Introduction to Stochastic Calculus

Here is a corresponding introduction I found very useful, for readers with advanced undergraduate &#x2F; graduate level math knowledge.<p><a href="https:&#x2F;&#x2F;almostsuremath.com&#x2F;stochastic-calculus&#x2F;" rel="nofollow">https:&#x2F;&#x2F;almostsuremath.com&#x2F;stochastic-calculus&#x2F;</a>

Is stochastic calculus something that requires a computer to stimulate many possible unfolding of events, or is there a more elegant mathematical way to solve for some of the important final outputs and probability distributions if you know the distribution of dW? This is an awesome article. I&#x27;ve seen stochastic calculus before but this is the first time I really felt like I started to grok it.

A further step is Langevin Dynamics, where the system has damped momentum, and the noise is inserted into the momentum. This can be used in molecular dynamics simulations, and it can also be used for Bayesian MCMC sampling.<p>Oddly, most mentions of Langevin Dynamics in relation to AI that I&#x27;ve seen omit the use of momentum, even though gradient descent with momentum is widely used in AI. To confuse matters further, &quot;stochastic&quot; is used to refer to approximating the gradient using a sub-sample of the data at each step. You can apply both forms of stochasticity at once if you want to!

Own favorite source on stochastic calculus:<p><pre><code> Eugene Wong, {\it Stochastic Processes in Information and Dynamical Systems,\&#x2F;} McGraw-Hill, New York, 1971.\ \</code></pre>

I remember studying stochastiv calculus<p>And I remember noting that the standard deviation in regular statistics was that “quadratic variation” was slightly different than how variance is calculated. Off by one or squared or whatever. I made a note to eventually investigate why. Probably due to some stochastic volatility.

Here’s an example where I ran into this recently.<p>Let’s say we play a “game”. Draw a random number A between 0 and 1 (uniform distribution). Now draw a second number B from the same distribution. If A &gt; B, draw B again (A remains). What is the average number of draws required? (In other words, what is the average “win streak” for A?)<p>The answer is infinity. The reason is, some portion of the time A will be extremely high and take millions of draws to beat.

Question for HN readers: We have defined about 50 spots (loci) in the mouse genome that contain DNA differences that modulate mortality rates. Most of them have complex age-dependent “actuarial” effects. We would like to predict age at death.<p>Would stochastic calculus be a useful approach in actuarial prediction of life expectancies of mice?<p>(And this is why I am pleased to see this high on HN.)

Here&#x27;s my understanding of Ito calculus if it helps anyone:<p>1. The only random process we understand initially is Brownian motion.<p>2. Luckily, we can change coordinates.

This is such a good model for how to write a beginner friendly introduction. Especially the motivation for the Ito lemma, with the dW^2 term remaining important even though it disappears in regular calculus, and the conversion to Stratonovich is really nice.

Can someone please help me parse this sentence?<p>&gt; <i>Brownian motion and Itô calculare a notable example of fairly high-level mathematics that are applied to model the real world</i><p>What is “Itô calculare” supposed to have been? I am stumped. “Its calculation”?

For those in quant finance, how much of this is useful in your day to day?

I had to study quantum stochastic calculus for my PhD. Really crazy because you get totally different results for the same mathematical expression compared to normal calculus

Does it have applications in modeling decision-making?

still wild to me that diffusion models are fast becoming the secret sauce behind ai image generation, but their roots are buried deep in stochastic calculus<p>who knew brownian motion would eventually help create cat memes?

Seems like a great article. Having some prior experience with stochastic calculus, I think I understand almost everything here. Any other good introductory materials?