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3608OU Volatility Implied by a Conditional Variance Target 3An OU process satisfies dX t = 1.4(theta - X t)dt + sigma dW t. If Var(X 0.5 | X 0) should equal 0.6, what sigma is required?随机过程中等derivation未尝试面试订阅3621Quadratic Variation of an Itô Integral with Linear LoadingLet X t = integral 0 t (1+s) dW s. What is [X] 2?随机过程中等derivation未尝试面试订阅3622Quadratic Variation of an Itô Integral with Descending LoadingLet X t = integral 0 t (2-s) dW s. What is [X] 1?随机过程中等derivation未尝试面试订阅3623Quadratic Variation of a Piecewise-Volatility IntegralLet X t = integral 0 t sigma(s) dW s where sigma(s)=1 on [0,1], sigma(s)=2 on (1,2], and sigma(s)=0 on (2,3]. What is [X] 3?随机过程中等derivation未尝试面试订阅3624Quadratic Variation of a Square-Root LoadingLet X t = integral 0 t sqrt(1+s) dW s. What is [X] 3?随机过程中等derivation未尝试面试订阅3625Quadratic Variation of a Rescaled Time Polynomial LoadingLet X t = integral 0 t (s/3) dW s. What is [X] 3?随机过程中等derivation未尝试面试订阅3636Why Smooth Drift Terms Disappear from Quadratic VariationWhy do smooth finite-variation terms fail to contribute to quadratic variation even though they can dominate the path in level?随机过程中等essay未尝试面试订阅3637Why Independent Brownian Motions Have Zero CovariationWhy does independence of Brownian drivers force the covariation term to vanish?随机过程中等essay未尝试面试订阅3638Why Quadratic Variation Scales with the Square of a CoefficientWhy does multiplying a Brownian-driven process by c multiply its quadratic variation by c 2 rather than by c?随机过程中等essay未尝试面试订阅3639Why Quadratic Variation Identifies Diffusion StrengthWhy is quadratic variation often the cleanest object for reading off the local diffusion strength of a process?随机过程中等essay未尝试面试订阅3640Why a Time Change Alters Quadratic VariationWhy does replacing W t by W ct change quadratic variation even before any extra scaling coefficient is added?随机过程中等essay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