Brownian motion is the continuous limit of a scaled symmetric random walk. Its increments are independent and normally distributed, with \\(W_t-W_s\\sim N(0,t-s)\\). It is a martingale, but its paths are almost surely continuous and nowhere differentiable.
\nThe defining calculation for stochastic calculus is its quadratic variation:
\n$$[W,W]_t=t,$$
\nor, in differential notation, \\((dW_t)^2=dt\\). Cross variation with ordinary time is zero. The notes use this contrast to explain why the ordinary chain rule cannot simply be applied to a Brownian path.
\nFirst-passage times and the running maximum appear here as well. They connect Brownian motion to barrier and lookback payoffs later in the course.
","contentMarkdown":"Brownian motion is the continuous limit of a scaled symmetric random walk. Its increments are independent and normally distributed, with \\(W_t-W_s\\sim N(0,t-s)\\). It is a martingale, but its paths are almost surely continuous and nowhere differentiable.\n\nThe defining calculation for stochastic calculus is its quadratic variation:\n\n$$[W,W]_t=t,$$\n\nor, in differential notation, \\((dW_t)^2=dt\\). Cross variation with ordinary time is zero. The notes use this contrast to explain why the ordinary chain rule cannot simply be applied to a Brownian path.\n\nFirst-passage times and the running maximum appear here as well. They connect Brownian motion to barrier and lookback payoffs later in the course.","dataUrl":"https://sharifhsn.dev/api/posts/brownian-motion.json","date":"2024-09-26","datePublished":"2024-09-26","description":"Brownian motion is the continuous limit of a scaled symmetric random walk. Its increments are independent and normally distributed, with \\(W_t-W_s\\sim N(0,t-s)\\). It is a martingal…","site":"https://sharifhsn.dev","slug":"brownian-motion","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Brownian Motion","Quadratic Variation"],"title":"Brownian Motion","url":"https://sharifhsn.dev/blog/brownian-motion/","version":"1","wordCount":102}