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A stochastic process is a collection of random variables representing the evolution of a system over time, where the future state depends on both the present state and inherent randomness. It is widely used in fields like finance, physics, and biology to model phenomena that evolve unpredictably over time.
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A Markov process is a stochastic process that satisfies the Markov property, meaning the future state is independent of the past given the present state. It is widely used in various fields to model random systems that evolve over time, where the next state depends only on the current state and not on the sequence of events that preceded it.
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A Markov Chain is a mathematical system that undergoes transitions from one state to another on a state space, where the probability of each state depends only on the state attained in the previous step. This 'memoryless' property, known as the Markov property, makes them particularly useful for modeling random processes in various fields such as economics, genetics, and computer science.
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The Markov property is a fundamental characteristic of stochastic processes, where the future state of the process depends only on the present state and not on the sequence of events that preceded it. This memoryless property simplifies the analysis and modeling of complex systems in fields like physics, economics, and artificial intelligence.
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A transition matrix is a square matrix used to describe the transitions of a Markov chain, with each element representing the probability of moving from one state to another. It is fundamental in modeling stochastic processes where future states depend only on the current state, not on the sequence of events that preceded it.
State transition probabilities are fundamental to understanding how systems evolve over time in stochastic processes, describing the likelihood of moving from one state to another in a given time step. These probabilities are crucial in fields like Markov chains, where they define the dynamics and predict future states based on current conditions.
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A Markov Chain is a mathematical system that undergoes transitions from one state to another on a state space, with the probability of each state depending only on the previous state and not on the sequence of events that preceded it. This memoryless property is known as the Markov property, making Markov Chains particularly useful for modeling random processes in various fields such as finance, genetics, and computer science.
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Expected value is a fundamental concept in probability and statistics that represents the average outcome one would anticipate from a random event if it were repeated many times. It is calculated by summing all possible values, each weighted by their probability of occurrence, providing a measure of the center of a probability distribution.
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In the context of Markov chains, a recurrent state is one that the process will return to with probability one, meaning it is guaranteed to be visited infinitely often. recurrent states are pivotal in understanding the long-term behavior of stochastic processes, as they provide insights into the stability and ergodicity of the system.
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An absorbing state in a Markov chain is a state that, once entered, cannot be left, effectively trapping the process. This characteristic makes absorbing states crucial for understanding long-term behavior and stability in stochastic processes, as they often signify terminal or absorbing outcomes in various systems.
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A Gaussian Process is a collection of random variables, any finite number of which have a joint Gaussian distribution, used in machine learning to define a distribution over functions. It is particularly useful for regression tasks due to its ability to provide a probabilistic prediction with uncertainty quantification, making it a powerful tool for modeling complex data with inherent noise.
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Markov Models are mathematical frameworks used to model systems that transition from one state to another, where the probability of each future state depends only on the current state and not on the sequence of events that preceded it. They are widely used in various fields such as economics, genetics, and computer science for modeling random processes and decision-making under uncertainty.
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An autoregressive process is a type of stochastic time series model where future values are expressed as a linear function of past values and a stochastic term. It is widely used in time series forecasting and analysis, as it captures temporal dependencies and patterns in sequential data.
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A time-homogeneous process is a stochastic process whose transition probabilities are invariant over time, meaning the process behaves the same way regardless of the specific time at which it is observed. This property simplifies analysis and modeling, as it allows the use of time-independent transition matrices or kernels to describe the evolution of the process.
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Martingale Theory is a fundamental concept in probability theory and financial mathematics, describing a stochastic process where the conditional expectation of future values, given past and present values, is equal to the present value. It is crucial for modeling fair games and is widely used in financial markets to assess the fairness and efficiency of pricing strategies.
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A submartingale is a type of stochastic process where the expected future value, given all past information, is at least as large as the present value, highlighting a non-decreasing trend over time. This concept is crucial in financial mathematics and probability theory, as it models fair games and processes with a potential upward drift.
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Levy flight is a random walk in which the step lengths have a probability distribution that is heavy-tailed, often used to model the foraging patterns of animals and the movement of particles in turbulent fluids. This concept is significant in various fields due to its ability to describe processes that exhibit anomalous diffusion, where the mean squared displacement grows faster than linearly with time.
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Fractional Brownian Motion (fBm) is a generalization of classical Brownian motion that incorporates memory and self-similarity, characterized by the Hurst parameter, H, which dictates the roughness of the path. Unlike standard Brownian motion, fBm is neither a semimartingale nor has independent increments, making it useful in modeling phenomena with long-range dependence in fields like finance and telecommunications.
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A non-decreasing process is a stochastic process where the value of the process does not decrease over time, meaning it either stays the same or increases. This property is crucial in fields like finance and queueing theory, where it models phenomena such as stock prices or cumulative arrivals that naturally do not decrease.
The generator of a Markov process is a linear operator that describes the infinitesimal evolution of the process, providing a bridge between the process and its differential equations. It plays a crucial role in understanding the long-term behavior and the transition dynamics of continuous-time Markov processes.
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The infinitesimal generator is a crucial operator in the study of stochastic processes, particularly in the context of Markov processes, where it provides a way to describe the instantaneous rate of change of the process. It serves as a bridge between the probabilistic dynamics of the process and the analytical framework, often used in deriving partial differential equations that characterize the process's evolution.
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The Kolmogorov backward equation is a fundamental partial differential equation used to describe the time evolution of transition probabilities in continuous-time Markov processes. It is instrumental in various fields, including finance and physics, for modeling stochastic systems where future states depend only on the current state and not on the path taken to arrive there.
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A Lévy process is a stochastic process with stationary and independent increments, often used to model random phenomena with jumps or discontinuities, such as stock prices or natural events. It generalizes the Wiener process by allowing for jumps, making it a versatile tool in fields like finance, physics, and engineering.
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A transition semigroup is a family of operators that describe the evolution of probability distributions over time in a Markov process, providing a powerful framework for analyzing stochastic processes. These semigroups capture the dynamics of the system by mapping initial distributions to their future states, and are fundamental in the study of continuous-time Markov processes and their generators.
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A jump process is a type of stochastic process that incorporates sudden changes, or 'jumps', in value at random times, making it useful for modeling phenomena with abrupt shifts. It is widely used in fields like finance to model stock prices and insurance to assess risk, where continuous models like Brownian motion fall short.
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A Wiener Process, also known as Brownian motion, is a continuous-time stochastic process that serves as a mathematical model for random movement, often used in finance to model stock prices. It is characterized by having independent, normally distributed increments and continuous paths, making it a fundamental building block for stochastic calculus and the modeling of various random phenomena.
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Monte Carlo Simulation is a computational technique that uses random sampling to estimate complex mathematical models and assess the impact of risk and uncertainty in forecasting models. It is widely used in fields such as finance, engineering, and project management to model scenarios and predict outcomes where analytical solutions are difficult or impossible to derive.
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Random Search is a hyperparameter optimization technique that involves randomly sampling from the hyperparameter space and evaluating performance, offering a simple yet effective approach for exploring large search spaces. It can often find good solutions faster than grid search by not being constrained to a fixed search pattern, making it particularly useful when dealing with high-dimensional spaces or when computational resources are limited.
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A birth-death process is a specific type of continuous-time Markov chain that models systems where transitions occur between states representing the 'birth' and 'death' of entities. It is widely used in queueing theory, population dynamics, and other fields to describe systems where the future state depends only on the current state and not on the sequence of events that preceded it.
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Random Walk Theory suggests that stock market prices evolve according to a random walk and thus cannot be predicted based on past movements. This implies that the market is efficient, and any attempt to outperform it through analysis or timing is futile, as all known information is already reflected in stock prices.
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A martingale process is a stochastic process where the conditional expectation of the next value, given all prior values, is equal to the present value, implying no predictable trend over time. It is often used in financial modeling to represent fair games or market prices, where future movements are independent of past behavior.
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