An Introduction to Stochastic Calculus

Transcription

1 An Introduction to Stochastic Calculus Haijun Li Department of Mathematics Washington State University Weeks 3-4 Haijun Li An Introduction to Stochastic Calculus Weeks / 22

2 Outline 1 Conditional Expectation A Motivating Example σ-fields 2 The General Conditional Expectation The Conditional Expectation Given Known Information Rules for Calculation of Conditional Expectations The Projection Property of Conditional Expectations Haijun Li An Introduction to Stochastic Calculus Weeks / 22

3 Conditioning Based on Available Information Let X be a random variable defined on a probability space (Ω, F, P), and B Ω with P(B) > 0. The conditional probability of A given B is defined as P(A B) := P(A B). P(B) That is, P( B) : F [0, 1] is a probability measure given the information that the event B has occurred. The conditional distribution function of X given B is defined as A A {}}{{}}{ F(x B) := P( X x B) = P( {X x} B). P(B) Haijun Li An Introduction to Stochastic Calculus Weeks / 22

4 Conditional Average Let F(x) = P(X x) denote the distribution of X. The conditional expectation of X given B is the average of the conditional distribution F(x B). That is, B x df (x) E(X B) := x df (x B) = = E(XI B) P(B) P(B), where I B (ω) : Ω [0, 1] is the indicator function of the event B: { 1, if ω B I B (ω) = 0, if ω / B. E(X B) can be viewed as our estimate to X given the information that the event B has occurred. E(X B c ) is similarly defined. Together, E(X B) and E(X B c ) provide our estimate to X depending on whether or not B occurs. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

5 Conditional Expectation Under Discrete Conditioning Think I B as a random variable that carries the information on whether the event B occurs, and the conditional expectation E(X I B ) of X given I B is a random variable defined as { E(X B), if ω B E(X I B )(ω) := E(X B c ), if ω / B. The random variable E(X I B ) is our estimate to X based on the information provided by I B. Consider a discrete random variable Y on Ω taking distinct values y i, i = 1, 2,...,. Let A i = {ω Ω : Y (ω) = y i }. Note that Y carries the information on whether or not events A i s occur. Define the conditional expectation of X given Y : E(X Y )(ω) := E(X A i ) = E(X Y = y i ), if ω A i, i = 1, 2,.... The random variable E(X Y ) can be viewed as our estimate to X based on the information carried by Y. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

6 Example: Uniform Random Variable Consider the random variable X(ω) = ω on Ω = (0, 1], with density function given by f X (x) = 1 for all x (0, 1]. Assume that one of the events ( i 1 A i = n, i ], i = 1,..., n, n occurred. Then E(X A i ) = 1 P(A i ) A i xf X (x)dx = 1 2 2i 1 n (i.e., the center of A i ). The value E(X A i ) is the updated expectation on the new space A i, given the information that A i occurred. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

7 Haijun Li An Introduction to Stochastic Figure Calculus : Weeks / Estimating CONDITIONAL Uniform EXPECTATION Random Variable 59 L, I0 O W Figure Left: a unaform random variable X on (0,1] (dotted line) and its expectation (solid line); see Example Right: the random variable X (dotted line) and the conditional expectations E(X(Ai) (solid lines), where Ai = ((i - 1)/5, i/5], i = 1,...,5. These conditional expectations can be interpreted as the values of a discrete random variable E(X]Y) with distinct constant values on the sets Ai; see Example

8 Estimating Uniform Random Variable Consider ( i 1 A i = n, i ], i = 1,..., n, n E(X A i ) = 1 2i 1 (i.e., the center of A i ). 2 n Define Y (ω) := n i=1 i 1 n I A i (ω), i = 1,..., n. The conditional expectation E(X Y )(ω) = 1 2 2i 1, if ω A i, i = 1,..., n. n Since E(X Y )(ω) is the average of X given the information that ω ((i 1)/n, i/n], E(X Y ) is a coarser version of X, that is, an approximation to X, given the information that any of the A i s occurred. E(X Y ) X, when n is sufficiently large. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

9 Properties of Conditional Expectation The conditional expectation is linear: for random variables X 1, X 2 and constants c 1, c 2, EX = E[E(X Y )]. E([c 1 X 1 + c 2 X 2 ] Y ) = c 1 E(X 1 Y ) + c 2 E(X 2 Y ). If X and Y are independent, then E(X Y ) = EX. The random variable E(X Y ) is a (measurable) function of Y : E(X Y ) = g(y ), where g(y) = E(X Y = y i )I {yi }(y). i=1 Haijun Li An Introduction to Stochastic Calculus Weeks / 22

10 σ-fields Observe that the values of Y did not really matter for the definition of E(X Y ) under discrete conditioning, but it was crucial that conditioning events A i s describe the information carried by all the distinct values of Y. That is, we estimate the random variable X via E(X Y ) based on the information provided by observable events A i s and their composite events, such as A i A j and A i A j,... Definition of σ-fields A σ-field F on Ω is a collection of subsets (observable events) of Ω satisfying the following conditions: F and Ω F. If A F, then A c F. If A 1, A 2, F, then i=1 A i F, and i=1 A i F. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

11 Generated σ-fields For any collection C of events, let σ(c) denote the smallest σ-field containing C, by adding all possible unions, intersections and complements. σ(c) is said to be generated by C. The following are some examples. F = {, Ω} = σ({ }). F = {, Ω, B, B c } = σ({b}). F = {A : A Ω} = σ({a : A Ω}). Let C = {(a, b] : < a < b < }, then any set in B 1 = σ(c) is called a Borel subset in R. Let C = {(a, b] : < a i < b i <, i = 1,..., d}, then any set in B d = σ(c) is called a Borel subset in R d. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

12 σ-fields Generated By Random Variables Let Y be a discrete random variable taking distinct values y i, i = 1, 2,.... Define A i = {ω : Y (ω) = y i }, i = 1, 2,.... A typical set in the σ-field σ({a i }) is of this form A = i I A i, I {1, 2,... }. σ({a i }) is called the σ-field generated by Y, and denoted by σ(y ). Let Y be a random vector and A(a, b] = {ω : Y (ω) (a, b], < a i < b i <, i = 1,..., d}. The σ-field σ({a(a, b], a, b R d }) is called the σ-field generated by Y, and denoted by σ(y ). σ(y ) provides the essential information about the structure of Y, and contains all the observable events {ω : Y (ω) C}, where C is a Borel subset of R d. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

13 σ-fields Generated By Stochastic Processes For a stochastic process Y = (Y t, t T ), and any (measurable) set C of functions on T, let A(C) = {ω : the sample path (Y t (ω), t T ) belongs to C}. The σ-field generated by the process Y is the smallest σ-field that contains all the events of the form A(C). Example: For Brownian motion B = (B t, t 0), let F t := σ({b s, s t}) denote the σ-field generated by Brownian motion prior to time t. F t contains the essential information about the structure of the process B on [0, t]. One can show that this σ-field is generated by all sets of the form, 0 t 1 t 2 t n t, A t1,...,t n (C) = {ω : (B t1 (ω),..., B tn (ω)) C} for all the n-dimensional Borel sets C. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

14 Information Represented by σ-fields For a random variable, a random vector or a stochastic process Y on Ω, the σ-field σ(y ) generated by Y contains the essential information about the structure of Y as a function of ω Ω. It consists of all subsets {ω : Y (ω) C} for suitable sets C. Because Y generates a σ-field, we also say that Y contains information represented by σ(y ) or Y carries the information σ(y ). For any measurable function f acting on Y, since {ω : f (Y (ω)) C} = {ω : Y (ω) f 1 (C)}, measurable set C we have that σ(f (Y )) σ(y ). That is, a function f acting on Y does not provide new information about the structure of Y. Example: For Brownian motion B = (B t, t 0), consider the function f (B) = sup 0 t 1 B t. The σ(f (B)) σ({b s, s t}) for any t 1. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

15 The General Conditional Expectation Let (Ω, F, P) be a probability space, and Y, Y 1 and Y 2 denote random variables (or random vectors, stochastic processes) defined on Ω. The information of Y is contained in F or Y does not contain more information than that contained in F σ(y ) F. Y 1 contains more information than Y 2 σ(y 2 ) σ(y 1 ). Conditional Expectation Given the σ-field Let X be a random variable defined on Ω. The conditional expectation given F is a random variable, denoted by E(X F), with the following properties: E(X F) does not contain more information than that contained in F: σ(e(x F)) F. For any event A F, E(XI A ) = E(E(X F)I A ). By virtue of the Radon-Nikodym theorem, we can show the existence and almost sure (a.s.) uniqueness of E(X F). Haijun Li An Introduction to Stochastic Calculus Weeks / 22

16 Conditional Expectation Given Generated Information Let Y be a random variable (random vector or stochastic process) on Ω. The conditional expectation of X given Y, denoted by E(X Y ), is defined as E(X Y ) := E(X σ(y )). The random variables X and E(X F) are close to each other, not in the sense that they coincide for any ω, but averages (expectations) of X and E(X F) on suitable sets A are the same. The conditional expectation E(X F) is a coarser version of the original random variable X and is our estimate to X given the information F. Example: Let Y be a discrete random variable taking distinct values y i, i = 1, 2,.... Any set A σ(y ) can be written as A = i I A i = i I {ω : Y (ω) = y i }, for some I {1, 2,... }. Let Z := E(X Y ). Then σ(z ) σ(y ) and Z (ω) = E(X A i ), for ω A i. Observe that E(XI A ) = E(X i I I Ai ) = i I E(XI Ai ) = i I E(X A i )P(A i ) = E(ZI A ). Haijun Li An Introduction to Stochastic Calculus Weeks / 22

17 Remarks Classical Conditional Expectation: Let B be an event with P(B) > 0, P(B c ) > 0. Define F B := σ({b}) = {, Ω, B, B c }. Then E(X F B )(ω) = E(X B), for ω B. Classical Conditional Probability: If X = I A, then E(I A F B )(ω) = E(I A B) = P(A B), for ω B. P(B) The idea to write the conditional expectation as averages of a random variable given a σ-field (information set) goes back to Andrey Kolmogorov (1933, the Foundations of the Theory of Probability). Haijun Li An Introduction to Stochastic Calculus Weeks / 22

18 Rules for Calculation of Conditional Expectations Let X, X 1, X 2 denote random variables defined on (Ω, F, P). 1 For any two constants c 1, c 2, E(c 1 X 1 + c 2 X 2 F) = c 1 E(X 1 F) + c 2 E(X 2 F). 2 EX = E[E(X F)]. 3 If X and F are independent, then E(X F) = EX. In particular, if X and Y are independent, then E(X Y ) = EX. 4 If σ(x) F, then E(X F) = X. In particular, if X is a function of Y, then σ(x) σ(y ) and E(X Y ) = X. 5 If σ(x) F, then E(XX 1 F) = XE(X 1 F). In particular, if X is a function of Y, then σ(x) σ(y ) and E(XX 1 Y ) = XE(X 1 Y ). 6 If F and F are two σ-fields with F F, then E(X F) = E[E(X F ) F], and E(X F) = E[E(X F) F ]. The more information (F ) we know, the finer estimate (E(X F ) VS E(X F)) we can provide for X. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

19 Examples Example 1: If X and Y are independent, then E(XY Y ) = Y EX, and E(X + Y Y ) = EX + Y. Example 2: Consider Brownian motion B = (B t, t 0). The σ-fields F s = σ(b x, x s) represent an increasing stream of information about the structure of the process. Find E(B t F s ) = E(B t B x, x s) for s 0. If s t, then F s F t and thus E(B t F s ) = B t. If s < t, then E(B t F s ) = E(B t B s F s ) + E(B s F s ) = E(B s F s ) = B s. E(B t F s ) = B min(s,t). Haijun Li An Introduction to Stochastic Calculus Weeks / 22

20 Another Example: Squared Brownian Motion Consider again Brownian motion B = (B t, t 0), with the σ-fields F s = σ(b x, x s). Define X t := B 2 t t, t 0. If s t, then F s F t and thus E(X t F s ) = X t. If s < t, observe that X t = [(B t B s ) + B s ] 2 t = (B t B s ) 2 + B 2 s + 2B s (B t B s ) t. Since (B t B s ) and (B t B s ) 2 are independent of F s, we have E[(B t B s ) 2 F s ] = E(B t B s ) 2 = (t s), E[B s (B t B s ) F s ] = B s E(B t B s ) = 0. Since σ(b 2 s ) σ(b s ) F s, we have Thus, E(X t F s ) = X s. E(X t F s ) = X min(s,t). E(B 2 s F s ) = B 2 s. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

21 The Projection Property of Conditional Expectations We now formulate precisely the meaning of the statement that the conditional expectation E(X F) can be understood as the optimal estimate to X given the information F. Define L 2 (F) := {Z : σ(z ) F, EZ 2 < }. If F = σ(y ), then Z L 2 (σ(y )) implies that Z is a function of Y. The Projection Property Let X be a random variable with EX 2 <. The conditional expectation E(X F) is that random variable in L 2 (F) which is closest to X in the mean square sense: E[X E(X F)] 2 = min E(X Z ) 2. Z L 2 (F) If F = σ(y ), then E(X Y ) is that function of Y which has a finite second moment and which is closest to X in the mean square sense. Haijun Li An Introduction to Stochastic Calculus Weeks / 22

22 The Best Prediction Based on Available Information It follows from the projection property that the conditional expectation E(X F) can be viewed as the best prediction of X given the information F. For example, for Brownian motion B = (B t, t 0), we have, s t, E(B t B x, x s) = B s, and E(B 2 t t B x, x s) = B 2 s s. That is, the best predictions of the future values B t and B 2 t t, given the information about Brownian motion until the present time s, are the present values B s and B 2 s s, respectively. This property characterizes the whole class of martingales with a finite second moment. Haijun Li An Introduction to Stochastic Calculus Weeks / 22