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.
In the following questions, let Bt denote a Brownian motion with B0 = 0. 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 with N(-1,1) distribution. Show that MnX +2nXn +n(n - 1) is a martingale. 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 with N(-1,1) distribution. Show that...
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 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...
1. Let Wt denote a standard Brownian motion. Evaluate the following expectationE[|Wt+t Wt|],where | · | denote absolute value. V [(Wt -Ws)2]. 2. Let Wt denote a Brownian motion. Derive the stochastic dierential equation for dXt andgroup the drift and diusion coecients together for the following stochastic processes:(a) Xt = Wt2(b) Xt =t+eWt(c) Xt = Wt3 3tWt(d) Xt = et+Wt(e) Xt = e2t sin(Wt)(f) Xt =eWt2t
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
solve d, e, f ASP please Let Y denote the claim size for a certain type of insurance policy and suppose Y has an exponential distribution with mean $300,000. (a) Write the density of Y for all values of y. (b) Determine the probability that a claim is smaller than $400,000 Define X = -Y so that X gives claim sizes that are negative, representing a payout from the insurance company. The remaining parts of this problem involve analyzing the...
2. Let X be a Bernoulli random variable with probability of X -1 being a. a) Write down the probability mass function p(X) of X in terms of a. Mark the range of a (b) Find the mean value mx(a) EX] of X, as a function of a (c) Find the variance σ剤a) IX-mx)2) of X, as a function of a. (d) Consider another random variable Y as a function of X: Y = g(X) =-log p(X) where the binary...
b,c,d please 1. Let the RV X denote your course marks in STAB52 (out of 100). From the syllabus you know that X = .15Xw + 3x +.55XF, where Xw, XM, Xp are the worksheet, midterm, and final marks (all out of 100), respectively. (a) (5 points) Would you prefer the RVs Xw, XM, Xp to be independent, positively corre- lated, or negatively correlated? Justify your answer. (b) (5 points) If E(XW) = 80, E (XM) = 70, EXF|XM) =...