Introduction to Stochastic Processes
Expert-defined terms from the Professional Certificate in Stochastic Calculus for Finance course at London School of Business and Administration. Free to read, free to share, paired with a globally recognised certification pathway.
Absolute Continuity #
A concept in measure theory that describes the relationship between two measures, where one measure is dominated by the other. In the context of stochastic processes, absolute continuity is used to define the Radon-Nikodym derivative, which is essential for calculating expectations and probabilities. Related terms: measure theory, Radon-Nikodym theorem, stochastic integral.
Adapted Process #
A stochastic process that is adapted to a given filtration, meaning that the process is measurable with respect to the filtration at each point in time. Adapted processes are essential in stochastic calculus as they allow for the definition of stochastic integrals and the application of Itô's lemma. Related terms: filtration, stochastic integral, Itô's lemma.
Arbitrage #
A concept in financial economics that refers to the existence of a risk-free profit opportunity, where an investor can earn a positive return without taking on any risk. In the context of stochastic calculus, arbitrage is used to derive the fundamental theorem of asset pricing, which provides a framework for pricing financial securities. Related terms: financial economics, fundamental theorem of asset pricing, risk-neutral measure.
Augmented Filtration #
A filtration that is augmented with the null sets of the underlying probability space, ensuring that the filtration is right-continuous and complete. Augmented filtrations are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma.
Brownian Motion #
A stochastic process that is continuous, adapted, and has independent increments, with each increment being normally distributed. Brownian motion is a fundamental concept in stochastic calculus and is used to model stock prices and other financial securities. Related terms: stochastic process, adapted process, independent increments.
Cadlag #
A stochastic process that is right-continuous and has left-hand limits at each point in time. Cadlag processes are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, stochastic integral, Itô's lemma.
Complete Market #
A financial market where every contingent claim can be replicated using a self-financing trading strategy. Complete markets are essential in stochastic calculus as they allow for the pricing of financial securities using the fundamental theorem of asset pricing. Related terms: financial market, contingent claim, self-financing trading strategy.
Contingent Claim #
A financial security that pays off a stochastic amount at a future time, contingent on the outcome of a stochastic process. Contingent claims are used in stochastic calculus to model options and other financial derivatives. Related terms: financial security, stochastic process, option pricing.
Diffusion Process #
A stochastic process that is continuous and has a diffusion coefficient that determines its volatility. Diffusion processes are used in stochastic calculus to model stock prices and other financial securities. Related terms: stochastic process, diffusion coefficient, volatility.
Doob's Optional Stopping Theorem #
A theorem that provides a condition for a stopping time to be optional, allowing for the application of martingale theory to stochastic processes. Doob's optional stopping theorem is used in stochastic calculus to prove the fundamental theorem of asset pricing. Related terms: stopping time, martingale theory, fundamental theorem of asset pricing.
Drift #
A concept in stochastic processes that refers to the trend or mean of a stochastic process over time. Drift is used in stochastic calculus to model stock prices and other financial securities. Related terms: stochastic process, mean, trend.
Empirical Measure #
A measure that is defined using the empirical distribution of a random sample. Empirical measures are used in stochastic calculus to estimate parameters of stochastic processes. Related terms: measure, empirical distribution, random sample.
Filtration #
A family of sigma-algebras that represents the information available at each point in time. Filtrations are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: sigma-algebra, stochastic integral, Itô's lemma.
Financial Derivative #
A financial security that derives its value from the value of an underlying asset. Financial derivatives are used in stochastic calculus to model options and other financial securities. Related terms: financial security, underlying asset, option pricing.
Finite Variation Process #
A stochastic process that has finite variation over a given time interval, meaning that the sum of the absolute values of the increments of the process is finite. Finite variation processes are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma.
Gaussian Process #
A stochastic process that has normally distributed increments, with each increment being independent of the others. Gaussian processes are used in stochastic calculus to model stock prices and other financial securities. Related terms: stochastic process, normal distribution, independent increments.
Hedging #
A trading strategy that aims to reduce or eliminate the risk associated with a financial security. Hedging is used in stochastic calculus to price financial derivatives and to manage risk. Related terms: trading strategy, financial security, risk management.
Independence #
A concept in probability theory that describes the relationship between two or more random variables, where the occurrence of one variable does not affect the occurrence of the others. Independence is used in stochastic calculus to model stock prices and other financial securities. Related terms: probability theory, random variable, stochastic process.
Itô's Lemma #
A theorem that provides a formula for calculating the stochastic differential of a function of a stochastic process. Itô's lemma is used in stochastic calculus to define stochastic integrals and to price financial derivatives. Related terms: stochastic process, stochastic differential, stochastic integral.
Jump Process #
A stochastic process that has discontinuous paths, with the process jumps from one value to another at random times. Jump processes are used in stochastic calculus to model stock prices and other financial securities that exhibit jump behavior. Related terms: stochastic process, discontinuous, jump behavior.
Lebesgue Measure #
A measure that is defined on the real line and assigns a non-negative value to each subset of the real line. Lebesgue measure is used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: measure, real line, stochastic integral.
Local Martingale #
A stochastic process that is a martingale on each finite interval, but may not be a martingale on the entire time horizon. Local martingales are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, martingale, time horizon.
Markov Chain #
A stochastic process that has the Markov property, meaning that the future state of the process depends only on its current state and not on any of its past states. Markov chains are used in stochastic calculus to model stock prices and other financial securities. Related terms: stochastic process, Markov property, state space.
Martingale #
A stochastic process that has the martingale property, meaning that the expected value of the process at a future time is equal to its current value. Martingales are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, martingale property, expected value.
Measurable Function #
A function that is measurable with respect to a given sigma-algebra, meaning that the function can be expressed as a limit of simple functions. Measurable functions are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: function, sigma-algebra, simple function.
Measure #
A function that assigns a non-negative value to each subset of a given set, satisfying certain axioms such as countable additivity. Measures are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: function, set, countable additivity.
Merton's Theorem #
A theorem that provides a condition for a stopping time to be optional, allowing for the application of martingale theory to stochastic processes. Merton's theorem is used in stochastic calculus to prove the fundamental theorem of asset pricing.
Normal Distribution #
A probability distribution that is symmetric and has a bell-shaped curve, with the majority of the probability mass concentrated around the mean. Normal distributions are used in stochastic calculus to model stock prices and other financial securities. Related terms: probability distribution, symmetric, bell-shaped.
Null Set #
A set that has zero measure, meaning that the set is negligible in terms of probability. Null sets are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: set, zero measure, negligible.
Option Pricing #
A method for calculating the value of an option based on the value of the underlying asset. Option pricing is used in stochastic calculus to price financial derivatives and to manage risk. Related terms: option, underlying asset, financial derivative.
Optional Stopping Theorem #
A theorem that provides a condition for a stopping time to be optional, allowing for the application of martingale theory to stochastic processes. The optional stopping theorem is used in stochastic calculus to prove the fundamental theorem of asset pricing.
Partition #
A collection of sets that are mutually exclusive and exhaustive, meaning that each element of the underlying set belongs to exactly one of the sets in the partition. Partitions are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: set, mutually exclusive, exhaustive.
Poisson Process #
A stochastic process that has independent increments and a constant intensity, meaning that the probability of a jump occurring in a given time interval is proportional to the length of the interval. Poisson processes are used in stochastic calculus to model stock prices and other financial securities that exhibit jump behavior. Related terms: stochastic process, independent increments, constant intensity.
Predictable Process #
A stochastic process that is measurable with respect to the predictable sigma-algebra, meaning that the process can be expressed as a limit of simple functions. Predictable processes are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, predictable sigma-algebra, simple function.
Probability Measure #
A measure that assigns a non-negative value to each subset of a given set, satisfying certain axioms such as countable additivity and normalization. Probability measures are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: measure, set, countable additivity.
Quadratic Variation #
A measure of the volatility of a stochastic process over a given time interval, defined as the sum of the squared increments of the process. Quadratic variation is used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, volatility, stochastic integral.
Radon #
Nikodym Derivative: A function that represents the density of one measure with respect to another, used to define the stochastic integral of a stochastic process. The Radon-Nikodym derivative is used in stochastic calculus to apply Itô's lemma and to price financial derivatives. Related terms: measure, density, stochastic integral.
Random Variable #
A function that assigns a real number to each outcome of a random experiment. Random variables are used in stochastic calculus to model stock prices and other financial securities. Related terms: function, real number, random experiment.
Risk #
Neutral Measure: A probability measure that is equivalent to the physical measure but has the property that the expected return of a stochastic process is equal to the risk-free rate. Risk-neutral measures are used in stochastic calculus to price financial derivatives and to manage risk. Related terms: probability measure, equivalent, risk-free rate.
Self #
Financing Trading Strategy: A trading strategy that does not require any external funding, meaning that the strategy is financed solely by the trader's initial wealth and the returns from the strategy. Self-financing trading strategies are used in stochastic calculus to price financial derivatives and to manage risk. Related terms: trading strategy, external funding, initial wealth.
Sigma #
Algebra: A collection of sets that is closed under countable unions and complementation, used to define a measure on a given set. Sigma-algebras are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: set, countable unions, complementation.
Simple Function #
A function that takes on only a finite number of values, used to approximate more general functions in stochastic calculus. Simple functions are used to define stochastic integrals and to apply Itô's lemma. Related terms: function, finite number, stochastic integral.
Stochastic Differential Equation #
A mathematical equation that describes the evolution of a stochastic process over time, used to model stock prices and other financial securities. Stochastic differential equations are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: mathematical equation, stochastic process, evolution.
Stochastic Integral #
A mathematical object that represents the integral of a stochastic process with respect to another stochastic process, used to define stochastic differential equations and to price financial derivatives. Stochastic integrals are used in stochastic calculus to apply Itô's lemma and to manage risk. Related terms: mathematical object, stochastic process, integral.
Stochastic Process #
A family of random variables that are indexed by time, used to model stock prices and other financial securities. Stochastic processes are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: family of random variables, indexed, time.
Stopping Time #
A random variable that represents the time at which a stochastic process is stopped, used to define stochastic integrals and to apply Itô's lemma. Stopping times are used in stochastic calculus to prove the fundamental theorem of asset pricing. Related terms: random variable, time, stochastic process.
Submartingale #
A stochastic process that has the submartingale property, meaning that the expected value of the process at a future time is greater than or equal to its current value. Submartingales are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, submartingale property, expected value.
Supermartingale #
A stochastic process that has the supermartingale property, meaning that the expected value of the process at a future time is less than or equal to its current value. Supermartingales are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: stochastic process, supermartingale property, expected value.
Time Horizon #
A time interval over which a stochastic process is defined, used to model stock prices and other financial securities. Time horizons are used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: time interval, stochastic process, time.
Trading Strategy #
A rule for buying and selling financial securities over time, used to manage risk and to maximize returns. Trading strategies are used in stochastic calculus to price financial derivatives and to manage risk. Related terms: rule, financial securities, risk management.
Volatility #
A measure of the variability of a stochastic process over time, used to model stock prices and other financial securities. Volatility is used in stochastic calculus to define stochastic integrals and to apply Itô's lemma. Related terms: measure, variability, stochastic process.
Wiener Process #
A stochastic process that is continuous, adapted, and has independent increments, with each increment being normally distributed. Wiener processes are used in stochastic calculus to model stock prices and other financial securities. Related terms: stochastic process, continuous, adapted.