Conditional Expectation

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1 Chapter Conditional Expectation Please see Hull s book (ection 9.6.).1 Binomial Model for tock Price Dynamics tock prices are assumed to follow this simple binomial model: The initial stock price during the period under study is denoted 0. t each time step, the stock price either goes up by a factor of u or down by a factor of d. It will be useful to visualize tossing a coin at each time step, and say that the stock price moves up by a factor of u if the coin comes out heads (H), and down by a factor of d if it comes out tails (T ). Note that we are not specifying the probability of heads here. Consider a sequence of tosses of the coin (ee Fig..1) The collection of all possible outcomes (i.e. sequences of tosses of length ) is =fhhh HHT HTH HTT THH THH THT TTH TTTg: typical sequence of will be denoted!, and! k will denote the kth element in the sequence!. We write k (!) to denote the stock price at time k (i.e. after k tosses) under the outcome!. Note that k (!) depends only on!1! :::! k. Thus in the -coin-toss example we write for instance, 1(!) = 1(!1!!) = 1(!1) (!) = (!1!!) = (!1!): Each k is a random variable defined on the set. More precisely, let F = P(). Then F is a -algebra and ( F) is a measurable space. Each k is an F-measurable function!ir, that is, ;1 k is a function B!F where B is the Borel -algebra on IR. We will see later that k is in fact 9

2 50 (HH) = u 0 ω = Η (HHH) = u (H) = u0 ω 1 = Η ω 1 = Τ 1 (T) = d 0 ω = Η ω = Τ (HT) = ud 0 ω = Τ ω = Η (HHT) = u d 0 (HTH) = u d 0 (THH) = u d 0 (TH) = ud 0 ω = Τ ω = Η (HTT) = d u 0 (THT) = d u 0 (TTH) = d u 0 ω = Τ ω = Η (TT) = d 0 ω = Τ (TTT) = d 0 Figure.1: three coin period binomial model. measurable under a sub--algebra of F. Recall that the Borel -algebra B is the -algebra generated by the open intervals of IR. In this course we will always deal with subsets of IR that belong to B. For any random variable X defined on a sample space and any y IR, we will use the notation: fx yg = f! X(!) yg: The sets fx <yg fx yg fx = yg etc, are defined similarly. imilarly for any subset B of IR, we define fx Bg = f! X(!) Bg: ssumption.1 u>d>0.. Information Definition.1 (ets determined by the first k tosses.) We say that a set is determined by the first k coin tosses if, knowing only the outcome of the first k tosses, we can decide whether the outcome of all tosses is in. In general we denote the collection of sets determined by the first k tosses by F k. It is easy to check that F k is a -algebra. Note that the random variable k is F k -measurable, for each k =1 ::: n. Example.1 In the coin-toss example, the collection F1 of sets determined by the first toss consists of:

3 CHPTER. Conditional Expectation H = fhhh HHT HTH HTTg,. T = fthh THT TTH TTTg,.,.. The collection F of sets determined by the first two tosses consists of: 1. HH = fhhh HHTg,. HT = fhth HTTg,. TH = fthh THTg,. TT = ftth TTTg, 5. The complements of the above sets, 6. ny union of the above sets (including the complements), 7. and. Definition. (Information carried by a random variable.) Let X be a random variable!ir. We say that a set is determined by the random variable X if, knowing only the value X(!) of the random variable, we can decide whether or not!. nother way of saying this is that for every y IR, either X ;1 (y) or X ;1 (y) \ =. The collection of susbets of determined by X is a -algebra, which we call the -algebra generated by X, and denote by (X). If the random variable X takes finitely many different values, then (X) is generated by the collection of sets fx ;1 (X(!))j! g these sets are called the atoms of the -algebra (X). In general, if X is a random variable!ir, then (X) is given by (X) =fx ;1 (B) B Bg: Example. (ets determined by ) The -algebra generated by consists of the following sets: 1. HH = fhhh HHTg = f! (!) =u 0g,. TT = ftth TTTg = f = d 0g. HT [ TH = f = ud0g. Complements of the above sets, 5. ny union of the above sets, 6. = f(!) g, 7. =f(!) IRg.

4 5. Conditional Expectation In order to talk about conditional expectation, we need to introduce a probability measure on our coin-toss sample space. Let us define p (0 1) is the probability of H, q =(1; p) is the probability of T, the coin tosses are independent, so that, e.g., IP (HHT)=p q etc. P IP () =! IP (!), 8. Definition. (Expectation.) IEX = X! X(!)IP (!): If then and IE(I X) = I (!) = ( 1 if! 0 if! 6 XdIP = X! X(!)IP (!): We can think of IE(I X) as a partial average of X over the set...1 n example Let us estimate 1, given. Denote the estimate by IE(1j). From elementary probability, IE(1j) is a random variable Y whose value at! is defined by where y = (!). Properties of IE(1j): Y (!) =IE(1j = y) IE(1j) should depend on!, i.e., it is a random variable. If the value of is known, then the value of IE(1j) should also be known. In particular, If! = HHH or! = HHT, then (!) =u 0. If we know that (!) =u 0, then even without knowing!, we know that 1(!) =u0. We define IE(1j)(HHH)=IE(1j)(HHT)=u0: If! = TTT or! = TTH, then (!) =d 0. If we know that (!) =d 0, then even without knowing!, we know that 1(!) =d0. We define IE(1j)(TTT)=IE(1j)(TTH)=d0:

5 CHPTER. Conditional Expectation 5 If! = fhth HTT THH THTg, then (!) =ud0. If we know (!) = ud0, then we do not know whether 1 = u0 or 1 = d0. We then take a weighted average: IP () =p q + pq + p q + pq =pq: Furthermore, 1dIP = p qu0 + pq u0 + p qd0 + pq d0 = pq(u + d)0 For! we define Then IE(1j)(!) = R 1dIP IP () IE(1j)dIP = = 1 (u + d) 0: 1dIP: In conclusion, we can write where g(x) = IE(1j)(!) =g((!)) 8 >< >: u0 if x = u 0 1 (u + d)0 if x = ud0 d0 if x = d 0 In other words, IE(1j) is random only through dependence on. We also write IE(1j = x) =g(x) where g is the function defined above. The random variable IE(1j) has two fundamental properties: IE(1j) is ()-measurable. For every set (), IE(1j)dIP = 1dIP:.. Definition of Conditional Expectation Please see Williams, p.8. Let ( F IP ) be a probability space, and let G be a sub--algebra of F. Let X be a random variable on ( F IP ). Then IE(XjG) is defined to be any random variable Y that satisfies: (a) Y is G-measurable,

6 5 (b) For every set G, we have the partial averaging property YdIP = XdIP: Existence. There is always a random variable Y satisfying the above properties (provided that IEjXj < 1), i.e., conditional expectations always exist. Uniqueness. There can be more than one random variable Y satisfying the above properties, but if Y 0 is another one, then Y = Y 0 almost surely, i.e., IP f! Y (!) =Y 0 (!)g =1: Notation.1 For random variables X Y, it is standard notation to write IE(XjY ) = IE(Xj(Y )): Here are some useful ways to think about IE(XjG): random experiment is performed, i.e., an element! of is selected. The value of! is partially but not fully revealed to us, and thus we cannot compute the exact value of X(!). Based on what we know about!, we compute an estimate of X(!). Because this estimate depends on the partial information we have about!, it depends on!, i.e., IE[XjY ](!) is a function of!, although the dependence on! is often not shown explicitly. If the -algebra G contains finitely many sets, there will be a smallest set in G containing!, which is the intersection of all sets in G containing!. The way! is partially revealed to us is that we are told it is in, but not told which element of it is. We then define IE[XjY ](!) to be the average (with respect to IP ) value of X over this set. Thus, for all! in this set, IE[XjY ](!) will be the same... Further discussion of Partial veraging The partial averaging property is We can rewrite this as IE(XjG)dIP = XdIP 8 G: (.1) IE[I :IE(XjG)] = IE[I :X]: (.) Note that I is a G-measurable random variable. In fact the following holds: Lemma.10 If V is any G-measurable random variable, then provided IEjV:IE(XjG)j < 1, IE[V:IE(XjG)] = IE[V:X]: (.)

7 CHPTER. Conditional Expectation 55 Proof: To see this, first use (.) and linearity of expectations P to prove (.) when V is a simple n G-measurable random variable, i.e., V is of the form V = k=1 c ki K, where each k is in G and each c k is constant. Next consider the case that V is a nonnegative G-measurable random variable, but is not necessarily simple. uch a V can be written as the limit of an increasing sequence of simple random variables V n ; we write (.) for each V n and then pass to the limit, using the Monotone Convergence Theorem (ee Williams), to obtain (.) for V. Finally, the general G- measurable random variable V can be written as the difference of two nonnegative random-variables V = V + ; V ;, and since (.) holds for V + and V ; it must hold for V as well. Williams calls this argument the standard machine (p. 56). Based on this lemma, we can replace the second condition in the definition of a conditional expectation (ection..) by: (b ) For every G-measurable random-variable V, we have IE[V:IE(XjG)] = IE[V:X]: (.).. Properties of Conditional Expectation Please see Willams p. 88. Proof sketches of some of the properties are provided below. (a) IE(IE(XjG)) = IE(X): Proof: Just take in the partial averaging property to be. The conditional expectation of X is thus an unbiased estimator of the random variable X. (b) If X is G-measurable, then IE(XjG) =X: Proof: The partial averaging property holds trivially when Y is replaced by X. nd since X is G-measurable, X satisfies the requirement (a) of a conditional expectation as well. If the information content of G is sufficient to determine X, then the best estimate of X based on G is X itself. (c) (Linearity) IE(a1X1 + axjg) =a1ie(x1jg) +aie(xjg): (d) (Positivity) If X 0 almost surely, then IE(XjG) 0: Proof: Take = f! R IE(XjG)(!) < 0g. This R set is in G since IE(XjG) is G-measurable. Partial averaging implies IE(XjG)dIP = XdIP. The right-hand side is greater than or equal to zero, and the left-hand side is strictly negative, unless IP () = 0. Therefore, IP () =0.

8 56 (h) (Jensen s Inequality) If : R!R is convex and IEj(X)j < 1, then IE((X)jG) (IE(XjG)): Recall the usual Jensen s Inequality: IE(X) (IE(X)): (i) (Tower Property) If H is a sub--algebra of G, then IE(IE(XjG)jH) =IE(XjH): H is a sub--algebra of G means that G contains more information than H. If we estimate X based on the information in G, and then estimate the estimator based on the smaller amount of information in H, then we get the same result as if we had estimated X directly based on the information in H. (j) (Taking out what is known) If is G-measurable, then IE(XjG) =:IE(XjG): When conditioning on G, the G-measurable random variable acts like a constant. Proof: Let be a G-measurable random variable. random variable Y is IE(XjG) if and only if (a) R Y is G-measurable; (b) YdIP = XdIP 8 G. R Take Y = :IE(XjG). Then Y satisfies (a) (a product of G-measurable random variables is G-measurable). Y also satisfies property (b), as we can check below: YdIP = IE(I :Y ) = IE[I IE(XjG)] = IE[I :X] ((b ) with V = I = XdIP: (k) (Role of Independence) If H is independent of ((X) G), then In particular, if X is independent of H, then IE(Xj(G H)) = IE(XjG): IE(XjH) =IE(X): If H is independent of X and G, then nothing is gained by including the information content of H in the estimation of X.

9 CHPTER. Conditional Expectation Examples from the Binomial Model Recall that F1 = f H T g. Notice that IE(jF1) must be constant on H and T. Now since IE(jF1) must satisfy the partial averaging property, We compute H H IE(jF1)dIP = dip H IE(jF1)dIP = dip: T T IE(jF1)dIP = IP ( H ):IE(jF1)(!) = pie(jf1)(!) 8! H : On the other hand, Therefore, We can also write H dip = p u 0 + pqud0: IE(jF1)(!) =pu 0 + qud0 8! H : imilarly, Thus in both cases we have IE(jF1)(!) = pu 0 + qud0 = (pu + qd)u0 = (pu + qd)1(!) 8! H IE(jF1)(!) =(pu + qd)1(!) 8! T : IE(jF1)(!) =(pu + qd)1(!) 8! : similar argument one time step later shows that IE(jF)(!) =(pu + qd)(!): We leave the verification of this equality as an exercise. We can verify the Tower Property, for instance, from the previous equations we have This final expression is IE(jF1). IE[IE(jF)jF1] = IE[(pu + qd)jf] = (pu + qd)ie(jf1) (linearity) = (pu + qd) 1:

10 58. Martingales The ingredients are: probability space ( F IP ). sequence of -algebras F0 F1 ::: F n, with the property that F 0 F1 :::F n F. uch a sequence of -algebras is called a filtration. sequence of random variables M0 M1 ::: M n. This is called a stochastic process. Conditions for a martingale: 1. Each M k is F k -measurable. If you know the information in F k, then you know the value of M k. We say that the process fm k g is adapted to the filtration ff k g.. For each k, IE(M k+1jf k )=M k. Martingales tend to go neither up nor down. supermartingale tends to go down, i.e. the second condition above is replaced by IE(M k+1jf k ) M k ;asubmartingale tends to go up, i.e. IE(M k+1jf k ) M k. Example. (Example from the binomial model.) For k =1 we already showed that IE( k+1jf k )=(pu + qd) k : For k =0, we set F0 = f g, the trivial -algebra. This -algebra contains no information, and any F0-measurable random variable must be constant (nonrandom). Therefore, by definition, IE(1jF0) is that constant which satisfies the averaging property IE(1jF0)dIP = The right hand side is IE1 =(pu + qd)0, and so we have In conclusion, 1dIP: IE(1jF0) =(pu + qd)0: If (pu + qd) =1then f k F k k =0 1 g is a martingale. If (pu + qd) 1 then f k F k k =0 1 g is a submartingale. If (pu + qd) 1 then f k F k k =0 1 g is a supermartingale.

SDS 321: Introduction to Probability and Statistics

SDS 321: Introduction to Probability and Statistics Lecture 2: Conditional probability Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/ psarkar/teaching