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Numerical Simulation of a Density SPDE with Path-Dependent Coefficients

Article Quant Q&A · Author: Leoncino
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Summary

The document sets out a numerical problem for a density process on positive values. Given a Brownian path at finite precision, it describes a deterministic partial differential equation whose drift, diffusion, and Brownian-driven transport terms depend on a recursively updated quantity derived from the density's integral. The initial state is specified as a point mass, and the author seeks an approach or R package for simulation.

Its main contribution is the mathematical setup and the identification of a coupling challenge: the coefficients at each step depend on the prior step's integral. It gives no discretization scheme, package recommendation, numerical results, or validation. The question therefore highlights an implementation problem rather than providing a worked method. Any practical solution would need to address representing the point-mass initial condition, spatial and temporal discretization, and stability, but these details are not covered in the document.

Key ideas

  • The density evolves according to a PDE conditioned on a supplied Brownian trajectory.
  • The drift and diffusion coefficients depend on an integral-based quantity from the previous time step.
  • The initial density is specified as a point mass on the positive domain.
  • The document asks for R-based numerical guidance but provides no solver or numerical scheme.

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# Numerical approximation of SPDE


# Numerical approximation of SPDE












I've already posted this question on MSE, but I'm not quite sure if it's the right community so I'm posting it here as well.

Background I want to approximate an SPDE of adensity process $V_t$. The Brownian motion part is given (up to a certain precision level $\epsilon>0$) by a brownian trajectory $t\mapsto w_t$. The starting point being $V_0(x)=\delta_{x_0}(x)$ (i.e. $V_0(x)=1$ if $x=x_0$) and $V^{(n)}=u_{n\epsilon}$, (considering that we have a Brownian path) I'm given the following deterministic PDE

$$ du_t(x)=-\mu(t,x,L^{(n-1)})\partial_x u_t(x)dt + \frac{1}{2}\sigma(t,x)\rho(t,L^{(n-1)})\partial_{xx}u_t(x)dt\\ -\sigma(t,x)\sqrt{1-\rho(t,L^{(n-1)})^2}\partial_x u_t(x)dw_t, $$ and $L^{(n)}=1-\int_0^\infty V^{(n)}(x)dx$, for $x>0$.

Problem I'm in need of an approach to simulate this in R. I already looked up the package Sim.DiffProc and also others like simecol or the general packages for solving PDEs and ODEs didn't really help me (I don't have that much experience in simulating this).

Question I kindly ask you to either provide me with information on how to approach solving the (determinisitc) PDE in R or with some advice on which package I could use (considering that the parameters $\sigma, \mu, \rho$ are dependent on $L^{(n-1)}$.

Thank you in advance.

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