If a random process x(t) has no periodic components and if x(t) is of non-zero mean then Ryy (T) = [E(X)]² %3D
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- 1 Suppose that X is a stochastic process with dynamics dXt = µdt +σdWt , where W is a P-Brownian motion. The drift µ and the volatility σ are both constants. Find if there is a measure Q such that the drift of process X under Q is η(∈ R) instead of µ.Prove the following property of the compound Poisson process:1. E(xt) = λ t E(Y).If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?
- 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.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_tertB) 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.
- Let (Nt)t>0 be a Poisson process with parameter λ=2, Find the following: (a) E(X3X4)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 oneEvaluate the following initial value problem using the Laplace transform and the partial fraction decomposition