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The possibility of different interpretations of the stochastic term (orcalculi) in the overdamped Langevin equation for the motion of a particle in aninhomogeneous medium is often referred to as the "Ito--Stratonovich dilemma,"although there is, in fact, a continuum of choices. We introduce two MonteCarlo (MC) simulation approaches for studying such systems, with bothapproaches giving the choice between different calculi (in particular, Ito,Stratonovich, and "isothermal"). To demonstrate these approaches, we study thediffusion on a 1D interval of a particle released at an interface (in themiddle of the system) between two media where this particle has differentdiffusivities (e.g., two fluids with different viscosities). For reflectingboundary conditions at the ends of the interval, a discontinuity at theinterface in the stationary-state particle distribution is found, except forthe isothermal case, as expected. We also study the first-passage problem usingabsorbing boundary conditions. Good agreement is found when comparing the MCapproaches to theory as well as Brownian and Langevin dynamics simulations.Also, the results themselves turn out to be interesting. For instance: 1) forsome calculi, there are more particles on the low-viscosity side (LVS) atearlier times and then fewer particles at later times; 2) there is nopreference to end up on a particular wall for the Ito variant, but a biastowards the wall on the LVS in all other cases; 3) the mean first-passage timeto the wall on the LVS grows as the viscosity on the other side is increased,except for the isothermal case where it approaches a constant; 4) when theviscosity ratio is high, the first-passage-time distribution for the wall onthe LVS is much broader than for the other wall, with a power-law dependence ofthe former in a certain time interval whose exponent depends on the calculus.

Monte Carlo Approaches for Simulating a Particle at a Diffusivity Interface and the "Ito--Stratonovich Dilemma"