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3576Reciprocal Transform That Removes Drift CompletelyA diffusion satisfies dX t = X t 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3577Integrating Factor That Linearizes an Affine DiffusionA diffusion satisfies dX t = (2X t + 1) dt + 3 dW t. Define Y t = e -2t (X t + 1/2). What SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅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