7- Two continuous-time WSS zero mean random processes X(t) and Y(t) are uncorrelated, which means that E[X(t₁)Y(t₂)] = 0 for all të and t₂. Is the sum random process Z(t) = 2X(t) + 3Y (t) also WSS, and if so, what is its ACF and PSD?
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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.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.
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- 1. Find the natural cubic spline sN (x) passing through the 3 points (xj, yj) given by (0, 2), (2, 3), and (3, 1).Then evaluate sN (1).Consider the autonomous DE . dx/dt = x^2 - 8x + 7 (a) Its stable critical point is x= unstable critical point is x= (b) Applying five iterations of the Forward Euler Method to the above DE, using a step size h(0.1) and initial condition x(0)=3, yields: x(0.1)approximately x1 = x(0.2)approximately x2 = x(0.3)approximately x3 = x(0.4)approximately x4= x(0.5)approximately x5 =A certain brand of upright freezer is available in three different rated capacities: 16ft3, 18 ft3, and 20 ft3. Let X = the rated capacity of a freezer of this brand sold at acertain store. Suppose that X has pmfx 16 18 20p(x) .2 .5 3a. Compute E(X), E(X2), and V(X).b. If the price of a freezer having capacity X is 70X – 650, what is the expectedprice paid by the next customer to buy a freezer?c. What is the variance of the price paid by the next customer?d. Suppose that although the rated capacity of a freezer is X, the actual capacityis h(X) = X - .008X2. What is the expected actual capacity of the freezer purchasedby the next custom
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