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8.19. Consider Brownian motion with reflecting barriers of -B and A, A >0, B> 0. Let p,(x) denote the density function of X, (a) Compute a differential equation satisfied by p.(x) (b) Obtai...
Let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distributio...
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that
let {X(t), 1 2 0} denote a Brownian motion
8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show that if random variables X and Y are independent then they are 1. uncorrelated, Cov(X, Y) -0. (b) Let X have distribution P(X-1)- P(X 0) P(X- -1)-1/3, and Y-İX . Show that X and Y are uncorrelated, but not independent. (c) Let (X, Y) be a Gaussian vector. Show that if X and Y are uncorrelated then they are independent.
find the probability that a european put option with underlying s finishes in the money a) Let Y e*t, Find the stochastic differential equation satisfied by t. b) Let Zt -eatX. Find the...
find the probability that a european put option with underlying s
finishes in the money
a) Let Y e*t, Find the stochastic differential equation satisfied by t. b) Let Zt -eatX. Find the stochastic differential equation satisfied by Zt c) Find XtWdWs, where W, is a Brownian motion. 0 Hint: Set XtaW2 + bWt + ct. Find a, b, and c. 6) Find the probability that a European put option with underlying S
a) Let Y e*t, Find the stochastic...
In the following questions, let Bt denote a Brownian motion with B0 = 0.
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival probability, P(Un 〉 0, for all n 0, 1, 2, ). Justify your arguments.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival...
In the following questions, let Bt denote a Brownian motion with B0 = 0.
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival probability, P(Un 〉 0, for all n 0, 1, 2, ). Justify your arguments.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival...
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s< to}. Co...
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s< to}. Compute the cumulative distribution function of Min(to)
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for...
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for B(t). b) Show P(T, 0o)1. c) Use martingale methods to compute E [e-m] for any θ > 0 r>
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be...
In the following questions, let Bt denote a Brownian motion with B0 = 0.
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Let Xt be the solution of SDE dX, = 3X, dt + 2XtdBt and Xo = 1. (a) Write down the SDE for Yt-eatXt, where a is a constant. (b) Find the value of a such that Yt is a martingale, and give the mean and variance of Y, in this case.
Let Xt be the solution of SDE dX, = 3X, dt + 2XtdBt and...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the s...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stoc...
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation dS ơSdX+1Sdt. what stochastic differential equation does the stochastic process (a) Y 25, (b) Y = S (c) Y-es, (d) YeT-/S follow? In each cases express the coefficients of dX and dt in terms of Y rather than S. Use Ito's lemma
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic...
let {X(t), 1 2 0} denote a Brownian motion 8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion (d) Let Using (c) present an argument that
let {X(t), 1 2 0} denote a Brownian motion
8.1. Let Y(t) = tx(1/t). (a) What is the distribution of Y(t)? (b) Compute Cov(Y(s), Y()) (c) Argue that {Y(t), t 2 0] is also Brownian motion...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show that if random variables X and Y are independent then they are 1. uncorrelated, Cov(X, Y) -0. (b) Let X have distribution P(X-1)- P(X 0) P(X- -1)-1/3, and Y-İX . Show that X and Y are uncorrelated, but not independent. (c) Let (X, Y) be a Gaussian vector. Show that if X and Y are uncorrelated then they are independent.
find the probability that a european put option with underlying s
finishes in the money
a) Let Y e*t, Find the stochastic differential equation satisfied by t. b) Let Zt -eatX. Find the stochastic differential equation satisfied by Zt c) Find XtWdWs, where W, is a Brownian motion. 0 Hint: Set XtaW2 + bWt + ct. Find a, b, and c. 6) Find the probability that a European put option with underlying S
a) Let Y e*t, Find the stochastic...
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival probability, P(Un 〉 0, for all n 0, 1, 2, ). Justify your arguments.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival...
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival probability, P(Un 〉 0, for all n 0, 1, 2, ). Justify your arguments.
Consider a discrete time risk model U-2 + Σ Y" where {Yi) are iid random variables with N(2,1) distribution. Derive the lower bound for the survival...
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s< to}. Compute the cumulative distribution function of Min(to)
3. Let W(t be standard Brownian motion and let to > 0. Consider the random variable Min(to) min{W(s) 0 s
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for B(t). b) Show P(T, 0o)1. c) Use martingale methods to compute E [e-m] for any θ > 0 r>
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be...
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Let Xt be the solution of SDE dX, = 3X, dt + 2XtdBt and Xo = 1. (a) Write down the SDE for Yt-eatXt, where a is a constant. (b) Find the value of a such that Yt is a martingale, and give the mean and variance of Y, in this case.
Let Xt be the solution of SDE dX, = 3X, dt + 2XtdBt and...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic differential equation dS ơSdX+1Sdt. what stochastic differential equation does the stochastic process (a) Y 25, (b) Y = S (c) Y-es, (d) YeT-/S follow? In each cases express the coefficients of dX and dt in terms of Y rather than S. Use Ito's lemma
PROBLEM 2. [5 points] Let X(t) be a Brownian motion. Assume that the stock price follows the stochastic...
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