(Other variations of Brownian motion). Theorem 3.4.3 asserts...

(Other variations of Brownian motion). Theorem 3.4.3 asserts that if $T$ is a positive number and we choose a partition $\Pi$ with points $0=t_0<t_1<t_2<\cdots<t_n=T$, then as the number $n$ of partition points approaches infinity and the length of the longest subinterval $\|\Pi\|$ approaches zero, the sample quadratic variation $$ \sum_{j=0}^{n-1}\left(W\left(t_{j+1}\right)-W\left(t_j\right)\right)^2 $$ approaches $T$ for almost every path of the Brownian motion $W$. In Remark 3.4.5, we further showed that $\sum_{j=0}^{n-1}\left(W\left(t_{j+1}\right)-W\left(t_j\right)\right)\left(t_{j+1}-t_j\right)$ and $\sum_{j=0}^{n-1}\left(t_{j+1}-t_j\right)^2$ have limit zero. We summarize these facts by the multiplication rules $$ d W(t) d W(t)=d t, \quad d W(t) d t=0, \quad d t d t=0 . $$

(Other variations of Brownian motion). Theorem 3.4.3 asserts that if $T$ is a positive number and we choose a partition $\Pi$ with points $0=t_0<t_1<t_2<\cdots<t_n=T$, then as the number $n$ of partition points approaches infinity and the length of the longest subinterval $\|\Pi\|$ approaches zero, the sample quadratic variation
$$
\sum_{j=0}^{n-1}\left(W\left(t_{j+1}\right)-W\left(t_j\right)\right)^2
$$
approaches $T$ for almost every path of the Brownian motion $W$. In Remark 3.4.5, we further showed that $\sum_{j=0}^{n-1}\left(W\left(t_{j+1}\right)-W\left(t_j\right)\right)\left(t_{j+1}-t_j\right)$ and $\sum_{j=0}^{n-1}\left(t_{j+1}-t_j\right)^2$ have limit zero. We summarize these facts by the multiplication rules
$$
d W(t) d W(t)=d t, \quad d W(t) d t=0, \quad d t d t=0 .
$$

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Key Concepts

Brownian Motion

A Brownian motion is a continuous-time stochastic process with independent, normally distributed increments that model random movement. It serves as a key example of a martingale and is the cornerstone of many developments in stochastic calculus and financial mathematics.

Quadratic Variation

Quadratic variation measures the accumulated squared increments of a process over a time interval. For Brownian motion, its quadratic variation over an interval [0, T] is equal to T, demonstrating a unique path property that distinguishes it from processes with smooth paths.

Itô Calculus

Itô calculus is a branch of mathematics that extends traditional calculus to stochastic processes. The multiplication rules, such as dW(t)dW(t) = dt, dW(t)dt = 0, and dt dt = 0, formalize the manipulation of differentials in this context, reflecting the fact that increments of Brownian motion have non-trivial quadratic variation while their first-order terms vanish.

Partitioning and Limit Processes

Partitioning an interval into smaller subintervals and then taking the limit as the mesh size approaches zero is a critical technique in defining integrals and variations. This approach underpins the rigorous derivation of properties such as the quadratic variation of Brownian motion and the resulting differential rules in stochastic calculus.

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