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3394Recover the Leading O(h^2) Error CoefficientSuppose D h = L + c h 2 + O(h 4). If D 0.3 = 4.18 and D 0.15 = 4.045, what is c?数学简单derivation未尝试面试订阅3395Recover the Limit of a First-Order SchemeA first-order approximation obeys D h = L + c h + O(h 2). If D 0.4 = 1.9 and D 0.2 = 1.6, what is the extrapolated estimate of L?数学简单derivation未尝试面试订阅3396Max Mesh Under a Central-Difference Error BudgetA central-difference first-derivative formula has truncation error bounded by M h 2 / 6, and a third-derivative bound gives M=12. How large can h be if you want the truncation error below 0.002?数学中等derivation未尝试面试订阅3397Forward-Difference Mesh Under a Linear Error BoundA forward-difference derivative formula has truncation error bounded by M h / 2 with M=8. How large can h be if the truncation budget is 0.01?数学中等derivation未尝试面试订阅3398Second-Derivative Mesh From a Fourth-Derivative BoundA centered second-derivative formula has error bounded by M h 2 / 12 with M=24. What h keeps the truncation error at or below 0.005?数学中等derivation未尝试面试订阅3406Zero-Flux Ghost PointAt a left boundary, a zero-flux Neumann condition is approximated by (u 1 - u -1 )/(2h)=0. What relation does this impose on the ghost point u -1 ?数学中等derivation未尝试面试订阅3407Ghost Point Under a Positive Boundary SlopeAt x=0, approximate u x(0)=3 using (u 1-u -1 )/(2h) with h=0.1 and u 1=1.4. What is the ghost point value u -1 ?数学中等derivation未尝试面试订阅3408Boundary Curvature From a Nonzero Slope ConditionSuppose u(0)=4, u 1=5, h=0.25, and the left-boundary slope satisfies u x(0)=-2 approximated by (u 1-u -1 )/(2h). What second derivative estimate do you get from (u -1 - 2u 0 + u 1)/h 2?数学中等derivation未尝试面试订阅3409Boundary Curvature Under Zero FluxSuppose u(0)=7, u 1=6, h=0.5, and the left boundary has zero flux. What second derivative estimate follows from the ghost-point stencil (u -1 -2u 0+u 1)/h 2?数学中等derivation未尝试面试订阅3410Second-Order Forward Stencil at a BoundaryUse the second-order forward stencil (-3f(0)+4f(h)-f(2h))/(2h) with h=0.1, f(0)=1, f(0.1)=1.24, and f(0.2)=1.52. What derivative estimate do you get at 0?数学中等derivation未尝试面试订阅3566Cubic Brownian Polynomial MartingaleChoose a so that M t = W t 3 + a t W t is a martingale.随机过程中等derivation未尝试面试订阅3567Quartic Brownian Polynomial MartingaleChoose a and b so that M t = W t 4 + a t W t 2 + b t 2 is a martingale.随机过程中等derivation未尝试面试订阅3568Quintic Brownian Polynomial MartingaleChoose a and b so that M t = W t 5 + a t W t 3 + b t 2 W t is a martingale.随机过程中等derivation未尝试面试订阅3569Exponential-Cosine Local MartingaleChoose a so that M t = e a t cos(2W t) is a local martingale.随机过程中等derivation未尝试面试订阅3570Exponential-Sine Local MartingaleChoose a so that M t = e a t sin(3W t) is a local martingale.随机过程中等derivation未尝试面试订阅3571Nonzero Power Making S_t^p DriftlessA GBM satisfies dS t = 0.04 S t dt + 0.2 S t dW t. Find the nonzero p such that S t p is a local martingale.随机过程中等derivation未尝试面试订阅3572Discount Rate Making e^{-gamma t} S_t^2 DriftlessA GBM satisfies dS t = 0.03 S t dt + 0.2 S t dW t. What gamma makes e -gamma t S t 2 a local martingale?随机过程中等derivation未尝试面试订阅3573Time Shift Making log S_t DriftlessA GBM satisfies dS t = 0.07 S t dt + 0.3 S t dW t. What c makes log S t + c t a local martingale?随机过程中等derivation未尝试面试订阅3574Dynamics of the Reciprocal of GBMA GBM satisfies dS t = 0.05 S t dt + 0.3 S t dW t. If Y t = 1 / S t, what are the drift and diffusion coefficients of Y t?随机过程中等derivation未尝试面试订阅3575Dynamics of the Square Root of GBMA GBM satisfies dS t = 0.02 S t dt + 0.4 S t dW t. If Z t = S t 1/2 , what are the drift and diffusion coefficients of Z t?随机过程中等derivation未尝试面试订阅