ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1

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1 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Last compiled: November 6, Conditional expectation Exercise 1.1. To start with, note that P(X Y = P( c R : X > c, Y c or X c, Y > c = P( c Q : X > c, Y c or X c, Y > c [ ] P(X > c, Y c + P(X c, Y > c c Q where the second equality follows by the density of Q in R. So it is enough to show that for all c Q it holds P(X > c, Y c = P(X c, Y > c =. To this end, fix any c Q. Since {Y c} σ(y, by definition of conditional expectation E[X1 {Y c} ] = E[Y 1 {Y c} ]. It follows that = E[(X Y 1 {Y c} ] = E[(X Y 1 {X>c,Y c} ] + E[(X Y 1 {X c,y c} ]. and, by reversing the roles of X and Y, that = E[(Y X1 {X c} ] = E[(Y X1 {X c,y >c} ] + E[(Y X1 {X c,y c} ]. By adding these equations we then see that = E[(X Y 1 {X>c,Y c} ] + E[(Y X1 {X c,y >c} ]. Both of the above summands are nonnegative and so they must both equal. It follows that P(X > c, Y c = and P(X c, Y > c =. If we only know that E[X Y ] = Y a.s. then we cannot conclude as we have above. For example, suppose that Y = and that X takes values in { 1, 1} with equal probability. Then, trivially, X and Y are integrable. Moreover, since σ(y = {, Ω}, from E(X1(Ω = E(X = = Y E(X1( = = Y we conclude that E[X Y ] = E[X] = = Y a.s. So the assumptions hold, but X Y with probability 1. Comments and corrections should be sent to Vittoria Silvestri, vs358@cam.ac.uk. 1

2 2 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Exercise 1.2. Let X and Y be independent Bernoulli random variables of parameter p (, 1 and let us define Z := 1 {X+Y =}. Since Z {, 1} almost surely by definition, it follows that E[X Z] = E[X Z = ]1 {Z=} + E[X Z = 1]1 {Z=1} = E[X1 {Z=}] P(Z = 1 {Z=} + E[X1 {Z=1}] P(Z = 1 1 {Z=1} a.s. If Z = 1, then X =, and so the second summand equals. For the first summand, observe that P(Z = = 1 P(Z = 1 = 1 P(X =, Y = = 1 (1 p 2 = p(2 p and, additionally, that E[X1 {Z=} ] = P(Z =, X = 1 = P(X = 1 = p. It follows that E[X Z] = 1 {Z=} /(2 p a.s. Finally, by symmetry, E[Y Z] = E[X Z] a.s. Exercise 1.3. Let X and Y be independent exponential random variables of parameter θ and define Z := X + Y. In order to see that Z has a Γ(2, θ distribution, it will suffice for us to show that the density of Z, f Z, is such that f Z (z = θ 2 ze θz 1 {z }. If z then P(Z z = as X and Y are a.s. positive, so let us consider z >. As (X, Y has a density given by f X,Y (x, y = f X (xf Y (y = θ 2 e θ(x+y 1 {x,y }, we have that P(Z z = P(X + Y z = {x+y z} z z y = θ (e 2 θy = θ z θ 2 e θ(x+y 1 {x,y } dx dy e θx dx dy (1 e θ(z y e θy dy = 1 e θz θze θz. So the distribution function of Z is F Z (z = (1 e θz θze θz 1 {z }. Differentiating gives the p.d.f. of Z: F Z(z = f Z (z = θ 2 ze θz 1 {z } and hence Z Γ(2, θ. Remark. Alternatively, we could use the general fact that, if (X k : k = 1,..., n is a sequence of independent random variables with respective densities (f k : k = 1,..., n, then the sum X X n has a density given by the convolution f 1 f n.

3 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 3 For the second part of the exercise, take h: R R to be a non-negative Borel function. We aim to show that almost surely, E[h(X Z] = 1 Z Z h(x dx. Let us define Φ: R 2 R 2 : (x, y (x, x + y. This is a C 1 -diffeomorphism and, further, Φ 1 : R 2 R 2 : (x, z (x, z x is such that det(dφ 1 1. The change of variables formula with Φ applies and tells us that the density of (X, Z is given by f X,X+Y (x, z = f X,Y (x, z x = f X (xf Y (z x = θ 2 e θz 1 {z x }. Therefore and f X Z (x z = f X,Z(x, z 1 f Z (z {z:fz (z>} ( E[h(X Z] = h(x f X,Z(x, Z dx R f Z (Z = h(x θ2 e θz θ 2 Ze θz 1 {Z x>} dx = 1 Z R Z h(x dx a.s. 1 {fz (Z>} To answer the third part of the question, suppose that Z Γ(2, θ and that, for every non-negative Borel function h: R R, it holds E[h(X Z] = 1 Z Z h(u du Then we aim to show that X and Z X are independent exponential random variables of parameter θ. To this end, it is enough to show that the joint distribution of (X, Y with Y := Z X factorizes, and the marginal densities are Gamma densities with parameter θ. To this end we can apply the change of variables formula with Φ 1, which yields a.s. f X,Z X (x, y = f X,Z (x, x + y = θ 2 e θ(x+y 1 {x+y x } = θe θx 1 {x } θe θy 1 {y } = f X (xf Y (y with f X, f Y being p.d.f. of Gamma(θ. Exercise 1.4. Let X be a non-negative random variable on the probability space (Ω, F, P and G be a sub-σ-algebra of F. We first show that if X >, then E[X G ] > a.s. As A := {E[X G ] } G, we have that E[X1 A ] = E[E[X G ]1 A ] and hence that E[X1 A ] =. Since X > on A, we conclude that P(A =. To see that the event {E[X G ] > } is the smallest element of G that contains the event {X > } (up to null events, assume the contrary. Then

4 4 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 there exists a G -measurable event B such that {X > } B {E[X G ] > } and C := {E[X G ] > } \ B has positive measure. Then E[X1 C ] = E[E[X G ]1 C ] and so E[E[X G ]1 C ] =. As E[X G ] > on C, it must be the case that P(C =, which contradicts the assumptions. It follows that {E[X G ] > } B up to a null event. Exercise 1.5. Recall that P(X = n, Y t = b t (ay n e (a+by dy for n integer, t real. In order to compute E[h(Y 1 {X=n} ] we first compute the p.d.f. of Y conditional on the event X = n. f Y X=n (y X = n = d dt P(Y t X = n t=y = d ( P(X = n, Y t dt P(X = n Moreover = b P(X = n (ay n e (a+by. P(X = n = P(X = n, Y < = b = ba n (a + b n+1 (ay n e (a+by dy (a + b n+1 y n e (a+by dy = }{{} 1 ba n (a + b n+1 where in the last equality we have used that the p.d.f. of a Gamma(n+1, a+b integrates to 1, recalling that Γ(n + 1 = for n positive integer. Hence (a + bn+1 f Y X=n (y X = n = b ba n (ay n e (a+by = t=y (a + bn+1 Γ(n + 1 yn e (a+by 1 (,+ (y. In other words, the law of Y conditioned to the event {X = n} is Γ(n+1, a+b. Therefore if h: (, [, is a Borel function, then E[h(Y X = n] = E[h(G] where G is another random variable, defined on the same probability space, with law Gamma(n + 1, a + b. Hence E[h(Y X = n] = (a + bn+1 Next we compute E(Y/(X + 1: ( Y ( Y E = E X + 1 X + 1 P(X X = n = n = n= ( n = P(X = n = a + b n + 1 a + b n= h(yy n e (a+by dy. n= ( Y E P(X = n n + 1 P(X = n n= } {{ } 1 = 1 a + b.

5 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 5 For E(1 {X=n} Y we have: Since and f Y (y = E(1 {X=n} Y = P(X = n Y = f X,Y (n, Y f Y (Y f X,Y (n, y = d dt P(X = n, Y t t=y = b (ayn e (a+by 1 (, (y (ay n f X,Y (n, y = b e (a+by 1 (, (y = be by 1 (, (y, n= n= from which Y exponential(b, we conclude (ay n P(X = n Y = b e (a+by 1 be by = (ay n e ay. That is, the law of X conditional on Y is Poisson(aY. This also implies that E(X Y = ay (recall that the expected value of a Poisson random variable of parameter λ is λ. Exercise 1.6. Let us suppose that X and Y are random variables defined on the probability space (Ω, F, P and that G is a sub-σ-algebra of F. We say that X and Y are conditionally independent given G if, for all Borel functions f, g : R [,, E[f(Xg(Y G ] = E[f(X G ]E[g(Y G ] (1 almost surely. If G = {, Ω} in the above then this implies that E[f(Xg(Y ] = E[f(X]E[g(Y ] for all non-negative Borel functions f and g. In particular, if we take f = 1 A and g = 1 B for A, B B, this implies that P(X A, Y B = E[1 A (X1 B (Y ] = E[1 A (X]E[1 B (Y ] = P(X AP(Y B. That is to say, if X and Y are independent conditionally on {, Ω}, then they are independent. (The converse is also true by linearity and the monotone convergence theorem. We next show that the random variables X and Y are conditionally independent given G if and only if, for every non-negative G -measurable random variable Z and all non-negative Borel functions f and g, E[f(Xg(Y Z] = E[f(XZE[g(Y G ]]. (2 Suppose first that X and Y are independent conditionally on G. Then E[f(Xg(Y Z] = E[E[f(Xg(Y Z G ]] = E[ZE[f(Xg(Y G ]] = E[ZE[f(X G ]E[g(Y G ]].

6 6 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 with the second equality holding as Z is G -measurable and as Z, f and g are non-negative. Further, as ZE[g(Y G ] is G -measurable and as everything is a.s. non-negative, E[ZE[f(X G ]E[g(Y G ]] = E[f(XZE[g(Y G ]]. Now assume that (2 holds for every G -measurable Z and f, g non-negative Borel functions. We aim to show that this implies (1. Let us take A G and Z = 1 A. We have that E[E[f(Xg(Y G ]1 A ] = E[f(Xg(Y 1 A ] = E[f(X1 A E[g(Y G ]] = E[E[f(X G ]E[g(Y G ]1 A ]. The second equality follows from our hypothesis; the final equality holds as 1 A E[g(Y G ] is G -measurable and as everything is a.s. non-negative. Therefore, as E[f(X G ]E[g(Y G ] is G -measurable and a.s. non-negative it follows that, with probability 1, E[f(Xg(Y G ] = E ( E[f(X G ]E[g(Y G ] G = E[f(X G ]E[g(Y G ]. For the last part of the exercise, we have to show that E[f(Xg(Y Z] = E[f(XZE[g(Y G ]] (3 for every G -measurable random variable Z and all Borel functions f, g : R [, if and only if, for each Borel function g : R [,, E[g(Y σ(g, σ(x] = E[g(Y G ]. (4 Assume (3. It is immediate that E[g(Y G ] is σ(g, σ(x-measurable. We are to show that, for all A σ(g, σ(x, E[g(Y 1 A ] = E[E[g(Y G ]1 A ]. It suffices, by the theorem on the uniqueness of extensions, to prove this for all A B, where A G and B σ(x, as the set {A B : A G, B σ(x} is a generating π-system for σ(g, σ(x that contains Ω. So let A G and B σ(x. Then E[g(Y 1 A B ] = E[ 1 B }{{} f(x g(y 1 }{{} A ] = E[1 B 1 A E[g(Y G ]] = E[E[g(Y G ]1 A B ] Z where we have used (3 in the second equality. Assume now that (4 holds. As f(xz is non-negative and σ(g, σ(x- measurable and as g(y is non-negative, E[g(Y f(xz] = E[E[g(Y f(xz σ(g, σ(x]] = E[E[g(Y σ(g, σ(x]f(xz] = E[E[g(Y G ]f(xz] where, we have used (4 in the second equality.

7 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 7 Exercise 1.7. Recall that, given a probability space (Ω, F, P, two sub-σalgebras G, H of F are said to be independent if for every G G, H H it holds P(G H = P(GP(H. Moreover, we know (see Proposition 1.24 in the lecture notes that if σ(x, G is independent of H, then E(X σ(g, H = E(X G a.s.. Hence we seek for an example in which σ(x, G and H are dependent. Let X 1, X 2 be independent Bernoulli(1/2, and set H = σ(x 1, G = σ(x 2 and X = 1(X 1 = X 2. Then H and G are independent by construction. Moreover, X is itself a Bernoulli(1/2 random variable and it is independent of H, since: P(X =, X 1 = = P(X 1 =, X 2 = 1 = P(X 1 = P(X 2 = 1 = 1/4 = P(X = P(X 1 = and similarly one sees that P(X = x, X 1 = y = P(X = xp(x 1 = y for all x, y {, 1}. So the assumptions are satisfied. Now notice that σ(g, H = F and therefore X is σ(g, H -measurable, from which E(X σ(g, H = X a.s. On the other hand, E(X G = E(1(X 1 = X 2 X 2 = 1(X 2 = + E(1(X 1 = X 2 X 2 = 11(X 2 = 1 = P(X 1 = 1(X 2 = + P(X 1 = 11(X 2 = 1 = 1 2 1(X 2 = (X 2 = 1 = 1 2 almost surely, since with probability 1 exactly one of the indicator functions in the last line is non-zero. But then E(X σ(g, H E(X G on an event of probability 1, since X takes values in {, 1}.

8 8 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 2. Discrete-time martingales Exercise 2.1. Assume that X is a martingale with respect to its natural filtration. Then, since the event {X = i, X 1 = i 1,..., X n = i n } is F n - measurable for all n and i,..., i n E, we have E(X n+1 X = i,..., X n = i n = E(X n+11(x = i,..., X n = i n P(X = i,..., X n = i n almost surely. = E(E(X n+1 F n 1(X = i,..., X n = i n P(X = i,..., X n = i n = E(X n1(x = i,..., X n = i n P(X = i,..., X n = i n = E(i n1(x = i,..., X n = i n P(X = i,..., X n = i n = i n Assume, on the other hand, that for all n and all i, i 1,..., i n E it holds E(X n+1 X = i,..., X n = i n = i n Then, being E countable, we can write E(X n+1 F n = E(X n+1 X = i,..., X n = i n 1(X }{{} = i,..., X n = i n i,...,i n E i n = i n 1(X n = i n = X n a.s. i n E To conclude that this implies that X is a martingale, take any m > n. Then using the tower property of the expectation we get a.s. E(X m F n = E(E(X m F m 1 F n = E(X m 1 F n =... = E(X n+1 F n = X n a.s. Exercise 2.2. We say that a process C = (C n : n is previsible with respect to the filtration (F n : n if C n+1 is F n -measurable for all n. Suppose that X = (X n n is a martingale on the filtered probability space (Ω, F, (F n : n, P, and the previsible process C is bounded. We aim to show that Y := C X is a martingale. It is clear that Y is adapted and integrable. Moreover, for each n we have: E[Y n+1 Y n F n ] = E[(C X n+1 (C X n F n ] = E[C n+1 (X n+1 X n F n ] = C n+1 E[X n+1 X n F n ] = where the second equality holds as C n+1 is bounded and F n -measurable and X n+1 X n is integrable, and the final equality holds by the martingale

9 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 9 property of X. This shows that Y is itself a martingale. If, on the other hand, X is a supermartingale and C is bounded and non-negative, then E[Y n+1 Y n F n ] = E[C n+1 (X n+1 X n F n ] = C n+1 E[X n+1 X n F n ] where the second equality holds as C n+1 is bounded and F n -measurable and X n+1 X n is integrable, while the inequality holds as C is non-negative and as X is a supermartingale. It follows that Y is a supermartingale. Exercise 2.3. Let (X n : n 1 be a sequence of independent random variables such that P(X n = n 2 = n 2 and P(X n = n 2 /(n 2 1 = 1 n 2. We aim to show that if S n = X X n, then S n /n 1 a.s. To this end, define the collection of events A n := {X n = n 2 }. Then, by the first Borel Cantelli lemma, P(A n i.o. =. It follows that P(A c n a.a. = 1 and hence that P(X n = n 2 /(n 2 1 a.a. = 1. It is thus enough to prove that S n /n 1 on this lattermost set. By definition, for each ω in this set there is some N ω Z > such that X n (ω = n 2 /(n 2 1 for all n N ω. We thus see that S n (ω n = 1 n N ω 1 k=1 X k (ω + 1 n } {{ } C ω = C ω n + n N ω + 1 n n n k=n ω k2 k 2 1 = C ω n + 1 n n 1 k 2 1 = 1 + o(1 k=n ω n k=n ω ( k 2 1 and therefore S n (ω/n 1 as n. Since this holds for all ω in a set o measure 1, we conclude S n /n 1 almost surely. We now want to show that the process (S n : n 1 is a martingale with respect to its natural filtration, (F n : n 1, and that it converges to a.s. It is clear that (S n : n 1 is adapted to its natural filtration. Moreover, as S n is a finite sum of integrable random variables, it is integrable. For the martingale property we note that E[S n+1 S n F n ] = E[X n+1 F n ] = E[X n+1 ] =, where the second equality follows from the independence of X n+1 from F n. It follows that (S n : n 1 is a martingale. Finally, for all ω such that S n (ω/n 1, it is immediate that S n (ω. As the set of all such ω has full measure, S n a.s. Exercise 2.4. Let T := m1 A + m 1 A c. We see that {T = m} = A F n F m and {T = m } = A c F n F m. Moreover, if k is neither m nor m, then {T = k} = F k. It follows that T is a stopping time.

10 1 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 For the second part of the exercise, assume that X is a martingale. Then, if T is a bounded stopping time, E[X T ] = E[X ] by the Optional Stopping Theorem. Assume, on the other hand, that for every T bounded stopping time it holds E[X T ] = E[X ]. Then, since n is a bounded stopping time, it holds E[X n ] = E[X ] for all n. Moreover, we can consider any A F n and define T := (n + 11 A + n1 A c. By the previous proposition this is a bounded stopping time so, again by our hypothesis, E[X ] = E[X T ] = E[X n+1 1 A + X n 1 A c] = E[(X n+1 X n 1 A ] + E[X n ]. As E[X ] = E[X n ], the above implies that E[X n+1 1 A ] = E[X n 1 A ]. As we have assumed that X is integrable and adapted and as A F n was arbitrary, E[X n+1 F n ] = X n a.s. We conclude that X is a martingale. Exercise 2.5. Suppose that X = (X n : n is a martingale (respectively, a supermartingale and that T is an a.s. finite stopping time. We aim to show that if there is some M such that X M a.s. then E[X T ] = E[X ] (respectively, E[X T ] E[X ]. To see this, recall that X T is a martingale (respectively, a supermartingale. Since T < a.s., it follows that T n T a.s. and therefore [ ] E[X T ] = E lim X T n n = lim n E[X T n] = lim n E[X T ] = E[X ]. The second equality holds by the dominated convergence theorem with M as the dominating (degenerate random variable, the third equality holds as X T is a martingale. In the case where X is a supermartingale, the third equality becomes by the supermartingale property. For the second part of the exercise, assume that E(T < and that X is a martingale (respectively, a supermartingale with bounded increments. We aim to show that E[X T ] = E[X ] (respectively, E[X T ] E[X ]. Being T integrable, it is a.s. finite, so T n T a.s. Now, for all n, we a.s. have that T X T n = X n T + (X k X k 1 X n + X k X k 1 X + MT. k=1 The far right-hand side of the above is integrable, so we can apply the dominated convergence theorem to get [ ] E[X T ] = E lim X T n = lim E[X T n] = lim E[X T ] = E[X ], n n n where the third equality holds as X T is a martingale. As in the proof of the previous proposition, in the case where X is a supermartingale the third k=1 equality becomes by the supermartingale property.

11 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 11 Exercise 2.6. Let T be a F n -stopping time, and suppose that for some integer N > and some ε > it holds We aim to show that E(T <. P(T N + n F n ε n. To start with, we show that P(T > kn (1 ε k for all k. P(T > kn = P(T > N, T > 2N,..., T > kn = E(P(T > N, T > 2N,..., T > kn F N = E ( 1(T > N P(T > 2N,..., T > kn F N }{{} iterate = E (1(T > NE ( 1(T > 2N,..., T > kn F N = E (1(T > NE ( E(1(T > 2N,..., T > kn F 2N F N ( ( = E 1(T > NE 1(T > 2N P(T > 3N,..., T > kn F 2N }{{} F N iterate =... ( ( ( = E 1(T > NE 1(T > 2N P T > kn F (k 1N F N }{{} 1 ε (1 ε k. From this we can conclude, since E(T = P(T n n= P(T k n N = N n= where we have set k n = n/n. P(T kn N Exercise 2.7. We are playing a game in which our winnings per unit stake on the game at time n 1 is ε n, where (ε n : n 1 is an i.i.d. sequence of k= random variables such that 1, with probability p P(ε n = 1 = 1, with probability q = 1 p where p (1/2, 1. Let Z n denote our fortune at time n and let C n be our stake on the game at time n, with C n < Z n 1. Our goal in this game is to choose a strategy that maximises our expected interest rate E[log(Z N /Z ], where Z is our initial fortune and N is some fixed time corresponding to how long we are to play the game. (1 ε k = N ε < k=

12 12 ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Assume that C is a previsible process. Then we aim to show that (log Z n nα : n 1 is a supermartingale, where α denotes the entropy α := p log p + q log q + log 2. To see this, note that for each n, Z n+1 = Z n + ε n+1 C n+1 by definition. Therefore: E[log Z n+1 F n ] = E[log(Z n + ε n+1 C n+1 F n ] = E[log(Z n + C n+1 1(ε n+1 = 1 F n ] + E[log(Z n C n+1 1(ε n+1 = 1 F n ] = p log(z n + C n+1 + q log(z n C n+1 ( ( C n+1 = p log Z n q log Z n ( = log Z n + p log 1 + C n+1 Z n ( Z n ( 1 C n+1 Z n ( + q log 1 C n+1 Z n almost surely, where we have used that the quantity log(z n ± C n+1 is F n -measurable. Let us now define the function f(x := p log(1 + x + q log(1 x for x [, 1. By elementary calculus, f is maximised at p q and f(p q = log 2 + p log p + q log q = α. Therefore from which E[log Z n+1 F n ] = log Z n + f(c n+1 /Z n log Z n + α a.s. (5. E[log Z n+1 (n + 1α F n ] log Z n nα a.s. The function log Z n nα is clearly F n -measurable and integrable for each n, and hence we conclude that the process (log Z n nα : n is a supermartingale. Note that E[log(Z N /Z ] = E[log Z N ] log Z Nα. Finally, it is clear that if we take C n+1 := (p qz n in (5 then all inequalities above become equalities, and so (log Z n nα : n is in fact a martingale for this strategy. As p q is the unique maximiser of f, this strategy is the unique optimal strategy. References [Wil91] David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991.