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4985Infer Return Rate From Expected Hitting Time 19A CTMC has states 0,1,2 with state 2 absorbing. From 0 to 1 the rate is a=1.5, and from 1 to 2 the rate is b=0.5. If the expected hitting time of state 2 from 0 is 3, what is the return rate c from 1 back to 0?随机过程困难数值题未尝试面试订阅4986Why Fixed Waiting Times Break the CTMC PropertyA simulator gets the jump-chain routing probabilities right but replaces exponential waits by deterministic one-minute waits in every state. Why is the resulting calendar-time process generally not a CTMC anymore?随机过程困难essay未尝试面试订阅4987Why Uniformization Can Use One Poisson ClockWhy can a CTMC with different exit rates across states still be simulated using one common Poisson clock plus occasional virtual self-jumps?随机过程困难essay未尝试面试订阅4988Stationary Does Not Mean SlowWhy can a state have a small stationary probability even if it has a large exit rate?随机过程困难essay未尝试面试订阅4989Same Jump Chain, Different Calendar-Time BehaviorWhy can two jump processes share exactly the same jump chain but still look very different when observed in real time?随机过程困难essay未尝试面试订阅4990Why First-Step WorksWhy is first-step analysis so effective for expected hitting times in jump processes?随机过程困难essay未尝试面试订阅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6051Martingale Condition for W Squared Minus c tLet W t be standard Brownian motion. For what constant c is the process M t = W t 2 - c t a martingale?随机过程中等数值题未尝试免费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6053SDE for the Exponential of Brownian MotionLet W t be standard Brownian motion and define Y t = e W t . Use Ito's lemma to find the SDE satisfied by Y t.随机过程简单derivation未尝试免费