Stochastic calculus extends integration and differential calculus to stochastic processes such as Brownian motion and semimartingales.
A semimartingale is the sum of a local martingale and an adapted finite-variation process. This is the broad class of integrators for which the Itô stochastic integral is defined.
A process has finite variation on compact intervals when every sample path has finite total variation of a function there. Such a process can be integrated pathwise by the Lebesgue-Stieltjes integral.
A semimartingale has a decomposition into a local martingale and an adapted finite-variation process . Under standard normalizations the decomposition is unique.
A continuous semimartingale has a decomposition in which both the local-martingale and finite-variation parts are continuous.
The quadratic variation of a continuous semimartingale is the limit in probabilityFinite-variation processes have zero quadratic variation, while a Brownian motion satisfies .
The stochastic integral integrates a predictable process against a semimartingale. It extends pathwise integration against finite-variation processes and the Itô integral against local martingales.
Whenever the integrands are admissible,
For an Itô process and a twice differentiable function ,The second-order term reflects the nonzero quadratic variation of Brownian motion.
Local time measures how intensely a semimartingale visits a level. For a continuous semimartingale it appears as the increasing correction term in the Tanaka formula.
A stochastic differential equation specifies infinitesimal drift and random diffusion through a stochastic integral equation.
A weak solution may choose its probability space and driving Brownian motion as part of the solution. It is weaker than a strong solution, which must be adapted to a prescribed Brownian motion.
For a differential operator , the martingale problem asks for a process such thatis a local martingale for every test function in a suitable domain.
If a continuous local martingale satisfiesthen its stochastic exponential is a true martingale through time .
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Stochastic calculus is a branch of mathematics that deals with processes that involve randomness or uncertainty. It extends classical calculus to include stochastic processes, which are mathematical objects that evolve over time in a probabilistic manner. Stochastic calculus is particularly useful in fields such as finance, economics, physics, and engineering, where systems are influenced by random factors. Key concepts and components of stochastic calculus include: 1. **Stochastic Processes**: These are mathematical objects that describe a collection of random variables indexed by time.