The Itô integral is defined as the limit of a sum where time is divided into small intervals. In each interval, the value of the stochastic process at the beginning of the interval is multiplied by the increment of the process over that interval.
Conditions: The integrator is a martingale, typically Brownian motion.; The integrand must be adapted to the filtration of the integrator.
The Itô integral is defined as the limit of a sum where time is divided into small intervals. In each interval, the value of the stochastic process at the beginning of the interval is multiplied by the increment of the process over that interval.
Conditions: The integrator is a martingale, typically Brownian motion.; The integrand must be adapted to the filtration of the integrator.
Classical integration sums smooth and predictable quantities, like accumulating steady bond interest, and is visually represented by the area under a smooth curve. Stochastic integration sums highly irregular, random movements, such as wild stock price jumps, and is visually depicted as summing areas under a jagged, erratic path.
Conditions: Classical integration applies to deterministic, smooth functions.; Stochastic integration applies to processes with non-smooth, random paths.
Classical integration sums smooth and predictable quantities, like accumulating steady bond interest, and is visually represented by the area under a smooth curve. Stochastic integration sums highly irregular, random movements, such as wild stock price jumps, and is visually depicted as summing areas under a jagged, erratic path.
Conditions: Classical integration applies to deterministic, smooth functions.; Stochastic integration applies to processes with non-smooth, random paths.