In the following questions, let Bt denote a Brownian motion with B0 = 0.
Solution:
given that
Let Xn, n-0, 1, 2, denote a biased random walk given by X0 0 and Xn+1 = Xn+Yn+1, where {Yn} are i.i.d. random variables.
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. 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. 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...
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.
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.
2. Let Bt denote a Brownian motion. Consider the Black-Scholes model for the price of stock St, 2 So-1 and the savings account is given by β,-ea (a) Solve the equation for the price of the stock St and show that it is not a (b) Explain what is meant by an Equivalent Martingale Measure (EMM) martingale. State the Girsanov theorem. Give the expression for Bt under the EMM Q, hence derive the expression for St under the EMM, and...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale. 3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
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...
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...
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing bound- aries. Suppose X0 = 50. Let T = min{j : Xj = 0 or 100}. Let Fn denote the information contained in X1,··· ,Xn. (1) Verify that Xn is a martingale. (2) Find P (XT = 100). (3) Let Mn = Xn2 − n. Verify that Mn is also a martingale. (4) It is known that Mn and T...