Differentials, Derivatives, and Integrals for Continuous Processes
Summary
The document distinguishes a process value, its increments, an ordinary time integral, and an integral with respect to process increments. It presents dX as notation for small changes in a process and cautions against treating it as an ordinary derivative fraction. Integrating X with respect to time accumulates the area under its path, while integrating with respect to dX is a different operation associated with stochastic integration.
The answer connects the latter integral to a trading interpretation: if the integrand represents the number of shares held, the integral can represent cumulative gains from price increments. It also notes that weighted increments need not sum to the original process value. The explanation is informal and omits conditions needed to define stochastic integrals rigorously; its suggestion to increase holdings when future increments are likely positive is not developed into a valid trading method. The notation should therefore be supplemented with formal stochastic calculus for technical use.
Key ideas
- The process value X and its increments dX are distinct mathematical objects.
- Integrating X over time accumulates the area beneath its path.
- Integrating with respect to dX accumulates process changes and can include a weighting function.
- A holdings process integrated against price changes can represent cumulative trading gains.
- The informal discussion does not state the technical conditions required for stochastic integration.
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# Notation clarity on continous proesses
# Notation clarity on continous proesses
Can someone clarify differences between $dX_t,\frac{\partial X_t}{\partial t},\int_0^t X_{t'}dt',\int_0^tdX_{t'}$?
Does $\int_0^t\frac{\partial X_{t'}}{\partial{t'}}d{t'}=X_t$?
## Answer by nbbo2 (score 3)
https://quant.stackexchange.com/a/32849
$X_t$ is the observable state of a physical system, it could be the price of a stock, the position of one leaf of a tree as it shakes during a storm, etc.
$dX_t$ the "increments of $X_t$" is an abstract notion representing the tiny little random and nonrandom changes that affect $X_t$ from moment to moment. It cannot be drawn or visualized, but when artists try to represent it they usually show a messy spiky blur like this
which is not really accurate mathematically, it is a attractive fiction. $dX_t$ can be used in 2 ways: in an SDE, in which case it will be usually accompanied by other differentials such as $dt$ or $dW_t$ etc. Or under an integral sign for example $X_t = \int_0^tdX_{t}$, which by the way is essentially the definition of $dX_t$. It is considered bad form (unacceptable in polite company) to put it in a fraction like this $\frac{\partial X_t}{\partial t}$.
The two integrals you show are rather distinct. $\int_0^t X_{t}dt$ just computes the area under the curve $X_t$, for example the area between the stock price and the x axis. Of course because the stock price is random the area is also a random variable.
The other integral is the famous Ito integral, invented in Japan less than 100 years ago, relativelly recenty in the history of Maths. $\int_0^t \phi(t)dX_{t}$ essentailly means that when adding up the $dX_t$ together we enhance or dilute them according to the "amplitude" function $\phi(t)$ thus we will probably get something different from the $X_t$ that we started with. If $\phi(t)$ represents the number of shares of stock you own then the integral is computing your cumulative profit from time 0. To become rich, just make $\phi(t)$ big when the upcoming increments are likely to be positive.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.