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3586Why It\^o Needs the Second DerivativeIn one paragraph, explain why It\ o's lemma has a second-derivative term even though ordinary chain rule does not.随机过程中等essay未尝试面试订阅3587Why Log GBM Loses Half Sigma SquaredWhy does d\log S t for GBM contain the drift adjustment - 2/2?随机过程中等essay未尝试面试订阅3588How Drift Cancellation Creates Local MartingalesWhat is the core idea behind turning a transformed diffusion into a local martingale by choosing the right prefactor or discount rate?随机过程中等essay未尝试面试订阅3589Why Time-Dependent Prefactors Are UsefulWhy are time-dependent factors like e -at or (1+t) -1 so common in It\ o calculations?随机过程中等essay未尝试面试订阅3590It\^o Versus Ordinary Chain RuleGive a short explanation of how It\ o's lemma should be viewed as a stochastic version of Taylor expansion rather than as a minor tweak to ordinary chain rule.随机过程中等essay未尝试面试订阅3591GBM Drift for a Median Target Over Two YearsA geometric Brownian motion satisfies dS t = mu S t dt + 0.3 S t dW t with S 0 = 100. If the median of S 2 should equal 128, what drift mu is required?随机过程中等derivation未尝试面试订阅3594GBM Drift Implied by a Mild Median UpliftA geometric Brownian motion satisfies dS t = mu S t dt + 0.15 S t dW t with S 0 = 60. If the median of S 1.5 should equal 66, what drift mu is required?随机过程中等derivation未尝试面试订阅3596Volatility from a 95th-to-Median Ratio in One YearFor a GBM, the ratio of the 95th percentile of S 1 to its median is observed to be 1.9. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3598Volatility Matching a 97.5th-to-Median SpreadFor a GBM, the ratio of the 97.5th percentile of S 1.5 to its median is observed to be 2.2. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3599Short-Horizon Volatility from an Upper Quantile RatioFor a GBM, the ratio of the 90th percentile of S 0.5 to its median is observed to be 1.3. What volatility sigma does this imply?随机过程中等derivation未尝试面试订阅3601OU Long-Run Mean Implied by a Conditional Mean Target 1An OU process satisfies dX t = 0.8(theta - X t)dt + sigma dW t with X 0 = 7. If E[X 1] should equal 5.2, what theta is implied?随机过程中等derivation未尝试面试订阅3603OU Long-Run Mean Implied by a Conditional Mean Target 3An OU process satisfies dX t = 0.6(theta - X t)dt + sigma dW t with X 0 = 10. If E[X 0.5] should equal 8.9, what theta is implied?随机过程中等derivation未尝试面试订阅3606OU Volatility Implied by a Conditional Variance Target 1An OU process satisfies dX t = 1(theta - X t)dt + sigma dW t. If Var(X 1 | X 0) should equal 0.45, what sigma is required?随机过程中等derivation未尝试面试订阅3607OU Volatility Implied by a Conditional Variance Target 2An OU process satisfies dX t = 0.7(theta - X t)dt + sigma dW t. If Var(X 2 | X 0) should equal 0.3, what sigma is required?随机过程中等derivation未尝试面试订阅3608OU Volatility Implied by a Conditional Variance Target 3An OU process satisfies dX t = 1.4(theta - X t)dt + sigma dW t. If Var(X 0.5 | X 0) should equal 0.6, what sigma is required?随机过程中等derivation未尝试面试订阅3611Discounted GBM DynamicsA GBM satisfies dS t = 0.05 S t dt + 0.2 S t dW t. Define Y t = e -0.05 t S t. What SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅3612Log of GBMA GBM satisfies dS t = 0.08 S t dt + 0.3 S t dW t. If Y t = log S t, what drift and diffusion coefficients does Y t have?随机过程中等derivation未尝试面试订阅3613Centered OU DynamicsAn OU process satisfies dX t = 1.4(3 - X t)dt + 0.7 dW t. If Y t = X t - 3, what SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅3614OU Integrating-Factor TransformAn OU process satisfies dX t = 0.9(2 - X t)dt + 1.1 dW t. If Z t = e 0.9 t (X t - 2), what SDE does Z t satisfy?随机过程中等derivation未尝试面试订阅3615CIR Exponential RescalingA CIR process satisfies dX t = 1.2(4 - X t)dt + 0.5 sqrt(X t)dW t. If Y t = e 1.2 t X t, what SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅