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3681Upper-Hit Probability with Drift and Nonunit VolatilityA diffusion satisfies dX t = 0.4dt + 1.5dW t, starts at 1.5, 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3683Start Needed for a Drifted Hit Probability with Nonunit VolatilityA diffusion satisfies dX t = 0.3dt + 1.2dW t and stops on first exit from [0,6]. For what start x does the upper-exit probability equal 0.65?随机过程中等derivation未尝试面试订阅3684Barrier Needed for a 90% Drifted Hit Probability with Sigma OneA diffusion satisfies dX t = 0.5dt + 1dW t, starts at 2, and stops on first exit from [0,b]. What upper barrier b makes the upper-exit probability equal 0.9?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3691Density Tilt Making an Energy Factor Slow Down Under QUnder P, a factor satisfies dX t = 1.1 dt + 0.5 dW t. A new measure Q is defined by W t Q = W t + theta t. What theta makes the drift of X under Q equal 0.2?随机过程中等derivation未尝试面试订阅3692Density Tilt Turning a Positive Drift into a Mild Negative OneUnder P, a factor satisfies dX t = 0.4 dt + 0.8 dW t. A new measure Q is defined by W t Q = W t + theta t. What theta makes the drift of X under Q equal -0.08?随机过程中等derivation未尝试面试订阅3693Density Tilt Removing Carry from a Basis ProcessUnder P, a factor satisfies dX t = 0.9 dt + 0.3 dW t. A new measure Q is defined by W t Q = W t + theta t. What theta makes the drift of X under Q equal 0?随机过程中等derivation未尝试面试订阅3694Density Tilt Lifting a Negative Drift Above ZeroUnder P, a factor satisfies dX t = -0.2 dt + 0.4 dW t. A new measure Q is defined by W t Q = W t + theta t. What theta makes the drift of X under Q equal 0.1?随机过程中等derivation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3697Co-Moving Spread Drift After Re-Centering the BenchmarkUnder P, two processes share the same Brownian driver: dX t = 0.7dt + 0.4dW t and dY t = -0.2dt + 0.9dW t. If Q is chosen so that X has drift -0.1 under Q, what drift does Y have under Q?随机过程中等derivation未尝试面试订阅3698Macro Factor Drift After Removing a Reference PremiumUnder P, two processes share the same Brownian driver: dX t = 0.5dt + 0.25dW t and dY t = 0.15dt + 0.35dW t. If Q is chosen so that X has drift 0 under Q, what drift does Y have under Q?随机过程中等derivation未尝试面试订阅3700Auxiliary Process Drift After Calibrating a Target ProcessUnder P, two processes share the same Brownian driver: dX t = 0.3dt + 0.2dW t and dY t = 0.45dt + 0.1dW t. If Q is chosen so that X has drift -0.05 under Q, what drift does Y have under Q?随机过程中等derivation未尝试面试订阅3701Linear Desk PnL Drift Target from a Shared DriverUnder P, dX t = 0.8dt + 0.4dW t and dY t = 0.3dt + 0.2dW t share the same Brownian motion. Let L t = 2X t + 1Y t. If Q is chosen so that L has drift 0.6 under Q, what theta is required and what is the resulting Q-drift of X?随机过程中等derivation未尝试面试订阅3702Spread Drift Target via One Girsanov TiltUnder P, dX t = 0.7dt + 0.5dW t and dY t = 0.2dt + 0.4dW t share the same Brownian motion. Let L t = 1X t + -1Y t. If Q is chosen so that L has drift -0.1 under Q, what theta is required and what is the resulting Q-drift of X?随机过程中等derivation未尝试面试订阅