How Brownian Motion, Itô Processes, and Random Walks Differ
Summary
The document gives an intuitive comparison of three stochastic-process concepts used in quantitative finance. A random walk evolves in discrete steps, illustrated by repeated coin tosses that move a path up or down. Brownian motion is continuous in time and serves as a basic model for random fluctuations; it can be understood as a limit of suitably scaled symmetric random walks.
An Itô process combines a drift component, representing systematic movement, with a diffusion component driven by Brownian motion, representing uncertainty. Brownian motion is therefore a special case of an Itô process, while a discrete random walk is a different kind of process despite the analogy between their increments. The answers describe Brownian increments as independent and normally distributed with zero mean. This is a conceptual introduction rather than a formal account: precise definitions, scaling conditions, and the assumptions needed for financial models are outside its scope, and some brief answer formulations simplify the distinctions.
Key ideas
- A random walk evolves in discrete time through random increments.
- Brownian motion is a continuous-time model of random fluctuations and can arise as a limit of scaled random walks.
- An Itô process combines systematic drift with random diffusion.
- Brownian motion is a special case of an Itô process.
- The intuitive comparisons omit formal conditions and mathematical detail.
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# Difference between ito process, brownian motion and random walk # Difference between ito process, brownian motion and random walk Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out. 1.Ito process 2.Brownian motion 3.Random walk The wikipedia articles are too in depth (yes!) for me at this point. ## Answer by Egodym (score 4, accepted) https://quant.stackexchange.com/a/18057 Assuming no math at all: Using an Ito process we can describe the return of a stock with two components: an average level (the "drift") plus some uncertainty (the "volatility"). This uncertainty is represented by a Brownian Motion. As written in Wikipedia, > A random walk is a mathematical formalization of a path that consists of a succession of random steps. You can get the random steps by tossing a coin n times. If head, go up one step; if tail, go down one step. This is the "symmetric random walk". You can obtain a Brownian Motion from the symmetric random walk using a bit of mathematical machinery. ## Answer by SBF (score 7) https://quant.stackexchange.com/a/18067 First, for Ito processes and Brownian motion. Ito process is a continuous-time trajectory with random evolution, so non-smooth and very kinky - also has a fractal look: no matter how much you'd zoom in, it will look similar. Ito process consists in fact of two parts: the drift part (deterministic evolution) and the diffusion part (where all the kinkiness and fractalness comes from). If Ito process does not posses the latter, it just looks like a continuous smooth trajectory. For example, $y = x^2$ is such an (purely drift) Ito process. In contrast, if Ito process has only diffusion component, you won't be able to spot it. That is, when the diffusion component is present, it's hard to say whether there is a drift component or not, because of the noise diffusion provides. Brownian motion is a special case of an Ito process, and is the main building block for the diffusion component. In fact, any diffusion is just a time scaled Brownian motion. One important property of Brownian motion is that its increments are uncorrelated (in fact, they are independent) whereas in general Ito process there can be loads of cross-correlation happening. Finally, formally random walk is a discrete-time process - hence not comparable with Ito processes which are continuous-time things. On the other hand, random walk must also have independent increments - that's why Brownian motion sometimes referred to as a (continuous-time) random walk. ## Answer by emcor (score 2) https://quant.stackexchange.com/a/18066 A Brownian Motion is a continuous time series of random variables whose increments are i.i.d. normally distributed with 0 mean. An Ito Process is a Brownian Motion with possibly nonzero mean. A random walk is a discrete process whose increments are +/-1 with equal probability. ## Answer by joseprupi (score 1) https://quant.stackexchange.com/a/43246 I would have added this as a comment to one of the answers but I don't have enough reputation for it. I recommend below lectures for this (they have been pretty useful to me at least): - From Maths Partner channel: Starting at Building Brownian Motion from a Random Walk and following videos up to "Derivation of Ito's Lemma" - From MIT OpenCourseWare: Stochastic Processes I and Stochastic Processes II
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