Stochastic calculus

Stochastic integration

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? What if we have some noise when obtaining the data for the increments?
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} .

Here we are dealing with objects of a different nature than list of values or functions. We have stochastic processes. Both, the input in the increment column, and the output, the result, are stochastic processes, which can be interpreted as a curve-valued random variables, instead of single curves. The canonical example is Brownian motion as a limit of random walks.

Stochastic differential equations

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.