Stochastic calculus

Overview

When you have a function $X$ , interpreted as a table

$t$	$X (t)$
0	1
1	2
2	5

you can think of the "discrete derivative" $Δ X$ as the extended table

$t$	$Δ X (t)$	$X (t)$
0	-	1
1	1	2
2	3	5

For the understanding of stochastic calculus, it is better to think that the usual derivative is not a "real" thing: it emerges when you take increasingly smaller steps $Δ t$ for the $t$ column and you consider the ratio $Δ X (t) / Δ t$ .
If we stick to real life, with real measurements, we only have this kind of discrete tables, not functions. The problem of recovering $X$ from $Δ X$ , or even from $d X / d t$ , is called integration. And it is written as

\frac{d X}{d t} = g (t),

d X = g (t) d t,

where $g (t)$ is the known data. With this setup,

X = X_{0} + \int g (t) d t \approx X_{0} + \sum Δ X = X_{0} + \sum g (t) Δ t .

So far, so good. But, what if $Δ X (t)$ does not correspond to a fixed list of values but, instead, to a random list?
For example, suppose that the contents of the list are generated by multiplying a certain quantity $H$ times a random number from a Gaussian. This is usually represented as $H (t) Δ W$ , where $W$ stands for Wiener process. We enter into the realm of stochastic calculus. Indeed, the reconstruction of a function from this noisy derivative could be done in several different ways, for example, Itô integral,

X_{t} = X_{0} + \int_{0}^{t} H (s) d W_{s} \approx X_{0} + \sum_{i} H (t_{i}) Δ W_{i} .

or Stratonovich integral

X_{t} = X_{0} + \int_{0}^{t} H (s) \circ d W_{s} \approx X_{0} + \sum_{i} H (\frac{t_{i} + t_{i + 1}}{2}) Δ W_{i} .

The resulting object is a stochastic process, which can be interpreted as a curve-valued random variable, instead of a single curve. The canonical example is Brownian motion as a limit of random walks.

Coming back to classical analysis, it may happen that we want to recover a function $X$ from a table in which the column doesn't solely depend on $t$ , but also on $X (t)$ . That is, we would have a kind of 2D table with $t$ free in the first column, $X (t)$ free in the first row, and the values of $Δ X$ in the interior of the table:

$t ∖ X (t)$	0	1	2	3
0	0	1	2	3
1	1	2	3	4
2	2	3	4	5
3	3	4	5	6

This problem is called a first-order ODE. It can be expressed as $\frac{d X}{d t} = F (t, X (t))$ or

\begin{matrix} (*) & d X = F (t, X (t)) d t, \end{matrix}

and the idea is the same: recover $X$ from the known data $F$ .

Now, like above, the content of the table can also be randomized. For example, it could be a Gaussian, multiplied by $t$ , or by $X (t)$ , or in general $H (t, X_{t}) Δ W_{t}$ . This is known as stochastic ordinary differential equation, denoted as

\begin{matrix} (**) & d X_{t} = H (t, X_{t}) d W_{t} . \end{matrix}

The solution is again a stochastic process, but now is more difficult to find it out. In the same way that equation $(*)$ must be seen as encoding "any function/table $X (t)$ such that its increments correspond to $F (t, X (t)) d t$ ", equation $(* *)$ means "any stochastic process such that its increments are $H (t, X_{t}) d W_{t}$ , with $d W_{t}$ the increments of a Brownian motion".

Finally, we have more general stochastic ordinary differential equation, in which the increments have a deterministic part and a random part:

d X_{t} = F (t, X_{t}) d t + H (t, X_{t}) d W_{t} .

Main definitions

stochastic process: A collection of random variables indexed by time, representing systems evolving over time with inherent randomness. It is like the random version of a function, a random curve, may we say.
Markov process: A type of stochastic process where the future state depends only on the current state, not on the sequence of events that preceded it (memoryless property).
Markov chain: A discrete-time Markov process with a countable state space, often used to model systems that transition from one state to another with certain probabilities.
random walk: A specific type of Markov chain where each step is determined randomly, often used to model phenomena like stock prices or particle movement.
Brownian motion: A continuous-time stochastic process with continuous paths and stationary, independent increments, serving as a fundamental example of a Markov process and a semimartingale.
Semimartingales: A broad class of stochastic processes that generalize martingales and are suitable for defining stochastic integrals, including processes with jumps.
Itô integral: A construction used to integrate with respect to Brownian motion (or more general semimartingales) giving rise to new stochastic processes, foundational for stochastic calculus. It is defined in a way that accounts for the randomness in the integrator.
martingales: A class of stochastic processes where the future expected value, given all past information, is equal to the current value. They are often obtained as outcomes of Itô integrals.
Itô Diffusions: Solutions to stochastic differential equations (SDEs) driven by Brownian motion, representing systems influenced by both deterministic and random factors.
stochastic ordinary differential equation: Equations that describe the evolution of stochastic processes, incorporating both deterministic trends and random fluctuations. They are the formalism used to model Itô diffusion.

%%{init: {'themeVariables': { 'fontSize': '13px', 'nodeSpacing': 30, 'rankSpacing': 25 }}}%%
graph TD
    A["Stochastic Process"]
    subgraph Discrete-Time
        C["Markov Chain"]
        D["Random Walk"]
    end
    subgraph Continuous-Time
        G["Semimartingale"]
        E["Brownian Motion"]
    end
    A -->|Markov property| B["Markov Process"]
    B --> C
    C --> D
    B --> G
    G --> E
    %% Random Walk approximation to Brownian
    D -.->|scaling limit| E
    %% Itô Integral as operator
    G -.->|Itô integral operator| H["New Processes"]
    %% Outputs: Martingales, SDEs, Diffusions
    H --> F["Martingale"]
    H --> I["Itô Diffusion"]
    %% Style
    classDef operator fill:#f9f,stroke:#333,stroke-width:2px;
    class G,H,J operator

Solutions

See solutions of an SDE.

Symmetries

See overview on symmetries of SDEs.