < ∞, for 7. Discrete i = 1,..., n − 1. stoch. integral
8. Refl. princ., For each t ≤ T the Ito integral at time t for a simple integrand X is pass.times given by: 9. Quad.var., path prop. n−1 Z t X 10. Ito integral Xs dWs =: (X · W )t = ξi (Wti+1∧t − Wti ∧t ). 0 11. Ito’s i=0 formula 1. Stoch. pr., filtrations Remark 10.4
2. BM as If t < T is such that tk ≤ t < tk+1 ≤ tn = T , this means weak limit k−1 3. Gaussian p. X 4. Stopping (X · W )t = ξi (Wti+1 − Wti ) + ξk (Wt − Wtk ). times i=0 5. Cond. expectation For T this means 6. Martingales n−1 7. Discrete X stoch. integral (X · W )T = ξi (Wti+1 − Wti ) 8. Refl. princ., i=0 pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Theorem 10.5 (Properties of Ito integral for simple integrands) 1. Stoch. pr., filtrations 1 Linearity: X , Y simple, α, β constants, then 2. BM as weak limit Z T Z T Z T 3. Gaussian p. (αXt + βYt ) dWt = α Xt dWt + β Yt dWt . 4. Stopping 0 0 0 times 5. Cond. R T expectation 2 Let 0 ≤ a < b. Then 0 1I(a,b](t)dWt = Wb − Wa and T b 6. Martingales R R 0 1I(a,b](t)Xt dWt = a Xt dWt . 7. Discrete T stoch. integral R 3 Zero mean: E[ 0 Xt dWt ] = 0 8. Refl. princ., pass.times 2 R T R T 2 4 Isometry: E Xt dWt = E[X ]dt 9. Quad.var., 0 0 t path prop.
10. Ito integral
11. Ito’s formula (More) general integrands: Lemma 10.6 1. Stoch. pr., filtrations Let (Xt )0≤t≤T be a bounded and progressively measurable stochastic
2. BM as process. Then there exists a sequence of simple predictable processes weak limit m (Xt )0≤t≤T , m ≥ 1, such that 3. Gaussian p. "Z T # 4. Stopping m 2 times lim E |Xt − Xt | dt = 0. (10.1) m→∞ 5. Cond. 0 expectation
6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Lemma 10.7
Let (Xt )0≤t≤T be a progressively measurable stochastic process such 1. Stoch. pr., that filtrations "Z T # 2 2. BM as E Xt dt < ∞. (10.2) weak limit 0 3. Gaussian p. Then there exists a sequence of simple predictable processes 4. Stopping m times (Xt )0≤t≤T , m ≥ 1, such that (10.1) holds, i.e., 5. Cond. " # expectation Z T m 2 6. Martingales lim E |Xt − Xt | dt = 0. m→∞ 0 7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Theorem 10.8 (Ito integral for general integrands with (10.2))
Let (Xt )0≤t≤T be a progressively measurable stochastic process such 1. Stoch. pr., R T 2 filtrations that (10.2) holds, i.e.,E [ 0 Xt dt] < ∞. Then the stochastic R T 2. BM as integral Xt dWt is defined and has the following properties: weak limit 0
3. Gaussian p. 1 Linearity: X , Y as above, then
4. Stopping times Z T Z T Z T (αX + βY ) dW = α X dW + β Y dW . 5. Cond. t t t t t t t expectation 0 0 0
6. Martingales T 2 R 7. Discrete Let 0 ≤ a < b. Then 0 1I(a,b](t)dWt = Wb − Wa and stoch. integral R T R b 0 1I(a,b](t)Xt dWt = a Xt dWt . 8. Refl. princ., pass.times R T 3 Zero mean: E[ Xt dWt ] = 0 9. Quad.var., 0 path prop. 2 R T R T 2 10. Ito integral 4 Isometry: E 0 Xt dWt = 0 E[Xt ]dt 11. Ito’s formula Theorem 10.9 (Ito integral for even more general integrands) 1. Stoch. pr., filtrations Let (Xt )0≤t≤T be a progressively measurable stochastic process such that 2. BM as ! weak limit Z T 2 3. Gaussian p. P Xt dt < ∞ = 1. (10.3) 0 4. Stopping times R T Then the stochastic integral Xt dWt is defined and has satisfies 5. Cond. 0 expectation (1) and (2) of Theorem 10.8.
6. Martingales 7. Discrete Remark 10.10 stoch. integral Attention: in this case (3) Zero-Mean and (4) Isometry are not 8. Refl. princ., pass.times satisfied in general!!! If, additionally the stronger condition (10.2) 9. Quad.var., holds, then we are in the case of Theorem 10.8 and Zero-Mean and path prop. Isometry hold. 10. Ito integral
11. Ito’s formula 1. Stoch. pr., Corollary 10.11 filtrations R T Let X be continuous and adapted. Then Xt dWt exists. In 2. BM as 0 weak limit particular, for any continuous function f : R → R the stochastic T 3. Gaussian p. R integral 0 f (Wt )dWt exists. 4. Stopping times
5. Cond. expectation
6. Martingales
7. Discrete stoch. integral 8. Refl. princ., Remark 10.12 pass.times The Ito integral is not monotone. 9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Ito integral process: 1. Stoch. pr., filtrations Let X be progressively measurable such that (10.3) holds, i.e., 2. BM as weak limit Z T 3. Gaussian p. 2 P( Xs ds < ∞) = 1. 4. Stopping 0 times R t 5. Cond. Then Xs dWs is defined for all t ≤ T . This gives a stochastic expectation 0 process (It)0≤t≤T with 6. Martingales 7. Discrete Z t stoch. integral It = XsdWs. 8. Refl. princ., 0 pass.times
9. Quad.var., There exists a modification with continuous paths: we always take path prop. this modification. 10. Ito integral
11. Ito’s formula Definition 10.13
1. Stoch. pr., A martingale (Mt )0≤t≤T is called quadratic integrable on [0, T ] if filtrations 2 2. BM as sup E[Mt ] < ∞. weak limit t∈[0,T ] 3. Gaussian p.
4. Stopping times Theorem 10.14 5. Cond. expectation Let X be progressively measurable such that (10.2) holds, i.e., T t 6. Martingales R 2 R E[ 0 Xs ds] < ∞. Then (It)0≤t≤T, where It = 0 XsdWs, is a 7. Discrete quadratic integrable martingale. 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. Corollary 10.15 4. Stopping Let f : → be a continuous and bounded function. Then times R R (I ) , where I = R t f(W )dW , is a quadratic integrable 5. Cond. t 0≤t≤T t 0 s s expectation martingale. 6. Martingales
7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula Quadratic variation and covariation of Ito integrals: 1. Stoch. pr., filtrations Theorem 10.16 2. BM as Let X be progressively measurable such that (10.3 holds. The weak limit quadratic variation of R t X dW satisfies 3. Gaussian p. 0 s s 4. Stopping Z t Z t Z t times 2 Xs dWs , Xs dWs = Xs ds a.s. 5. Cond. 0 0 0 expectation 6. Martingales for all 0 ≤ t ≤ T. 7. Discrete stoch. integral
8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 1. Stoch. pr., R t ˜ R t ˜ filtrations Covariation: Let It = 0 XsdWs and It = 0 Xs dWs , for 0 ≤ t ≤ T , 2. BM as and progressively measurable processes X , X˜ such that (10.3) holds. weak limit 3. Gaussian p. The covariation of I and ˜I is given by 4. Stopping times ˜ 1 ˜ ˜ ˜ ˜ 5. Cond. [I, I]t := [I + I, I + I]t − [I, I]t − [I, I]t . expectation 2
6. Martingales
7. Discrete Corollary 10.17 stoch. integral R t It holds that [I,˜I]t = XsX˜ sds a.s. 8. Refl. princ., 0 pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 1. Stoch. pr., filtrations Remark 10.18 2. BM as weak limit Analogously to the quadratic variation the covariation of two 3. Gaussian p. stochastic processes Y and X can be defined as follows: 4. Stopping m times Xn 5. Cond. [Y , X ] = lim (Y n − Y n )(X n − X n ), t ti ti−1 ti ti−1 expectation n→∞ i=1 6. Martingales in probability where 0 = tn < tn < . . . tn = t and 7. Discrete 0 1 mn stoch. integral n n maxi=1,...,mn (ti − ti−1) → 0 for n → ∞. 8. Refl. princ., pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 1. Stoch. pr., filtrations Integration by parts: 2. BM as weak limit Theorem 10.19 (Integration by parts) 3. Gaussian p. Let X , Y be continuous and adapted processes such that (10.3) holds 4. Stopping times for both processes. Then the following holds, for each 0 ≤ t ≤ T,
5. Cond. t t expectation Z Z Xt Yt = X0Y0 + Xs dYs + Ys dXs + [X , Y ]t . 6. Martingales 0 0 7. Discrete stoch. integral
8. Refl. princ., Notation: d(Xt Yt ) = Xt dYt + Yt dXt + d[X , Y ]t pass.times
9. Quad.var., path prop.
10. Ito integral
11. Ito’s formula 11. Ito’s formula: change of variables
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 Theorem 11.1 1. Stoch. pr., filtrations Let (Wt )t≥0 be a Brownian motion and let f : R → R be a twice 2 2. BM as continuously differentiable function (f ∈ C (R)). Then, for each t, weak limit 3. Gaussian p. Z t Z t 0 1 00 4. Stopping f (Wt ) − f (W0) = f (Ws )dWs + f (Ws )ds. (11.1) times 0 2 0
5. Cond. expectation
6. Martingales
7. Discrete stoch. integral In differential notation (11.1) reads as follows: 8. Refl. princ., pass.times 0 1 00 9. Quad.var., df (Wt ) = f (Wt )dWt + f (Wt )dt path prop. 2
10. Ito integral
11. Ito’s formula