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.
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
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:
Example of a 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.
In trading, an asset is anything that be bought and sold (ie. traded) in a 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
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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
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.
The Ito integral is the stochastic calculus version of the Riemann integral.
It's lemma is the stochastic calculus version of the chain rule.
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.
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.
Rings are very important in pure mathematics, cryptography, and computer science.
- 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
4294967296elements. Notice that the integer
4294967296is the 32-nd power of 2.
- The set of all unsigned 64-bit integers forms a (finite) ring denoted
18446744073709551616elements. Notice that the integer
18446744073709551616is the 64-th power of 2.
- The set of all continuous functions from the real numbers to the real numbers forms a ring.
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).
- The set of all rational numbers forms a field, denoted
- The set of all constructible numbers forms a field.
- The set of all real numbers forms a field, denoted
- The set of all complex numbers forms a field, denoted
- The ring
Z/3Zis 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
pis a prime number. At each prime
p, the ring
Z/pZis a finite field, also denoted
Fp(for field with
GF(p)(for Galois field with
- An infinite family of fields are given by global fields, which are finite-dimensional extensions of the field
Qof rational numbers or finite-dimensional extensions of the field
Fq(t)of rational functions over the finite field
qis a prime power).
Absolute Galois group¶
The absolute Galois group of a field
Gal(F), is a group that encodes data for all the separable extensions of
The absolute Galois group of the finite field
Gal(GF(p)), is the Prufer group.