Kolmogorov backward equation
E48986
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
Aliases (1)
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Kolmogorov equation
→
equation in stochastic processes → partial differential equation → |
| appliesTo |
Itô diffusion
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Markov processes → diffusion processes → |
| associatedWith |
Markov semigroup
→
transition function of a Markov process → |
| characterizes |
evolution of conditional expectations
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|
| complements |
Fokker–Planck equation
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Kolmogorov forward equation → |
| contrastedWith |
forward equation for probability density
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|
| describes |
time evolution of expected values of functionals of Markov processes
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|
| equivalentTo |
backward Fokker–Planck equation
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|
| field |
mathematical physics
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probability theory → stochastic analysis → stochastic processes → |
| hasComponent |
first-order spatial derivative terms
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second-order spatial derivative terms → time derivative term → |
| hasForm |
∂u/∂t + Lu = 0
→
|
| historicalPeriod |
20th century
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|
| involves |
boundary conditions
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diffusion coefficient → drift coefficient of the diffusion → terminal condition → |
| mathematicalNature |
linear partial differential equation
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|
| namedAfter |
Andrey Kolmogorov
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|
| relatedTo |
Dynkin formula
→
Itô calculus → generator of a Markov process → infinitesimal generator of a diffusion → parabolic partial differential equation → semigroup of operators → stochastic differential equation → |
| solutionMethod |
probabilistic representation via Feynman–Kac formula
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|
| solutionType |
value function of a stochastic process
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|
| timeDirection |
backward in time
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|
| usedFor |
characterizing transition probabilities of Markov processes
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computing conditional expectations of functionals of stochastic processes → optimal control of stochastic systems → pricing of derivative securities in mathematical finance → |
| usedIn |
chemical reaction kinetics
→
epidemiological modeling → neuroscience modeling of membrane potentials → population dynamics modeling → queueing theory → reliability theory → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Fokker–Planck equation
→
Markov process → |
relatedTo |
|
Andrei Kolmogorov
("Kolmogorov equations")
→
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notableWork |