The NNParamKolmogorov algorithm
Problems Supported:
NNParamKolmogorov obtains a
- terminal solution for parametric families of Forward Kolmogorov Equations of the form:
\[\partial_t u(t,x) = \mu(t, x, γ_mu) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x, γ_sigma) \Delta_x u(t,x)\]
with initial condition given by g(x, γ_phi)
- or an initial condition for parametric families of Backward Kolmogorov Equations of the form:
\[\partial_t u(t,x) = - \mu(t, x) \nabla_x u(t,x) - \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x)\]
with terminal condition given by g(x, γ_phi)
We can use the Feynman-Kac formula :
\[S_t^x = \int_{0}^{t}\mu(S_s^x)ds + \int_{0}^{t}\sigma(S_s^x)dB_s\]
And the solution is given by:
\[f(T, x) = \mathbb{E}[g(S_T^x, γ_phi)]\]