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Term in stochastic calculus
In stochastic calculus, stochastic logarithm of a semimartingale Y {\displaystyle Y} such that Y ≠ 0 {\displaystyle Y\neq 0} and Y − ≠ 0 {\displaystyle
Stochastic_logarithm
2.71828...; base of natural logarithms
constant, approximately equal to 2.71828, that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after
E_(mathematical_constant)
Continuous stochastic process
an exponential Brownian motion, is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion
Geometric_Brownian_motion
Mathematical theorem
iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to
Law_of_the_iterated_logarithm
Binary logarithm Bode plot Henry Briggs Bygrave slide rule Cologarithm Common logarithm Complex logarithm Discrete logarithm Discrete logarithm records
Index_of_logarithm_articles
Branch of mathematics
subtraction, multiplication, division, exponentiation, extraction of roots, and logarithm. For example, the operation of addition combines two numbers, called the
Algebra
Probabilistic optimal control
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or
Stochastic_control
Replacing a number with a simpler value
arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed
Rounding
Type of stochastic process
strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not
Stationary_process
Representation of a type of random process
dependent linearly on their own previous values on a stochastic basis. The model is in the form of a stochastic difference equation (or recurrence relation) which
Autoregressive_model
Unique strong solution of a stochastic differential equation
the existence of the quadratic variation term [X] in the solution. Stochastic logarithm Doléans-Dade, C. (1970). "Quelques applications de la formule de
Doléans-Dade_exponential
Average uncertainty in variable's states
possible values. The choice of base for log {\displaystyle \log } , the logarithm, varies for different applications. Base 2 gives the unit of bits (or
Entropy_(information_theory)
Partial differential equation
by getting rid of the time-dependent prefactor in the argument of the logarithm, to u ( x , t ) = − 2 ν ∂ ∂ x ln { ∫ − ∞ ∞ exp [ − ( x − x ′ ) 2 4
Burgers'_equation
Identity in Itô calculus analogous to the chain rule
the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically
Itô's_lemma
Mathematical operation in calculus
values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have ( log u v ) ′ = ( log
Logarithmic_derivative
Application of mathematical and statistical methods in finance
The latter focuses on applications and modeling, often with the help of stochastic asset models, while the former focuses, in addition to analysis, on building
Mathematical_finance
Bet sizing formula for long-term growth
a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected geometric
Kelly_criterion
Scientific study of digital information
binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. In
Information_theory
Topics referred to by the same term
communication sciences Geometric Brownian motion, continuous stochastic process where the logarithm of a variable follows a Brownian movement, that is a Wiener
GBM
Stochastic volatility model used in derivatives markets
model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta
SABR_volatility_model
Stochastic process modeling random walk with friction
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics, the physical sciences, and evolutionary
Ornstein–Uhlenbeck_process
Overview of and topical guide to probability
and a 'concrete' illustration Berry–Esséen theorem Law of the iterated logarithm Random walk Poisson process Compound Poisson process Wiener process Geometric
Outline_of_probability
Type of random mathematical object
is often defined on the real number line, where it can be viewed as a stochastic process. It is used, for example, in queueing theory to model random events
Poisson_point_process
Exponential representation for differential equations
deterministic case with alterations due to the stochastic setting the corresponding matrix logarithm will turn out as an Itô-process, whose first two
Magnus_expansion
{\displaystyle \epsilon } -covering, the entropy of T {\displaystyle T} is the logarithm of the minimum number of balls of radius ϵ {\displaystyle \epsilon } required
Dudley's_entropy_integral
Model in finance
describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset
Heston_model
Process forming a path from many random steps
Branching random walk – Stochastic process Brownian motion – Random motion of particles suspended in a fluid Law of the iterated logarithm – Mathematical theorem
Random_walk
Method of mathematical differentiation
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic
Logarithmic_differentiation
N-th root of the product of n numbers
log x {\displaystyle f(x)=\log x} . A logarithm of any base can be used in place of the natural logarithm. For example, the geometric mean of 1 {\displaystyle
Geometric_mean
and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable
Continuous-time stochastic process
Continuous-time_stochastic_process
Stochastic process generalizing Brownian motion
real-valued continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary
Wiener_process
Solution to a stochastic differential equation
sample paths. Diffusion processes are stochastic in nature and hence are used to model many real-life stochastic systems. Brownian motion, reflected Brownian
Diffusion_process
Croatian–American mathematician
theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical
William_Feller
model that is formed by maximizing a function that is related to the logarithm of the likelihood function, but in discussing the consistency and (asymptotic)
Quasi-maximum likelihood estimate
Quasi-maximum_likelihood_estimate
Mathematical concept
theory of stochastic processes. Intuitively, a measure should be infinitely divisible provided it has a well-defined "convolution logarithm." The natural
Convolution_power
Type of mathematical function
0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, log f ( θ x + ( 1 − θ
Logarithmically concave function
Logarithmically_concave_function
Grammar model in linguistics
in the tree. A special case of WCFGs are PCFGs, where the weights are (logarithms of ) probabilities. An extended version of the CYK algorithm can be used
Probabilistic context-free grammar
Probabilistic_context-free_grammar
Functional relationship between two quantities
the others. This behavior is what produces the linear relationship when logarithms are taken of both f ( x ) {\displaystyle f(x)} and x {\displaystyle x}
Power_law
Concept in Fourier analysis
is the result of computing the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. The method is a tool for investigating
Cepstrum
Probability distribution
frequency data on a log-log graph, with the axes being the logarithm of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent
Zipf's_law
Probability distribution with high skewness or kurtosis
especially in phonographic markets. The probability density function for logarithm of weekly record sales changes is highly leptokurtic and characterized
Fat-tailed_distribution
Economic principle
cities: The law of proportionate effect will therefore imply that the logarithms of the variable will be distributed following the log-normal distribution
Gibrat's_law
Ways inequality is measured
redistribution. Therefore, the Hoover index is the "non-stochastic" counterpart to the "stochastic" Theil index. Applying the Theil index to allocation processes
Income_inequality_metrics
Statistical model for count data
assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown
Poisson_regression
Mathematical approximation of a function
functions x ↦ ln(1 + x) and x ↦ cos x − 1. The Taylor series for the natural logarithm is (using big O notation) ln ( 1 + x ) = x − x 2 2 + x 3 3 + O ( x 4
Taylor_series
Divergent sum of positive unit fractions
} is the natural logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0.577} is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large
Harmonic_series_(mathematics)
Complexity class of problems
isomorphism problem, and decision versions of factoring and the discrete logarithm. Under the exponential time hypothesis, there exist natural problems that
NP-intermediate
Stochastic process in time series analysis
statistical analysis of time series, a trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be
Trend-stationary_process
process Increasing process Itô's lemma Jump diffusion Law of the iterated logarithm Lévy flight Lévy process Loop-erased random walk Markov chain Examples
List_of_probability_topics
Course designed to prepare students for calculus
The general logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function. Then the natural logarithm is obtained
Precalculus
Fundamental theorem in probability theory and statistics
{\displaystyle n\to \infty } . The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random
Central_limit_theorem
Method in statistics
method applies to Hadamard directionally differentiable functionals of stochastic processes that converge to a limiting process. The delta method was derived
Delta_method
Probability distribution
distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally
Log-normal_distribution
Probability distribution
distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters
Log-Laplace_distribution
Function related to statistics and probability theory
showing that the difference in the logarithm of the likelihood generated by the estimate's parameter values and the logarithm of the likelihood generated by
Likelihood_function
Computer scientist
of Engineering is Studies on Stochastic Modeling of Neurons. There, he contributed to the spiking neurons with stochastic pulse-frequency modulation. Advisors
Yasuo_Matsuyama
Method for evaluating indefinite integrals
integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because
Risch_algorithm
Data evaluation test
of data to see whether it can be described as random (patternless). In stochastic modeling, as in some computer simulations, the hoped-for randomness of
Randomness_test
Continuous probability distribution
Barndorff-Nielsen, Ole (1977). "Exponentially decreasing distributions for the logarithm of particle size". Proceedings of the Royal Society of London. Series
Normal-inverse Gaussian distribution
Normal-inverse_Gaussian_distribution
Economic model of endogenous growth
and the y-axis represents the natural logarithm of output ln y {\displaystyle \ln y} . Growth is stochastic in nature, as innovations t {\displaystyle
Aghion–Howitt_model
List of factorial and binomial topics List of fractal topics List of logarithm topics List of mathematical properties of points List of numeral system
Lists_of_mathematics_topics
Probability distribution
evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated
Normal_distribution
Power spectrum of a noise signal
refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly different properties
Colors_of_noise
Probability distribution
both Pickand's and Hill's tail-index estimators commonly make use of logarithm of the order statistics. The ratio estimator (RE-estimator) of the tail-index
Heavy-tailed_distribution
Random process of binary (boolean) random variables
infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component
Bernoulli_process
Model of future interest rates
analytically tractable, and with potentially negative rates), the natural logarithm (extension of the Black–Karasinski model, not analytically tractable,
Hull–White_model
Statistical measure of time series variability
rescaled range can be characterized by making a plot of the logarithm of R/S vs. the logarithm of the number of samples. The slope of this line gives the
Rescaled_range
Statistical test that compares goodness of fit
significantly different from one, or equivalently whether its natural logarithm is significantly different from zero. The likelihood-ratio test, also
Likelihood-ratio_test
Square of numbers with equal row, column and diagonal totals
integer) to the power of each element, because the logarithm of the product of 2 numbers is the sum of logarithm of each. Alternatively, if any 3 numbers in
Magic_square
Parameter of some vertical wind profile equations
neutral conditions. In reality, the wind at this height no longer follows a logarithm. It is so named because it is typically related to the height of terrain
Roughness_length
Study of discrete mathematical structures
discrete calculus, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete
Discrete_mathematics
Estimator for quality of a statistical model
want to compare a model of the response variable, y, with a model of the logarithm of the response variable, log(y). More generally, we might want to compare
Akaike_information_criterion
combine multiplicatively, the components are made additive by taking the logarithm of the image intensity, so that these multiplicative components of the
Homomorphic_filtering
Mathematical model for neuron networks
model is a mathematical model for a network of neurons with intrinsic stochasticity. In the most general definition, a GL network consists of a countable
Galves–Löcherbach_model
Instantaneous rate of change (mathematics)
{\frac {d}{dx}}x^{a}=ax^{a-1}} Functions of exponential, natural logarithm, and logarithm with general base: d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}
Derivative
Message encoded with more bits than needed
merely the entropy of each symbol, while, in the most general case of a stochastic process, it is r = lim n → ∞ 1 n H ( M 1 , M 2 , … M n ) , {\displaystyle
Redundancy (information theory)
Redundancy_(information_theory)
Interdisciplinary field of research
mathematics of stochastic processes. Coleman embodied this idea in his 1964 book Introduction to Mathematical Sociology, which showed how stochastic processes
Mathematical_sociology
Mathematical function
{1}{2c^{2}}}} ) The Gaussian functions are thus those functions whose logarithm is a concave quadratic function. The parameter c is related to the full
Gaussian_function
Fourier transform of the probability density function
the cumulant generating function as the logarithm of the moment-generating function, and call the logarithm of the characteristic function the second
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Probability distribution
distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random
Log-Cauchy_distribution
Set of random variables
conflicting terminology is in use: the word potential is often applied to the logarithm of φ C {\displaystyle \varphi _{C}} . This is because, in statistical
Markov_random_field
Method of estimating the parameters of a statistical model, given observations
supremum value. In practice, it is often convenient to work with the natural logarithm of the likelihood function, called the log-likelihood: ℓ ( θ ; y ) = ln
Maximum_likelihood_estimation
Inverse relationship between the intensity and duration
example when measuring a Hurter and Driffield curve (optical density versus logarithm of total exposure) for a photographic emulsion. Total exposure of the
Reciprocity_(photography)
Concept in statistics
v t e Stochastic processes Discrete time Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically
Gaussian_random_field
hypercube sampling Law (stochastic processes) Law of averages Law of comparative judgment Law of large numbers Law of the iterated logarithm Law of the unconscious
List_of_statistics_articles
American mathematician
457–462, 1965. Statistical methods related to the law of the iterated logarithm, "The Annals of Mathematical Statistics", 41(5), 1397–1409, 1970. Optimal
Herbert_Robbins
Discrete probability distribution
maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function: ℓ ( λ ) = ln ∏ i = 1 n f ( k i ∣ λ ) = ∑
Poisson_distribution
Probabilistic inequality
^{2}}}\right)\right),} where h(u) = (1 + u)log(1 + u) – u and log denotes the natural logarithm. For generalizations see Freedman (1975) and Fan, Grama and Liu (2012)
Bennett's_inequality
Type of scatter plot
statistical significance. It is constructed by plotting the negative logarithm (base 10) of the p-value on the y-axis, ensuring that data points with
Volcano_plot_(statistics)
Type of calculus problem
( t ) | = 0.85 t + B {\displaystyle \ln |y(t)|=0.85t+B} Eliminate the logarithm with exponentiation on both sides | y ( t ) | = e B e 0.85 t {\displaystyle
Initial_value_problem
hypotheses given the evidence for them, and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former
History_of_probability
Method of evaluating certain integrals along paths in the complex plane
a branch cut. This affects our choice of the contour C. Normally the logarithm branch cut is defined as the negative real axis, however, this makes the
Contour_integration
Class of statistical models
termed an exponential-response model (or log-linear model, since the logarithm of the response is predicted to vary linearly). Similarly, a model that
Generalized_linear_model
Statistical transform
output random number. The basic form requires two multiplications, 1/2 logarithm, 1/2 square root, and one trigonometric function for each normal variate
Box–Muller_transform
Statistical test
log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications
Anderson–Darling_test
Probability distribution
distribution; conversely, if X has a log-Laplace distribution, then its logarithm has a Laplace distribution. Let X , Y {\displaystyle X,Y} be independent
Laplace_distribution
School of macroeconomics
y t {\displaystyle \,y_{t}\,} is the logarithm of real GDP, y t ∗ {\displaystyle y_{t}^{*}} is the logarithm of potential output, and b y {\displaystyle
New_Keynesian_economics
Operation in calculus
The case n = −1 required the invention of a function, the hyperbolic logarithm, achieved by quadrature of the hyperbola in 1647. Further steps were made
Integral
logarithm / (S:R) Maximal ergodic theorem / (S:R) Op (statistics) / (S:R) Optional stopping theorem / (FS:R) Stationary process / (SU:R) Stochastic convergence /
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Technique used to increase the number of structures a microchip may contain
a single wafer exposure. The resolution limit may also originate from stochastic effects, as in the case of EUV. Consequently, 20 nm linewidth still requires
Multiple_patterning
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