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3651Classifying -|W_t|Classify X t = -|W t|.随机过程中等essay未尝试面试订阅3652Classifying a Stopped Brownian MotionLet tau = inf t : |W t| = 1 . Classify X t = W t wedge tau .随机过程中等essay未尝试面试订阅3653Choose c for a Cubic Brownian Local MartingaleFor standard Brownian motion, choose c so that X t=W t 3-c\int 0 t W s\,ds is a local martingale.随机过程中等derivation未尝试面试订阅3654Classifying exp(W_t-t/2)+tClassify X t = exp(W t - t/2) + t.随机过程中等essay未尝试面试订阅3655Choose c in the Quartic Brownian PolynomialFor standard Brownian motion, choose c so that X t=W t 4-6tW t 2+ct 2 is a martingale.随机过程中等derivation未尝试面试订阅3656Classify an Exponentially Weighted Stochastic IntegralClassify X t=\int 0 t e -s \,dW s as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3657Classify a Purely Increasing Adapted ProcessClassify X t=\int 0 t e -s \,ds as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3658Brownian Motion Plus Accumulated Variance BudgetClassify X t=W t+\int 0 t W s 2\,ds as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3659Classifying exp(W_t-t/2)-tClassify X t = exp(W t - t/2) - t.随机过程中等essay未尝试面试订阅3660Stochastic Integral with a Positive Deterministic CushionClassify X t=\int 0 t W s\,dW s+0.2t as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3661Why Drift Sign Dominates Martingale ClassificationWhen you already have an Itô decomposition, why does the sign of the drift usually decide martingale versus submartingale versus supermartingale?随机过程中等essay未尝试面试订阅3662Why a Stochastic Integral Is the Default Martingale CandidateWhy are square-integrable stochastic integrals usually the first objects you test as martingales in continuous time?随机过程中等essay未尝试面试订阅3663Why Compensation Terms Keep ReappearingWhy do compensation terms like -t or - frac12 a 2 t appear so often when building martingales from Brownian motion?随机过程中等essay未尝试面试订阅3666Upper-Hit Probability on [-3,5] from x=1A Brownian motion starts at x = 1 and is stopped at the first exit from [-3,5]. Using OST on W t, what is the probability it exits through 5?随机过程中等derivation未尝试面试订阅3667Start Required for a 70% Upper Exit Chance on [0,6]Brownian motion starts at x inside [0,6] and stops on first exit. For what x does OST imply a 70% chance of exiting through 6?随机过程中等derivation未尝试面试订阅3668Lower Barrier for a 40% Upper Exit ChanceBrownian motion starts at 1 with upper barrier 4 and lower barrier a. Using OST on W t, what a gives a 40% chance of exiting through 4?随机过程中等derivation未尝试面试订阅3669Symmetric Half-Width for a 75% Upper Exit ChanceBrownian motion starts at 2 inside [-L,L]. Using OST on W t, what L gives a 75% chance of hitting +L before -L?随机过程中等derivation未尝试面试订阅3670Average Upper-Exit Probability with Uniform Start on [0,8]The starting point X is uniform on [0,8]. Brownian motion starts at X and stops on first exit from [0,8]. What is the average probability of exiting through 8?随机过程中等derivation未尝试面试订阅3671Expected Exit Time from [0,5] Starting at 2Brownian motion starts at 2 and stops on first exit from [0,5]. Using OST on W t 2-t, what is E[tau]?随机过程中等derivation未尝试面试订阅3672Expected Exit Time from [-2,4] Starting at 1Brownian motion starts at 1 and stops on first exit from [-2,4]. Using OST on W t 2-t, what is E[tau]?随机过程中等derivation未尝试面试订阅