Mathematics / advanced
Diffusions and stochastic differential equations
Construct stochastic evolution through Ito integration, existence theory, generators, transition laws, and changes of measure.
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Module 01 advanced
Diffusions and stochastic differential equations
Construct stochastic evolution through Ito integration, existence theory, generators, transition laws, and changes of measure.
- 01
Brownian paths and quadratic variation
Specify the filtration and convergence mode behind Brownian roughness and its quadratic clock.
28 min - 02
Itô integral construction and isometry
Complete predictable step integrands in mean square and retain the information restriction.
29 min - 03
Itô formula and semimartingales
Derive the curvature correction and state the local integrability behind the differential shorthand.
30 min - 04
Strong and weak solutions
Separate a solution built from given noise from a solution specified jointly with its probability space.
30 min - 05
Lipschitz existence and pathwise uniqueness
Use Picard iteration and stopped moment estimates without confusing local solvability with nonexplosion.
31 min - 06
Diffusion generators and martingale problems
Read local stochastic dynamics through test functions while keeping the operator domain explicit.
32 min - 07
Kolmogorov backward and forward equations
Distinguish evolution of observables from evolution of laws and qualify every density claim.
32 min - 08
Feynman–Kac representations
Represent bounded PDE solutions by weighted paths while checking integrability, killing, and boundary data.
33 min - 09
Girsanov theorem and drift changes
Choose a valid density martingale and track the Brownian and drift signs under the new measure.
33 min - 10
Stochastic flows and boundary classification
Track initial-state dependence with shared noise and classify one-dimensional endpoints using fixed scale and speed conventions.
36 min