Variance of Brownian Motion Increments

Summary

The document explains the variance of an infinitesimal increment of standard Brownian motion. Since the process value at time t has a normal distribution with variance t, the difference between its values at times s and t is normally distributed with variance t minus s. In differential notation, an increment over a small interval dt is informally modeled as normal with variance dt.

This is a concise conceptual answer rather than a derivation. The differential notation is a shorthand for increments over small time intervals, not an ordinary random variable defined at a literal infinitesimal step. The explanation assumes standard Brownian motion and does not discuss other processes, drift, or applications to trading models.

Key ideas

• A standard Brownian motion value at time t has variance t.
• The increment between times s and t has variance equal to the length of that interval.
• The differential dB_t informally represents a normal increment with variance dt.
• The differential description is shorthand for small finite increments.

Tags

# What is Variance of delta of brownian motion


# What is Variance of delta of brownian motion












I am new to this.

If variance of Brownian motion b is t, what is the variance of db?

db is delta of b

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/50733

Let $(B_t)$ be a standard Brownian motion. Then, $B_t\sim N(0,t)$ and $B_t-B_s\sim N(0,t-s)$.

Informally, you can say $\mathrm{d}B_t\sim N(0,\mathrm{d}t)$ where $\mathrm{d}B_t=B_{t+\mathrm{d}t}-B_t$ is an infinitesimal increment.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.