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3580Reciprocal Dynamics for a Quadratic-Drift DiffusionA diffusion satisfies dX t = X t 2 dt + X t dW t. If Y t = 1 / X t, what SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅3581Time-Weighted Square of Brownian MotionApply It\ o to X t=e -t W t 2. What is dX t?随机过程困难derivation未尝试面试订阅3582Time-Weighted Exponential of Brownian MotionApply It\ o to X t=e -2t e W t . What is dX t?随机过程困难derivation未尝试面试订阅3583Brownian Motion Divided by a Deterministic ClockApply It\ o to X t=W t/(1+t). What is dX t?随机过程困难derivation未尝试面试订阅3584Time-Weighted Squared StockIf dS t=0.04S t\,dt+0.2S t\,dW t, what is d(S t 2/(1+t))?随机过程困难derivation未尝试面试订阅3585Discounted Log-Stock ProcessIf dS t=0.05S t\,dt+0.2S t\,dW t, what is d(e -0.03t \log S t)?随机过程困难derivation未尝试面试订阅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