Glossary


Stellar

Asset

An asset in Stellar is a security issued by a specific account that can be traded in the Stellar exchange.

Assets can be "cashed out" with the asset issuer for whatever it is that the asset is intended to represent. Accounts that issue assets are informally called anchors.

Any account can issue any asset to the network and call it anything they like, so you must trust an account before you can hold assets it has issued.

An asset is given by the following data:

  • the asset issuer
  • the asset type
  • the asset code.

The only exception to this rule is the native asset of the network, called the lumen, which only has an asset type: native.

Anchor

An anchor in Stellar is an account that issues assets and honors withdrawals for those assets.

By honoring withdrawals, anchors effectively serve as the link between the Stellar network and the physical world. They're the entities that allow you, for instance, to hold cows in the Stellar network (which are sequences of zeros and ones) and "cash them out" for actual cows (which produce milk).

Anchors have total control over:

  • which accounts can hold their assets,
  • who the holders of their assets can trade those assets with, and
  • when the holders can withdraw those assets.

The notion of an anchor in the Stellar network is a generalization of the notion of a bank in the physical world. A traditional bank holds cash and honors withdrawals for cash. A Stellar anchor

Account

An account in Stellar is a special kind of entry in the ledger that can hold balances, issue assets, and trade assets.

An account is publicly identified by its public key. Control over an account rests with whoever knows the secret seed or the secret key.

Example of a public key: GAPR4YI2IPPL7OBA6KG24OCQ4N35WHKNJEJIDTFOLNHLJUPDHACRX5AU Example of a secret seed: SBUNGVT2C4PHIC446T4QOFZ7LFD5NFXSJ46BIAME25R5DEXP24J765H5

Secret seed

The secret seed is a cryptographic object associated to an account from the which you can recover both the account's public key and private key.

In particular, this means that whoever knows the secret seed has control over the account.


Trading

Asset

In trading, an asset is anything that be bought and sold (ie. traded) in a market.

Market

A market is a place (physical or digital) where buyers meet sellers.

A market for an asset X is a place (physical or digital) where buyers of X meet sellers of X.

Limit order

A limit order is an order to trade at a given price, and it lives in the orderbook until a market order clears it.

Limit orders provide liquidity to the market, by exposing themselves to be fulfilled by market orders.

Market order

A market order is an order (to buy or sell) that is to be executed immediately, regardless of price.

Market orders take liquidity away from the market, by fulfilling limit orders.

Orderbook

The orderbook, also called the limit orderbook or the book of limit orders, is the list of all limit orders (both bids and asks). The amount of limit orders in the orderbook is a rough/naive measure of liquidity.

Bid

A bid offer is a limit order for buying an asset.

A bid price is the price of a bid offer, typically the lowest one.

Ask

An ask offer is a limit order for selling an asset.

An ask price is the price of an ask offer, typically the highest one.

Price

In a market, there's no such thing as the price of an asset. There's only the bid price, the ask price, and the last price (which is the price at which the last market order cleared a limit order).

The problem of executing a large market order at the best average price given the available orderbook is known to be a stochastic control problem. This is a problem in the first place because large market orders tend to walk the orderbook, so the optimal rate of which to execute a large market order (either or buy or sell an asset) to maximize profits is not obvious at all.

Walking the orderbook

If a market order is larger then the best matching limit order, the market order executes the best limit order first and then the limit orders that come above (or below) it, which have worse prices (for the issuer of the market order) than the best limit order.

This process of clearing worse and worse limit orders with a market order is known as walking the orderbook.

Liquidity

Intuitively, liquidity is a measure of how easy/hard it is to trade an asset.

The amount of limit orders in the orderbook is a rough/naive measure of liquidity, because anyone can place any limit order on the orderbook and withdraw it at will without any intention to allow the order to be fulfilled.

Liquidity is one of the key attributes of an asset that trades in a market, and it can heavily affect the price of an asset.

Market depth

Market depth is a measure of a market's capacity to absorb large market orders (which are liquidity-taking) without affecting the price of an asset. The greater the market depth a market has, the larger volume of market orders it can sustain without affecting the price of an asset. Market depth measures the volume of all limit orders in the orderbook. A market with a lot of depth is said to be deep. A market with little depth is said to be shallow.

Market depth is closely related to liquidity.


Probability

Random variable

A random variable is a function from the sample space to the state space.

Random variables are not variables in the sense of high-school algebra, but functions in the sense of calculus.

Stochastic process

A stochastic process is a function whose values are random variables.

Random variables are to stochastic processes as numbers are to functions.
One observation/sample of a random variable yields a (single) number.
One observation/sample of a random process yields a (single) function.

Copula

A copula is a probability density that measures dependence among random variables. A (2-dimensional) copula takes 2 marginal/unconditional densities and "glues" them together to produce a joint cumulative density that encodes all dependency data between the densities. That is, the copula of 2 random variables is a joint cumulative density.


Calculus

Differential

The differential of function f at a given point a is a linear function df_a which best approximates f (among all other linear functions) near the point a.

Differentials are very important in integral calculus because you can only integrate differentials. They're also very important in stochastic calculus because a continuous-time stochastic process is always nowhere differentiable, but it may have a differential, and so you may be able to integrate it.


Stochastic calculus

Ito integral

The Ito integral is the stochastic calculus version of the Riemann integral.

Ito's lemma

It's lemma is the stochastic calculus version of the chain rule.

Brownian motion

Brownian motion is the random motion of particles in certain media.

Brownian motion also refers to the mathematical machinery used to model such movement, in which case Brownian motion means Wiener process.

Wiener process

The Wiener process is a continuous-time stochastic process with independent Gaussian increments and continuous paths.

The Wiener process can be constructed as some limit of discrete-time stochastic processes, such as the scaling limit of a random walk. This is possible due to Donsker's theorem, an extension to the central limit theorem which involves convergence in distribution of a sequence of stochastic processes.

The Wiener process underlies the Black-Scholes model used in European options pricing.

Markov process

Levy process

Gaussian process


Algebra

Ring

A ring is a set where you can always add, subtract, and multiply elements (but not necessarily divide).

Rings are very important in pure mathematics, cryptography, and computer science.

Some examples:

  • The set of all integers forms a ring denoted Z (from the German word Zahlen).
  • The set of all unsigned 32-bit integers forms a (finite) ring denoted Z/4294967296Z, with 4294967296 elements. Notice that the integer 4294967296 is the 32-nd power of 2.
  • The set of all unsigned 64-bit integers forms a (finite) ring denoted Z/18446744073709551616Z, with 18446744073709551616 elements. Notice that the integer 18446744073709551616 is the 64-th power of 2.
  • The set of all continuous functions from the real numbers to the real numbers forms a ring.

Field

A field is a set where you can always add, subtract, and multiply elements (but not necessarily divide).

Fields are very important in pure mathematics, cryptography, and computer science.

Finite fields are important in cryptography because logarithms is thought to be very expensive in large finite fields. More precisely, no one has found a subexponential algorithm to compute these logarithms (except in special cases).

Some examples:

  • The set of all rational numbers forms a field, denoted Q
  • The set of all constructible numbers forms a field.
  • The set of all real numbers forms a field, denoted R.
  • The set of all complex numbers forms a field, denoted C.
  • The ring Z/3Z is a (finite) field, also denoted F3 (for field with 3 elements) or GF(3) (for Galois field with 3 elements).
  • An infinite family of finite fields is given by fields of the form Z/pZ, where p is a prime number. At each prime p, the ring Z/pZ is a finite field, also denoted Fp (for field with p elements) or GF(p) (for Galois field with p elements).
  • An infinite family of fields are given by global fields, which are finite-dimensional extensions of the field Q of rational numbers or finite-dimensional extensions of the field Fq(t) of rational functions over the finite field Fq (where q is a prime power).

Absolute Galois group

The absolute Galois group of a field F, denoted Gal(F), is a group that encodes data for all the separable extensions of F.

The absolute Galois group of the finite field GF(p), denoted Gal(GF(p)), is the Prufer group.