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3641Coefficient Making W_t^2-a t a MartingaleChoose a so that X t = W t 2 - a t is a martingale.随机过程中等derivation未尝试面试订阅3642Coefficient Making exp(aW_t-2t) a MartingaleChoose a so that X t = exp(aW t - 2t) is a martingale.随机过程中等derivation未尝试面试订阅3643Coefficient Making e^{a t} cos W_t a MartingaleChoose a so that X t = e a t cos W t is a martingale.随机过程中等derivation未尝试面试订阅3645Classifying an Exponentially Weighted Stochastic IntegralClassify X t = integral 0 t e -s dW s as a martingale, submartingale, or supermartingale.随机过程中等essay未尝试面试订阅3646Classifying W_t^2+tClassify X t = W t 2 + t.随机过程中等essay未尝试面试订阅3647Classifying 3-W_t^2-tClassify X t = 3 - W t 2 - t.随机过程中等essay未尝试面试订阅3648Classifying exp(W_t)Classify X t = exp(W t).随机过程中等essay未尝试面试订阅3649Classifying -exp(W_t)Classify X t = -exp(W t).随机过程中等essay未尝试面试订阅3650Classifying max(W_t,0)Classify X t = max(W t,0).随机过程中等essay未尝试面试订阅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