4. Consider the process X+ = Vaw (t/a), where a is a positive constant. Calculate Var[X/(t+u)...
Consider a diffusion process {Xt,t > 0} with constant drift μ(z)-a and constant volatility σ(z) > 0 (a) Please derive the conditional distribution of Xt| Xo.
Let W - {Wi,0< t < ) represent a standard Brownian motion Show that the process Z(s)-(zt-W f.0 < t-1) is a standard Brownian motion, where s > 0 is fixed
Assume an asset price St follows the geometric Brownian motion, dSt = u Stdt+oS+dZt, So =s > 0 where u and o are constants, r is the risk-free rate, and Zt is the Brownian motion. 1. Using the Ito's Lemma find to the stochastic differential equation satisfied by the process X+ = St. 2. Compute E[Xt] and Var[Xt]. 3. Using the Ito's Lemma find the stochastic differential equation satisfied by the process Y1 = Sert'. 4. Compute E[Y] and Var[Y].
8.2. Let W()-X(at)la for a >0. Verify that W(t is also Brownian motion
3. An infinite bar has initial temperature distribution: T(x,0)-T0 [s(x-l) +δ(x + 1)] Find T(x,t) for >0. The free space Green's function for the 1-D diffusion equation is G(x -x)- e 4DI 4TDt
u(x,0)= Consider the following wave equation U, = U23 -00<x<00,t> 0 (0, -0<x<-1, _x+1, -1<< <0, 1-x, 0<x<1, 1<x<00 (0, -00<x<-1, u,(x,0) = 1, -15xs1, (0, 1<x<0. Find u(1,0.5) and u(-1,0.5).
PROBLEM 4. Determine the function u = u(t, x) if Ut = Uzz, t> 0, x € (0, 7), and u(0, x) = cos (x), uz(t, 0) = uz(t, 7) = 0.
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) = {pos, li - j| > 1. Let X and S2 be the sample mean and variance, respectively. a. Show that X is a consistent estimator for u. b. Is S2 unbiased for 02? Justify. - c. Show that S2 is asymptotically unbiased for 02.
Help please! Let Be be Brownian motion and fix to > 0. Prove that By: = Bto+t - Blo; t o is a Brownian motion.
1. Consider the heat flow problem on the real line, where u(x,t), t > 0 is the temperature at point x at time t: ди 1 a2u t>O (*) at 2 ar2 u(x,0) = sin(7x) = > (a) What is the thermal diffusitivity constant ß? (b) Find the intervals of x where the temparature will increase at t = 0. (c) Sketch the graph of the temperature at t = 0. (d) On the same axes as in (c), sketch...