![3. Consider a filtration (F) and an (F)-adapted stochastic process (X) such that Xo 0 and E X] oo for all n 2 0. Also, let () be a sequence of constants. Define Mo 0 and rt 7t j-1)j-1, for n 1 Prove that (Ms n (F)-martingale.](http://img.homeworklib.com/questions/066edf90-599c-11eb-850c-c53009577f2c.png?x-oss-process=image/resize,w_560)
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3. Consider a filtration (F) and an (F)-adapted stochastic process (X) such that Xo 0 and...
(6 marks) Consider a filtered probability space (2,F,P, Ftte.). a. (2 marks) Let the stochastic p...
(6 marks) Consider a filtered probability space (2,F,P, Ftte.). a. (2 marks) Let the stochastic process (Xo.7] have independent increments and sat- b. (2 marks) Let eo.] be a stochastic process with Ep[X] Xo for all t E [0,T]. Is c. (2 marks) Let (W be a Brownian motion. Given c 0, and define the stochastic isfies Ep[IXll < oo fort [0,T]. Is the stochastic process {Ztieo.r], where z, = xt-EP[Xt] is a martingale with respect to {Ft}120 ? Explain....
[8 marks] Consider a discrete time stochastic process {Xn,n 2 0j defined by the equation with...
[8 marks] Consider a discrete time stochastic process {Xn,n 2 0j defined by the equation with Xo1 and Rn,n21 are random variables taking their values in (-1,00). Denote Sn-Li-1 Rk for n 〉 1 and So-0 i) [3 marks] Briefly explain why the filtration {F,:n 〉 0} gener- 0 generated by ated by Xo, X1,.. . , Xn and the filtration , n So, S1, , Sn should be identical ії) [5 marks] Show that {X,,n 〉 0} is a...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . ....
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
Help please! Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic...
Help please!
Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic set E, and assume that for some probability vector X matriz P E(0, 1ExE we have prove that Xn ~ Markov(λ, P)
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some z...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
Power function sample with joint pdf (or pmf) f (x |0), 0 e 0 c R. Suppose Let X1,..., X,n be a that {f(xn0) : 0 E 0} h...
Power function
sample with joint pdf (or pmf) f (x |0), 0 e 0 c R. Suppose Let X1,..., X,n be a that {f(xn0) : 0 E 0} has monotone likelihood ratio (MLR) in T(X). Consider test function if T(xn)> c 1 if T(Xn) (Xn) C if T(xn)c 0 where y E [0, 1] and c > 0 are constants. Prove that the power function of ø(X,,) is non-decreasing in 0
sample with joint pdf (or pmf) f (x |0),...
+20 Problem 7. Let f :D + R, xo be an accumulation point of D and...
+20 Problem 7. Let f :D + R, xo be an accumulation point of D and assume lim f(x) = L. Use the e-8 definition of the limit (not theorems or results from class or the text) to prove the following: (a) The function f is “bounded near xo”: there is an M ER and a 8 >0 such that for x E D, 0 < l< – xo<8 = \f(x) < M. Hint: compare with the proof that a...
(6 marks) Consider a filtered probability space (2,F,P, Ftte.). a. (2 marks) Let the stochastic process (Xo.7] have independent increments and sat- b. (2 marks) Let eo.] be a stochastic process with Ep[X] Xo for all t E [0,T]. Is c. (2 marks) Let (W be a Brownian motion. Given c 0, and define the stochastic isfies Ep[IXll < oo fort [0,T]. Is the stochastic process {Ztieo.r], where z, = xt-EP[Xt] is a martingale with respect to {Ft}120 ? Explain....
[8 marks] Consider a discrete time stochastic process {Xn,n 2 0j defined by the equation with Xo1 and Rn,n21 are random variables taking their values in (-1,00). Denote Sn-Li-1 Rk for n 〉 1 and So-0 i) [3 marks] Briefly explain why the filtration {F,:n 〉 0} gener- 0 generated by ated by Xo, X1,.. . , Xn and the filtration , n So, S1, , Sn should be identical ії) [5 marks] Show that {X,,n 〉 0} is a...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
Help please!
Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic set E, and assume that for some probability vector X matriz P E(0, 1ExE we have prove that Xn ~ Markov(λ, P)
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
Power function
sample with joint pdf (or pmf) f (x |0), 0 e 0 c R. Suppose Let X1,..., X,n be a that {f(xn0) : 0 E 0} has monotone likelihood ratio (MLR) in T(X). Consider test function if T(xn)> c 1 if T(Xn) (Xn) C if T(xn)c 0 where y E [0, 1] and c > 0 are constants. Prove that the power function of ø(X,,) is non-decreasing in 0
sample with joint pdf (or pmf) f (x |0),...
+20 Problem 7. Let f :D + R, xo be an accumulation point of D and assume lim f(x) = L. Use the e-8 definition of the limit (not theorems or results from class or the text) to prove the following: (a) The function f is “bounded near xo”: there is an M ER and a 8 >0 such that for x E D, 0 < l< – xo<8 = \f(x) < M. Hint: compare with the proof that a...
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