Time discretization in the Feynman-Kac formula with boundary conditions

I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.

For the SDE simulation I use the Euler-approximation, which introduces a time-step $$h$$ for the Brownian Motion, and since the PDE has some boundary conditions, we need to compute its hitting time $$\tau$$.

Numerically speaking, I start with $$\tau = 0$$ and increase it progressively.

So at each step:

• the process position is increased by a normal depending (also) on $$h$$ (the SDE);
• the variable $$\tau$$ is increased by some $$k$$;
• the hitting condition is checked.

My question is: is there a specific connection between $$h$$ and $$k$$? Below my reasoning.

EDIT: I am adding more details with the hope of expressing better my question.

Consider the 1-dimensional heat on $$[0,1]$$, $$u_t = u_{xx}$$ with zero boundary conditions. So $$u_0$$ a known initial state, and $$v(t,0)=v(t,1)=0$$ for each time. Suppose we want to compute the solution at time $$T=0.05$$.

The stopping time to insert into Feynman-Kac expectation is:

$$\tau = \inf \{ k \geq 0 : T-k \leq 0,$$ or $$X_s\geq 1$$, or $$X_s \leq 0 \}$$

where with $$X_t$$ solves $$dX_t = \sqrt{2} {dB}_t$$.

I approximate the SDE say with a time-step $$h=0.01$$, so if we start from $$x \in [0,1]$$ (point on which we are going to compute the PDE solution) and set $$k=0$$ (hitting time approximation), at every step we:

1. move by adding a normal $$N(0, 2h)$$
2. increase $$k$$ by $$\sqrt{h}$$

Again: when (1) goes to 0 or 1, or $$T - k$$ goes to 0, the hit is considered done.

Now, (1) works great: the process take its time for hitting the boundary. Conversely, since $$\sqrt(0.01)=0.1$$, $$T - \sqrt{h} = 0.05-0.1 = -0.05$$ so the boundary is immediately hit!

As a result, in the simulation the boundary is hit always after one step and I obtain just a small shift of my initial condition $$u_0$$.

If, conversely, in (2) I subtract something like $$h^{1.8}$$, rather than $$\sqrt{h}$$, the results are pretty nice...

For a random walk with position $$x_{k+1}=x_k+\delta x_k$$, incremented by a normally distributed stochastic variable $$\delta x_k$$ with variance $$h$$, the simplest hitting time estimate is $$\tau=Kh$$ with $$K$$ the smallest $$k$$ such that $$x_k$$ is outside of the boundary. This is an overestimate, because it is possible that both $$x_k$$ and $$x_{k+1}$$ are inside of the boundary, and yet the Brownian motion crossed the boundary between times $$kh$$ and $$(k+1)h$$. Buchmann constructs interpolants that test for this intermediate excursion.