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6024Persistence and the Covariance-Stationarity VerdictA GARCH(1,1) has =0.20, =0.75. Compute the persistence + and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.统计简单derivation未尝试面试订阅6026ARCH(1) as the Beta-Zero Special CaseA GARCH(1,1) reduces to ARCH(1) when =0: h t=\omega+ r t-1 2. With \omega=0.7 and =0.3, compute the unconditional variance h as a decimal.统计简单derivation未尝试面试订阅6027When GARCH(1,1) Becomes EWMARiskMetrics EWMA updates variance as h t=(1- )r t-1 2+ h t-1 . State the constraints on (\omega, , ) that make GARCH(1,1) coincide exactly with EWMA, and give the implied when =0.06. Report as a decimal.统计中等数值题未尝试面试订阅6029News-Impact Update from a Signed ReturnA GARCH(1,1) has \omega=0.00001, =0.08, =0.90. Today's conditional variance is h t=0.0004 and today's return is r t=-0.03. Compute tomorrow's conditional variance h t+1 as a decimal.统计中等derivation未尝试面试订阅6030One-Step Prediction with a Persistence CoefficientA latent state evolves as x t=0.9\,x t-1 +w t with w t\sim N(0,2). At time t-1 the filtered state is N(4,3). Compute the one-step-ahead predicted mean and predicted variance of x t (before any observation at time t).统计简单derivation未尝试免费6031Kalman Gain AloneIn a scalar measurement update y=x+\varepsilon with \varepsilon\sim N(0,4), the prior (predicted) state variance is P -=12. What fraction of the innovation is incorporated into the updated estimate, i.e. compute the Kalman gain K.统计简单数值题未尝试免费6032How Much Does One Print Shrink Uncertainty?A predicted state has variance P -=10. A single observation arrives with noise variance R=6 in the model y=x+\varepsilon. By how much does the posterior (updated) variance fall below P -? Give the updated variance P +.统计简单数值题未尝试免费6033Innovation Variance and a Standardized SurpriseIn the model y t=x t+v t with v t\sim N(0,3), the predicted state at time t is N(5,7). You then observe y t=11. Compute the innovation (one-step forecast error) variance S, and the standardized innovation (y t-m -)/ S .统计中等derivation未尝试面试订阅6034Where Does the Estimate Land After the Print?The predicted state is N(8,6) and you observe y=14 with measurement noise variance R=2 in y=x+\varepsilon. Compute only the updated (posterior) mean of the state.统计简单数值题未尝试免费6041One-Year Conditional Variance of an OU ProcessAn OU process has mean-reversion speed kappa = 0.5 and diffusion coefficient sigma = 0.3. Given X 0, what is the conditional variance Var(X 1 | X 0)?随机过程中等derivation未尝试面试订阅6042Long-Run Variance of a Mean-Reverting RateA short rate follows an OU process with kappa = 1.5 and sigma = 0.12. What is its long-run (stationary) variance?随机过程简单derivation未尝试面试订阅6043Autocorrelation of a Stationary OU Process at a LagA stationary OU process has mean-reversion speed kappa = 0.7. What is the autocorrelation between X t and X t+2 ?随机过程简单derivation未尝试面试订阅6044Differential of W_t SquaredLet W t be standard Brownian motion. Apply Ito's lemma to f(W t) = W t 2 and write the resulting SDE d(W t 2) in terms of dt and dW t.随机过程简单derivation未尝试免费6045Expected Value of a GBMA stock follows dS t = 0.1 S t dt + 0.4 S t dW t with S 0 = 50. Compute E[S 3].随机过程简单数值题未尝试免费6046Differential of t Times W_tLet W t be standard Brownian motion. Using the Ito product rule, find d(t W t) and express the resulting SDE in terms of dt and dW t.随机过程中等derivation未尝试免费6047Variance of a Geometric Brownian MotionA GBM satisfies dS t = 0.06 S t dt + 0.25 S t dW t with S 0 = 1. Compute Var(S 2).随机过程困难数值题未尝试面试订阅6048Ito Isometry for a Time-Weighted IntegralLet W t be standard Brownian motion. Using the Ito isometry, compute E[(integral from 0 to 2 of s dW s) 2].随机过程中等数值题未尝试免费6049Quadratic Variation of a Stochastic IntegralDefine X t = integral from 0 to t of W s dW s. Compute the quadratic variation [X] T at T = 4, expressed as E[[X] T] (i.e. the expected accumulated quadratic variation).随机过程困难数值题未尝试面试订阅6050Closed-Form Solution of a Linear Multiplicative SDESolve the SDE dX t = a X t dt + b X t dW t with X 0 given, where a and b are constants. Write the explicit closed-form expression for X t.随机过程中等derivation未尝试免费6052Direction of Mean Reversion in a CIR ProcessA CIR process satisfies dX t = 2(0.04 - X t) dt + 0.1 sqrt(X t) dW t. The current value is X t = 0.07. Is the instantaneous drift positive or negative, and what is its numerical value?随机过程简单数值题未尝试免费