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模块2.7.1 · 数学与统计能力 · 随机分析

布朗运动与伊藤积分

stochastic-calculus · brownian-motion · random-walk · donsker · gaussian-increments · filtration · martingale · quadratic-variation

题目3635 · 随机过程

Coefficient Making [aW_t-B_t, W_t+B_t]_2 Equal 6

Choose a so that [aW_t - B_t, W_t + B_t]_2 = 6 with independent W and B.

题目3634 · 随机过程

Coefficient Making [aW_t+B_t]_3 Equal 15

Choose a so that [aW_t + B_t]_3 = 15 with independent W and B.

题目3632 · 随机过程

Coefficient Making [W_t+aB_t, 2W_t-B_t]_1 Equal Zero

Choose a so that [W_t + aB_t, 2W_t - B_t]_1 = 0 with independent W and B.

题目3631 · 随机过程

Coefficient Making [W_t+aB_t]_2 Equal 10

Choose a so that [W_t + aB_t]_2 = 10 where W and B are independent Brownian motions.

题目3628 · 随机过程

Covariation of 0.5W_t+B_t with 3W_t+2B_t

Let X_t = 0.5W_t + B_t and Y_t = 3W_t + 2B_t with independent W and B. What is [X,Y]_4?

题目3627 · 随机过程

Covariation of 2W_t-B_t with W_t+4B_t

Let X_t = 2W_t - B_t and Y_t = W_t + 4B_t with independent W and B. What is [X,Y]_2?

题目3630 · 随机过程

Covariation of 2W_t+3B_t with -W_t+2B_t

Let X_t = 2W_t + 3B_t and Y_t = -W_t + 2B_t with independent W and B. What is [X,Y]_{0.5}?

题目3629 · 随机过程

Covariation of W_t-B_t with W_t+B_t

Let X_t = W_t - B_t and Y_t = W_t + B_t with independent W and B. What is [X,Y]_3?

题目3626 · 随机过程

Covariation of W_t+2B_t with 3W_t-B_t

Let X_t = W_t + 2B_t and Y_t = 3W_t - B_t where W and B are independent Brownian motions. What is [X,Y]_1?

题目3617 · 随机过程

Quadratic Variation of -3W_t Plus an Exponential Trend

Let X_t = -3W_t + e^t. What is [X]_1?

题目3618 · 随机过程

Quadratic Variation of 0.5W_t Plus an Integrated Drift

Let X_t = 0.5W_t + integral_0^t s ds. What is [X]_4?

题目3620 · 随机过程

Quadratic Variation of 1.2W_t Plus a Rational Drift

Let X_t = 1.2W_t + t/(1+t). What is [X]_5?

题目3616 · 随机过程

Quadratic Variation of 2W_t Plus a Smooth Drift

Let X_t = 2W_t + t^3 - sin t. What is [X]_2?

题目3623 · 随机过程

Quadratic Variation of a Piecewise-Volatility Integral

Let 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?

题目3625 · 随机过程

Quadratic Variation of a Rescaled Time Polynomial Loading

Let X_t = integral_0^t (s/3) dW_s. What is [X]_3?

题目3624 · 随机过程

Quadratic Variation of a Square-Root Loading

Let X_t = integral_0^t sqrt(1+s) dW_s. What is [X]_3?

题目3622 · 随机过程

Quadratic Variation of an Itô Integral with Descending Loading

Let X_t = integral_0^t (2-s) dW_s. What is [X]_1?

题目3621 · 随机过程

Quadratic Variation of an Itô Integral with Linear Loading

Let X_t = integral_0^t (1+s) dW_s. What is [X]_2?

题目3619 · 随机过程

Quadratic Variation of Brownian Motion with Two Smooth Corrections

Let X_t = W_t - t + cos t. What is [X]_3?

题目3633 · 随机过程

Scale Making [cW_{4t}]_1 Equal 1

Choose c so that [cW_{4t}]_1 = 1.

题目3640 · 随机过程

Why a Time Change Alters Quadratic Variation

Why does replacing W_t by W_{ct} change quadratic variation even before any extra scaling coefficient is added?

题目3637 · 随机过程

Why Independent Brownian Motions Have Zero Covariation

Why does independence of Brownian drivers force the covariation term to vanish?

题目3639 · 随机过程

Why Quadratic Variation Identifies Diffusion Strength

Why is quadratic variation often the cleanest object for reading off the local diffusion strength of a process?

题目3638 · 随机过程

Why Quadratic Variation Scales with the Square of a Coefficient

Why does multiplying a Brownian-driven process by c multiply its quadratic variation by c^2 rather than by c?

题目3636 · 随机过程

Why Smooth Drift Terms Disappear from Quadratic Variation

Why do smooth finite-variation terms fail to contribute to quadratic variation even though they can dominate the path in level?

课程布朗运动与伊藤积分 · 随机分析

从随机游走到布朗运动

上海某私募的量化研究员在白板上为沪深300 指数搭一个日内连续时间价格模型。她先画出一条平滑、处处可微的候选价格曲线公式，立刻被同事打断：「只要公式处处可微，你看到斜率为正的时刻就买入、转负就卖出，几秒内便能锁定无风险收益——这与无套利冲突。」结论是，连续时间随机模型背后的噪声源必须连续，但处处不可微。本节按龚光鲁《随机微分方程引论》的顺...

课程布朗运动与伊藤积分 · 随机分析

伊藤公式与随机微分方程

钩子：周五下午两点四十分，私募衍生品桌上的一个数周五下午两点四十分，你在一家沪深300指数增强私募的衍生品桌上，手里挂着一张以 300ETF 期权（300ETF options）对冲的指数风险敞口。模拟引擎用几何布朗运动（geometric Brownian motion, GBM）跑了 10 万条 60 个交易日的路径，你发现一个让人不安的现象：输入的年...

课程布朗运动与伊藤积分 · 随机分析

伊藤积分的构造

钩子：公式究竟在写什么？上海某量化私募的衍生品研究组周一例会上，桌上摆着一份连续时间 Delta 对冲方案的初稿：策略对沪深300 股指期货持有动态头寸公式，结算时账上的对冲腿累计盈亏被写成公式。主管把上一课的笔记翻到边角，红笔划了一行——只要公式在背后由布朗运动（Brownian motion）公式驱动，几乎每条样本路径的总变差（total ...

课程布朗运动与伊藤积分 · 随机分析

几何布朗运动与量化金融应用

路演桌上的两条收益率周三下午，沪深300 量化对冲产品的路演会。某私募管理人把净值幻灯片翻到第四页：年化预期收益 8%，年化波动率 40%。代销渠道的合规突然问：「按这个预期收益，投资者持有三年的中位数到底是多少？」管理人愣了三秒。这正是几何布朗运动（geometric Brownian motion, GBM）在路演桌上现形的瞬间——算术意义的期望收益与...