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Stochastic calculus is a branch of mathematics that deals with stochastic processes, which are random processes that evolve over time. It is used to model and analyze systems with uncertainty, such as financial markets, biological systems, and physical systems.
Stochastic calculus has a wide range of applications in various fields, including finance, physics, biology, and engineering. It is used to model and analyze complex systems with random fluctuations, and to make predictions and decisions based on probabilistic outcomes.
Stochastic calculus differs from traditional calculus in that it deals with random variables and processes, while traditional calculus deals with deterministic functions. Stochastic calculus also takes into account the uncertainty and randomness in a system, while traditional calculus assumes a predictable and stable environment.
The main tools used in stochastic calculus include Brownian motion, Itô's lemma, and stochastic differential equations. These tools allow for the analysis and modeling of random processes and the calculation of probabilities and expectations in such systems.
Stochastic calculus has many real-world applications, such as pricing financial derivatives, modeling stock prices and interest rates, predicting weather patterns, and analyzing biological systems. It is also used in engineering for modeling and optimizing systems with random fluctuations, such as in telecommunications and control systems.
