Example 5: For the random process X (t) = A cos w t + B sin ot, where A and B are random variables with E [A] = E [B] = 0, E [A?] = E [B?] > 0 and E [AB] = 0, prove that the process is mean-ergodic. %3D %3D
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- Consider the geometric Brownian motion with σ = 1: dS = μSdt + SdX, and consider the function F(S) = A + BSα. Find any necessary conditions on A, B, and α such that the function F(S) follows a stochastic process with no drift.Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).B) Let dP/dt =.5P - 50. Find the equilibrium solution for P. Furthermore, determine whether P is intially increasing faster if the initial population is 120 or 200.
- At any given time, a subatomic particle can be in one of two states, and it moves randomly from one state to another when it is excited. If it is state 1 on one observation, then it is 3 times as likely to be in state 1 as state 2 on the next observation. Likewise, if it is in state 2 on one observation, then it is 3 as likely to be in the state 2 as state 1 on the next observation. (a) If the particle is in state 1 on the fourth observation, what is the probability that it will be in state 2 on the sixth observation and state 1 on the seventh observation?Assume an asset price S_t follows the geometric Brownian motion, dS_t = µS_tdt + σS_dW_t, where µ and σ are constants and r is the risk-free rate. 1. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Xt = S_t^n , where n is a constant. 2. Compute E[X_t] and Var[X_t]. 3. Using the Ito’s Lemma find the stochastic differential equation satisfied by the process Y_t = S_tertA geometric Brownian motion has parameters mu= 0.2 and sigma=2. What is the probability that the process is above 4 at time t=5, given that it starts at 3. Round answer to 3 decimals.
- Which of the following processes (Xt)t is weakly stationary? A: Xt = 1:6 + Xt 1 + V tB: Xt = 0:6 Xt-1 +V tC: Xt = 0:8 Xt-1 + V tD: Xt = 0:8 t + 0:6 V t – 1 The term (t) is always assumed to be white noise with variance oneTwo interacting populations of hares and foxes can be modeled by the recursive equations h(t + 1) = 4h(t) − 2 f (t) f(t+1)=h(t)+ f(t). For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f (t). a. h(0)= f(0)=100 b. h(0)=200, f(0)=100 c. h(0)=600, f(0)=500Suppose that g = g(q, p, t), and that H is the Hamiltonian. Show that:a) (See the Figure)b) if any quantity does not explicitly depend on time and its Poisson parenthesis with the Hamiltonian is null, such quantity is a constant of motion for the system.