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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3676Drifted Brownian Upper-Hit Probability with Positive DriftA drifted Brownian motion satisfies dX t = 0.5dt + dW t, starts at 1, and stops on first exit from [0,4]. Using the exponential martingale, what is the probability it exits through 4?随机过程中等derivation未尝试面试订阅3677Drifted Brownian Upper-Hit Probability with Negative DriftA drifted Brownian motion satisfies dX t = -0.3dt + dW t, starts at 2, and stops on first exit from [0,5]. What is the probability it exits through 5?随机过程中等derivation未尝试面试订阅3680Drifted Brownian Upper-Hit Probability on a Short IntervalA drifted Brownian motion satisfies dX t = 0.2dt + dW t, starts at 2, and stops on first exit from [0,3]. What is the upper-exit probability?随机过程中等derivation未尝试面试订阅3682Upper-Hit Probability with Negative Drift and Lower VolatilityA diffusion satisfies dX t = -0.2dt + 0.8dW t, starts at 1, and stops on first exit from [0,4]. What is the upper-exit probability?随机过程中等derivation未尝试面试订阅3685Short-Interval Hit Probability with Small Drift and Small VolatilityA diffusion satisfies dX t = 0.1dt + 0.6dW t, starts at 1.2, and stops on first exit from [0,3]. What is the probability of hitting the upper barrier first?随机过程中等derivation未尝试面试订阅3686Why Bounded Stopping Times Make OST SaferWhy are bounded stopping times the cleanest setting for applying optional stopping in interview problems?随机过程中等essay未尝试面试订阅3687Why Drifted Brownian Motion Needs an Exponential MartingaleWhy is the exponential martingale the natural OST tool once Brownian motion has nonzero drift?随机过程中等essay未尝试面试订阅3688Why Expected Exit Time Comes from W_t^2-tWhy does the compensated square W t 2-t become the natural OST object for expected exit times?随机过程中等essay未尝试面试订阅3689Why OST Can Fail for Unbounded Stopping RulesWhy is it dangerous to apply optional stopping mechanically when the stopping rule is unbounded?随机过程中等essay未尝试面试订阅3690Why OST and Reflection Solve Different Brownian QuestionsWhy should OST problems not be confused with reflection-principle problems even though both often involve Brownian barriers?随机过程中等essay未尝试面试订阅3696Second Factor Drift After Neutralizing the First CarryUnder P, two processes share the same Brownian driver: dX t = 1.2dt + 0.6dW t and dY t = 0.5dt + 0.25dW t. If Q is chosen so that X has drift 0.3 under Q, what drift does Y have under Q?随机过程中等derivation未尝试面试订阅