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3513Why Conditioning on an Endpoint Creates a BridgeWhy does conditioning Brownian motion on a future endpoint naturally produce bridge-style residuals?随机过程中等essay未尝试面试订阅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3522Barrier Matching a Unit-Horizon Risk Across Horizon 4A barrier of 1.2 over horizon 1 gives some touch or survival risk. What barrier over horizon 4 preserves that same standardized ratio?随机过程中等derivation未尝试面试订阅3524Barrier Matching a Unit-Horizon Risk Across Horizon 9A barrier of 1 over horizon 1 gives some touch or survival risk. What barrier over horizon 9 preserves that same standardized ratio?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3529Endpoint Cap Needed for a 10% Joint TailFor standard Brownian motion over t = 1 with barrier a = 1, what endpoint cap b gives P(max 0<=s<=1 W s >= 1, W 1 <= b) = 0.10?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3536Why Reflection Solves a Path Event Using an Endpoint TailWhy does the reflection principle let a maximum-over-time event collapse to a Gaussian endpoint tail?随机过程中等essay未尝试面试订阅3537Finite-Horizon Touching Is Not the Same as Eventual HittingWhy is a finite-horizon barrier-touch probability conceptually different from the statement that Brownian motion eventually hits every level?随机过程中等essay未尝试面试订阅