Stochastic Integral

ProbabilityTheoryStochasticCalculus

1. Finite Variation Process integrators (Note: this is really a Lebesgue-Stieltjes Integral, defined pathwise)
   • Integrand: a Progressive Process
   • Integrator: a Finite variation process
   • Where (if this only holds a.s. then on with the complete filtration on the measure zero set where the condition does not hold will be progressive)where the defined process has Finite Variation Function sample paths.
2. Martingale bounded in L2 integrators,
   • Integrand: an Elementary Process
   • Integrator: a Martingale bounded in where the defined
   • process is the unique Martingale bounded in s.t. note that the LHS has a stochastic integral, whereas the RHS has a Lebesgue-Stieltjes Integral
   • For any stopping time
3. Martingale bounded in integrators, martingale integrads
   • Integrand:
   • Integrator: a Martingale bounded in
   • By density of Elementary Processes in , can be extended to
   • For and , : where the last equality is the Itô isometry