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3578Ito Dynamics of X_t^2 for a Reciprocal-Drift DiffusionA diffusion satisfies dX t = (1/X t) dt + dW t. What is d(X t 2)?随机过程中等derivation未尝试面试订阅3579Drift Correction for an Exponential of Brownian Motion With DriftA process satisfies dX t = 1 dt + 2 dW t. For M t = exp(0.5 X t - c t), what c makes M t a local martingale?随机过程中等derivation未尝试面试订阅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