Doob’s Decomposition. Let (Xm)m≥0 be a submartingale relative to the filtration F0 ⊂ F1 ⊂· ·...
Doob’s Decomposition. Let (Xm)m≥0 be a submartingale relative to the filtration F0 ⊂ F1 ⊂· · · . Show that there is a martingale (Mm)m≥0 and a predictable sequence (Am)m≥0 such that Xm =Mm+Am ∀m≥0 and 0=A0 ≤A1 ≤A2 ≤···.
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Doob’s Decomposition. Let (Xm)m≥0 be a submartingale relative to
the filtration F0 ⊂ F1 ⊂· ·...
Doob’s Decomposition: Let be a submartingale relative to the filtration . Show that there is a...
Doob’s Decomposition: Let be a submartingale relative to the filtration . Show that there is a martingale and a predictable sequence such that for all and . (Xm)m>o We were unable to transcribe this image(Mm)m>o (Am)m20 We were unable to transcribe this imagem>0 We were unable to transcribe this image
Let f0, f1, f2, . . . be the Fibonacci sequence defined as f0 = 0,...
Let f0, f1, f2, . . . be the Fibonacci sequence defined as f0 = 0, f1 = 1, and for every k > 1, fk = fk-1 + fk-2. Use induction to prove that for every n ? 0, fn ? 2n-1 . Base case should start at f0 and f1. For the inductive case of fk+1 , you’ll need to use the inductive hypothesis for both k and k ? 1.
5. (8 points) For a Markov chain {Xm, m 2 0, the Markov property says that:...
5. (8 points) For a Markov chain {Xm, m 2 0, the Markov property says that: Use (1) to show that where, ni 〈 n2 〈 n. 6. (8 points) Let {Zn, n-1) be lID with P(Zn-J)-Pi , J-0, ±1,±2, Let Sn-Σ zi. Show that {Sn, n-1} is a Markov chain.
Suppose X1, .. ,XM are independent, identically distributed random variables with mean a and variance b2. Let aM ≡ (1/M)Σi=1M aM and bM2≡ (1/(M-1)) Σi=1M (Xi-aM)2. a) Show that aM is an unbiased estim...
Suppose X1, .. ,XM are independent, identically distributed random variables with mean a and variance b2. Let aM ≡ (1/M)Σi=1M aM and bM2≡ (1/(M-1)) Σi=1M (Xi-aM)2. a) Show that aM is an unbiased estimator of E[X]: that is, E[aM] = a. b) Assume that the identity E[ Σi=1M (Xi-aM)2 ] = (M-1) b2 is correct. Show that bM2 is an unbiased estimator of var(X): that is, E[bM2] = b2
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Wr...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
(5) Fibonacci sequences in groups. The Fibonacci numbers Fn are defined recursively by Fo 0, F1...
(5) Fibonacci sequences in groups. The Fibonacci numbers Fn are defined recursively by Fo 0, F1 -1, and Fn - Fn-1+Fn-2 forn 2 2. The definition of this sequence only depends on a binary operation. Since every group comes with a binary operation, we can define Fibonacci- type sequences in any group. Let G be a group, and define the sequence {fn in G as follows: Let ao, a1 be elements of G, and define fo-ao, fi-a1, and fn-an-1an-2 forn...
solve 6 please . step by step. #5. Given a sequence fa pao, a , a,...
solve 6 please . step by step.
#5. Given a sequence fa pao, a , a, , define the sequence lbne bobt b2, by 田 b.-ao + a1 +a2 + + ak If a (x) is the generating function for (an) and b(x) is the generating function for tbnj, then show (1-x)a. This lets us write b)4 #6. Let(anpao, ai, a2, given by a,-0, ai-1, an+2-an+1 + an be the sequence of Fibonacci numbers; recall that the ordinary generating function...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with ou...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with out using negative words (you do not have to prove it): The lim, 10 10if R).Consider the following subsets of P: (b) Let P2-(f(t)- ao at + azt | ao, a1, a2 and Notice that Y...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m =...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
5. (8 points) For a Markov chain {Xm, m 2 0, the Markov property says that: Use (1) to show that where, ni 〈 n2 〈 n. 6. (8 points) Let {Zn, n-1) be lID with P(Zn-J)-Pi , J-0, ±1,±2, Let Sn-Σ zi. Show that {Sn, n-1} is a Markov chain.
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
(5) Fibonacci sequences in groups. The Fibonacci numbers Fn are defined recursively by Fo 0, F1 -1, and Fn - Fn-1+Fn-2 forn 2 2. The definition of this sequence only depends on a binary operation. Since every group comes with a binary operation, we can define Fibonacci- type sequences in any group. Let G be a group, and define the sequence {fn in G as follows: Let ao, a1 be elements of G, and define fo-ao, fi-a1, and fn-an-1an-2 forn...
solve 6 please . step by step.
#5. Given a sequence fa pao, a , a, , define the sequence lbne bobt b2, by 田 b.-ao + a1 +a2 + + ak If a (x) is the generating function for (an) and b(x) is the generating function for tbnj, then show (1-x)a. This lets us write b)4 #6. Let(anpao, ai, a2, given by a,-0, ai-1, an+2-an+1 + an be the sequence of Fibonacci numbers; recall that the ordinary generating function...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with out using negative words (you do not have to prove it): The lim, 10 10if R).Consider the following subsets of P: (b) Let P2-(f(t)- ao at + azt | ao, a1, a2 and Notice that Y...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
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