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3559Lower Barrier and Hit Probability From a Target Exit TimeBrownian motion starts at x = 1 with upper barrier 5. The lower barrier a is chosen so that the expected exit time is 8. What is the resulting probability of hitting 5 before a?随机过程中等derivation未尝试面试订阅3560Start Needed to Match Both a Probability and an Exit Time on [0, 10]Brownian motion starts at x inside [0, 10]. If the probability of hitting 10 before 0 is 0.6, what is the expected exit time?随机过程中等derivation未尝试面试订阅3561Why the Hitting Probability Is HarmonicWhy does the probability of hitting the upper barrier before the lower barrier end up being a linear function of the start x for Brownian motion?随机过程中等essay未尝试面试订阅3562Why Expected Exit Time Is Quadratic in the StartWhy does expected exit time from a bounded interval depend quadratically on the starting point rather than linearly?随机过程中等essay未尝试面试订阅3563Finite-Horizon Touching vs Eventual HittingWhy should finite-horizon barrier-touch probabilities not be confused with the interval-hitting probabilities used in bounded first-passage problems?随机过程中等essay未尝试面试订阅3564Why Optional Stopping Needs a Bounded Setup HereWhy are bounded intervals the natural place where optional-stopping arguments for Brownian hitting questions are most reliable?随机过程中等essay未尝试面试订阅3566Cubic Brownian Polynomial MartingaleChoose a so that M t = W t 3 + a t W t is a martingale.随机过程中等derivation未尝试面试订阅3567Quartic Brownian Polynomial MartingaleChoose a and b so that M t = W t 4 + a t W t 2 + b t 2 is a martingale.随机过程中等derivation未尝试面试订阅3568Quintic Brownian Polynomial MartingaleChoose a and b so that M t = W t 5 + a t W t 3 + b t 2 W t is a martingale.随机过程中等derivation未尝试面试订阅3569Exponential-Cosine Local MartingaleChoose a so that M t = e a t cos(2W t) is a local martingale.随机过程中等derivation未尝试面试订阅3570Exponential-Sine Local MartingaleChoose a so that M t = e a t sin(3W t) is a local martingale.随机过程中等derivation未尝试面试订阅3571Nonzero Power Making S_t^p DriftlessA GBM satisfies dS t = 0.04 S t dt + 0.2 S t dW t. Find the nonzero p such that S t p is a local martingale.随机过程中等derivation未尝试面试订阅3572Discount Rate Making e^{-gamma t} S_t^2 DriftlessA GBM satisfies dS t = 0.03 S t dt + 0.2 S t dW t. What gamma makes e -gamma t S t 2 a local martingale?随机过程中等derivation未尝试面试订阅3573Time Shift Making log S_t DriftlessA GBM satisfies dS t = 0.07 S t dt + 0.3 S t dW t. What c makes log S t + c t a local martingale?随机过程中等derivation未尝试面试订阅3574Dynamics of the Reciprocal of GBMA GBM satisfies dS t = 0.05 S t dt + 0.3 S t dW t. If Y t = 1 / S t, what are the drift and diffusion coefficients of Y t?随机过程中等derivation未尝试面试订阅3575Dynamics of the Square Root of GBMA GBM satisfies dS t = 0.02 S t dt + 0.4 S t dW t. If Z t = S t 1/2 , what are the drift and diffusion coefficients of Z t?随机过程中等derivation未尝试面试订阅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