Stochastic calculus

Introduction.

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

$x$	$f (x)$
0	1
1	2
2	5

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

$x$	$f (x)$	$Δ f (x)$
0	1	-
1	2	1
2	5	3

The usual derivative is not a "real" thing: it emerges when you take increasingly smaller steps $Δ x$ for the $x$ column and you consider the ratio $Δ f (x) / Δ x$ .
The problem of recover $f$ from $Δ f$ , or even from $d f / d x$ is called integration. So far, so good. But, what if $Δ f (x) / Δ x$ does not correspond to a fixed list of values but, instead, to a random list? We enter into the realm of stochastic calculus. Indeed, the reconstruction of a function from this noisy derivative is Itô integral or Stratonovich integral.

General diagram

%%{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

stochastic process: A collection of random variables indexed by time, representing systems evolving over time with inherent randomness.
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 diffusions.