Monte-Carlo algorithms for a forward Feynman–Kac-type representation for semilinear nonconservative partial differential equations

Proof of Lemma 3.5.

Let us fix ε > 0 , N ∈ ℕ ⋆ and t ∈ [ 0 , T ] . We first recall that for almost all x ∈ ℝ d ,

for which V ¯ t is given by (3.7). Let us fix i ∈ { 1 , … , N } .

Proof of (3.12). We only give details for the proof of the first inequality since the second one can be established through similar arguments.

From equation (A.1) we have

where for the second step above we have used the fact that K is in particular Lipschitz. The same arguments lead also to

which concludes the proof of (3.12).

Proof of (3.13). From

we deduce, for almost all x ∈ ℝ d ,

Since K is Lipschitz with related constant L K = ∥ ∇ ⁡ K ∥ ∞ , for almost all x ∈ ℝ d we obtain

where the second term in (A.3) comes from inequality (2.3). Since Λ is bounded, taking the supremum with respect to x and the expectation on both sides of inequality (A.3), we have

where we have used the fact that

since Φ and g are bounded.

The bound of 𝔼 ⁢ [ ∥ ∇ ⁡ u ¯ t ε , N - ∇ ⁡ u ¯ r ⁢ ( t ) ε , N ∥ ∞ ] is obtained by proceeding exactly in the same way as above, starting with

instead of (A.2), where x ℓ denotes the ℓ -th coordinate of x ∈ ℝ d . Then

follows. ∎