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3514Why Independent Increments Matter in Quant PracticeWhy is the independent-increment property operationally important when reasoning about multi-horizon risk or signal aggregation?随机过程中等essay未尝试面试订阅3515Why Brownian Paths Feel Smooth but Are NotWhy can Brownian paths be continuous everywhere and still fail to behave like ordinary smooth curves?随机过程中等essay未尝试面试订阅3516Touch Probability for Barrier 1 Over Horizon 1For standard Brownian motion, what is P(max 0<=s<=1 W s >= 1)?随机过程中等derivation未尝试面试订阅3518Barrier Giving Touch Probability 0.1 Over Horizon 9For standard Brownian motion over horizon t = 9, what barrier a gives touch probability 0.1?随机过程中等derivation未尝试面试订阅3519Barrier Giving Touch Probability 0.02 Over Horizon 0.25For standard Brownian motion over horizon t = 0.25, what barrier a gives touch probability 0.02?随机过程中等derivation未尝试面试订阅3520Horizon Needed for Touch Risk 0.2 Under Barrier 2For standard Brownian motion, what horizon t gives touch probability P(M t >= 2) = 0.2?随机过程中等derivation未尝试面试订阅3521Horizon Needed for Touch Risk 0.05 Under Barrier 1.8For standard Brownian motion, what horizon t gives touch probability P(M t >= 1.8) = 0.05?随机过程中等derivation未尝试面试订阅3525Joint Max-End Probability With t = 1, a = 1, b = 0For standard Brownian motion, what is P(max 0<=s<=1 W s >= 1, W 1 <= 0)?随机过程中等derivation未尝试面试订阅3528Barrier Needed for a 5% Joint Tail With Endpoint Cap 0For standard Brownian motion over t = 1, what barrier a gives P(max 0<=s<=1 W s >= a, W 1 <= 0) = 0.05?随机过程困难derivation未尝试面试订阅3530Barrier Survival Probability for a = 1 Over t = 1For standard Brownian motion, what is P(max 0<=s<=1 W s < 1)?随机过程困难derivation未尝试面试订阅3532Barrier Survival Probability for a = 2 Over t = 0.5For standard Brownian motion, what is P(max 0<=s<=0.5 W s < 2)?随机过程中等derivation未尝试面试订阅3533Barrier Needed for 90% Survival Over One Time UnitFor standard Brownian motion over t = 1, what barrier a gives P(max 0<=s<=1 W s < a) = 0.90?随机过程中等derivation未尝试面试订阅3534Horizon Giving 80% Survival Under Barrier 1.5For standard Brownian motion, what horizon t gives P(max 0<=s<=t W s < 1.5) = 0.80?随机过程中等derivation未尝试面试订阅3535Ratio of Touch Probabilities for Barriers 1 and 1.5Over horizon t = 1 for standard Brownian motion, what is P(M 1 >= 1) / P(M 1 >= 1.5)?随机过程中等derivation未尝试面试订阅3541Start Needed for a 70% Upper-Hit Probability on [0, 6]For Brownian motion on [0, 6], at what starting point x is the probability of hitting 6 before 0 equal to 0.7?随机过程中等derivation未尝试面试订阅3542Upper-Hit Probability From x = 1 on [-3, 5]For Brownian motion started at x = 1 inside [-3, 5], what is the probability of hitting 5 before -3?随机过程中等derivation未尝试面试订阅3543Symmetric Interval Size for a 2/3 Upper-Hit ProbabilityBrownian motion starts at x = 1 inside [-L, L]. What L makes the probability of hitting +L before -L equal to 2/3?随机过程中等derivation未尝试面试订阅3544Lower Barrier Needed for a 40% Upper-Hit ProbabilityBrownian motion starts at x = 1 and has upper barrier 4 and lower barrier a < 1. What a makes the probability of hitting 4 before a equal to 0.4?随机过程中等derivation未尝试面试订阅3545Upper Barrier Needed for a 1/3 Hit Probability From x = 2Brownian motion starts at x = 2 with lower barrier 0 and upper barrier b > 2. What b makes the probability of hitting b before 0 equal to 1/3?随机过程中等derivation未尝试面试订阅3546Expected Exit Time From [0, 5] Started at 2For Brownian motion started at x = 2 inside [0, 5], what is the expected exit time from the interval?随机过程困难derivation未尝试面试订阅