PIDEProblem(
    μ,
    σ,
    x0,
    tspan,
    g,
    f;
    p,
    x0_sample,
    neumann_bc,
    kw...
)

Defines a Partial Integro Differential Problem, of the form

\[\begin{aligned} \frac{du}{dt} &= \tfrac{1}{2} \text{Tr}(\sigma \sigma^T) \Delta u(x, t) + \mu \nabla u(x, t) \\ &\quad + \int f(x, y, u(x, t), u(y, t), ( \nabla_x u )(x, t), ( \nabla_x u )(y, t), p, t) dy, \end{aligned}\]

with $u(x,0) = g(x)$.

Arguments

• g : initial condition, of the form g(x, p, t).
• f : nonlinear function, of the form f(x, y, u(x, t), u(y, t), ∇u(x, t), ∇u(y, t), p, t).
• μ : drift function, of the form μ(x, p, t).
• σ : diffusion function σ(x, p, t).
• x: point where u(x,t) is approximated. Is required even in the case where x0_sample is provided. Determines the dimensionality of the PDE.
• tspan: timespan of the problem.
• p: the parameter vector.
• x0_sample : sampling method for x0. Can be UniformSampling(a,b), NormalSampling(σ_sampling, shifted), or NoSampling (by default). If NoSampling, only solution at the single point x is evaluated.
• neumann_bc: if provided, Neumann boundary conditions on the hypercube neumann_bc[1] × neumann_bc[2].

ParabolicPDEProblem(
    μ,
    σ,
    x0,
    tspan;
    g,
    f,
    p,
    xspan,
    x0_sample,
    neumann_bc,
    payoff,
    kw...
)

Defines a Parabolic Partial Differential Equation of the form:

\[\begin{aligned} \frac{du}{dt} &= \tfrac{1}{2} \text{Tr}(\sigma \sigma^T) \Delta u(x, t) + \mu \nabla u(x, t) \\ &\quad + f(x, u(x, t), ( \nabla_x u )(x, t), p, t) \end{aligned}\]

• Semilinear Parabolic Partial Differential Equation
  • f -> f(X, u, σᵀ∇u, p, t)
• Kolmogorov Differential Equation
  • f -> nothing
  • x0 -> nothing, xspan must be provided.
• Obstacle Partial Differential Equation
  • f -> nothing
  • g -> nothing
  • discounted payoff function provided.

Arguments

• μ : drift function, of the form μ(x, p, t).
• σ : diffusion function σ(x, p, t).
• x: point where u(x,t) is approximated. Is required even in the case where x0_sample is provided. Determines the dimensionality of the PDE.
• tspan: timespan of the problem.
• g : initial condition, of the form g(x, p, t).
• f : nonlinear function, of the form f(X, u, σᵀ∇u, p, t)

Optional Arguments

• p: the parameter vector.
• x0_sample : sampling method for x0. Can be UniformSampling(a,b), NormalSampling(σ_sampling, shifted), or NoSampling (by default). If NoSampling, only solution at the single point x is evaluated.
• neumann_bc: if provided, Neumann boundary conditions on the hypercube neumann_bc[1] × neumann_bc[2].
• xspan: The domain of the independent variable x
• payoff: The discounted payoff function. Required when solving for optimal stopping problem (Obstacle PDEs).