Strong convergence in infinite time interval of tamed-adaptive Euler–Maruyama scheme for Lévy-driven SDEs with irregular coefficients

Abstract

A tamed-adaptive Euler–Maruyama approximation scheme is proposed for Lévy-driven stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. The new scheme converges in both finite and infinite time intervals under some suitable conditions on the regularity and the growth of the coefficients.

Appendix

In this appendix, we will prove the theorem Theorem 2.1. Without loss of generality, we assume that $\gamma$ is a positive constant.

1.1 Existence of solution

For each $N>0$, set

$$\begin{aligned} b_{N}(x)=\left\{ \begin{array}{ll} b(x), &{} \text{ if } |x| \le N, \\ b\left( \dfrac{N x}{|x|}\right) (N+1-|x|), &{} \text{ if } N<|x|<N+1, \\ 0, &{} \text{ if } |x| \ge N+1, \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \sigma _{N}(x)=\left\{ \begin{array}{ll} \sigma (x), &{} \text{ if } |x| \le N, \\ \sigma \left( \dfrac{N x}{|x|}\right) (N+1-|x|), &{} \text{ if } N<|x|<N+1, \\ 0, &{} \text{ if } |x| \ge N+1, \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} c_{N}(x)=\left\{ \begin{array}{ll} c(x), &{} \text{ if } |x| \le N, \\ c\left( \dfrac{N x}{|x|}\right) (N+1-|x|), &{} \text{ if } N<|x|<N+1, \\ 0, &{} \text{ if } |x| \ge N+1. \end{array}\right. \end{aligned}$$

It is clearly that $b_{N}$, $c_{N}$ and $\sigma _{N}$ satisfy Assumptions of Theorem 2.2 in Li and Mytnik (2011). Thus, the equation

$$\begin{aligned} X^N_t=x_{0}+\int _{0}^{t} b_{N}\left( X^N_s\right) \mathrm{d} s+\int _{0}^{t} \sigma _{N}\left( X^N_s\right) \mathrm{d} W_{s} + \int _0^t\int _{\mathbb {R}_0} c_N\left( X^N_{s-}\right) z{\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) \end{aligned}$$

(49)

has a unique strong solution $X^N_t$. We will show that when $N \rightarrow \infty , X^{N}_t$ converges in probability to a process $X_t$ which satisfies Eq. (1).

For each $N>0$, set

$$\begin{aligned} \tau _N=T \wedge \inf \left\{ t \in [0 ; T]:\left| X^N_t\right| \ge N\right\} . \end{aligned}$$

Due to the pathwise uniqueness of solution to Eq. (49), $X^N_t=X^{M}_t$ almost surely for any $t<\tau _N$ and $N<M .$ Then, we will show that $\tau _N=T$ almost surely for all N large enough.

It is straightforward to show that the coefficients $b_{N}(x)$ , $c_{N}(x)$ and $\sigma _{N}(x)$ satisfy Condition C2’:

$$\begin{aligned}&p_0 xb(x) + \dfrac{p_0(p_0-1)}{2} \sigma ^2(x) +\dfrac{c^2(x)}{4L_0^2} \int _{\mathbb {R}_0} \left( (1+2L_0|z|)^{p_0}-1-2p_0L_0|z|\right) \nu (\mathrm{d}z) \\&\le 2\gamma |x|^2+2\eta , \end{aligned}$$

for any $x \in \mathbb {R}$.

Secondly, for any $p \in (1;2)$, using (55) to get

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le T}\left| X^N_t\right| ^{p}\right] \le C \quad \text{ for } \text{ any } N>0. \end{aligned}$$

Thus,

$$\begin{aligned} C \ge \mathbb {E}\left[ \sup _{0 \le t \le T}\left| X^N_t\right| ^{p}\right] \ge \mathbb {E}\left[ \sup _{0 \le t \le T}\left| X^N_t\right| ^{p} \mathbb {I}_{\left[ \tau _N<T\right] }\right] \ge N^{p} \mathbb {P}\left[ \tau _N<T\right] . \end{aligned}$$

It leads to $\sum _{N=1}^{\infty } \mathbb {P}\left( \tau _N<T\right) <\infty$. Thanks to Borel–Cantelli’s lemma, we obtain

$$\begin{aligned} \mathbb {P}\left[ \limsup _{N}\left\{ \tau _N<T\right\} \right] =0. \end{aligned}$$

Since $\left( \tau _N\right) _{N}$ is increasing, $\tau _N=T$ for all N large enough. It means $\lim _{N \rightarrow \infty } X^N_t=X_t$ exists almost surely and $X_t=X^{M}_t$ almost surely for any $t<\tau _N$ and $M \ge N$. On the other hand, for any $\kappa >0$, $k>1$, $2<q \le p_0$,

$$\begin{aligned} \mathbb {E}\left[ \left| X^{N+k}_{t \wedge \tau _{N+k}} - X^{N}_{t \wedge \tau _N} \right| ^{2}\right]&\le 2\mathbb {E}\left[ \left( \left| X^{N+k}_t \right| ^{2}+\left| X^N_t\right| ^{2}\right) \mathbb {I}_{\left\{ \tau _N<T\right\} }\right] \\&\le C_{q}\left( \kappa \mathbb {E}\left[ \left| X^{N+k}_t \right| ^{q}\right] +\kappa \mathbb {E}\left[ \left| X^N_t\right| ^{q}\right] +\frac{\mathbb {P}\left[ \tau _N<T\right] }{\kappa ^{2/(q-2)}}\right) . \end{aligned}$$

First, let $N \rightarrow \infty$ and then let $\kappa \rightarrow 0$, we get

$$\begin{aligned} \mathbb {E}\left[ \left| X^{N+M}_{t \wedge \tau _{N+M}} -X^{N}_{t \wedge \tau _N} \right| ^{2}\right] \rightarrow 0 \quad \text{ as } N \rightarrow \infty . \end{aligned}$$

(50)

It means that $(X^N_{t \wedge \tau _N})_{N\ge 1}$ is a Cauchy sequence in $L^2$ space. This implies

$$\begin{aligned} X^{N}_{t \wedge \tau _N} {\mathop {\longrightarrow }\limits ^{L^{2}}} X_t \quad \text{ as } N \rightarrow \infty . \end{aligned}$$

Furthermore, for any $p \in (0;p_0]$, since coefficients $b_N,c_N$ and $\sigma _N$ satisfy Condition C2’, there exists a constant $C_{p}>0$ such that

$$\begin{aligned} \sup _{0 \le t \le T} \mathbb {E}\left[ |X^N_t|^{p}\right] \le C_{p}. \end{aligned}$$

Thanks to Fatou’s lemma, there exists a constant $C_{p}>0$ such that

$$\begin{aligned} \sup _{0 \le t \le T} \mathbb {E}\left[ |X_t|^{p}\right] \le C_{p}. \end{aligned}$$

From the definition of $b_{N}(x)$, we have

$$\begin{aligned} \mathbb {E}\left[ \left| \int _{0}^{t \wedge \tau _N}\left[ b_{N}\left( X^N_s\right) -b( X_s)\right] \mathrm{d} s\right| ^{2}\right] =0. \end{aligned}$$

Moreover, using the fact that $|b(x)| \le C\left( 1+|x|^{\ell }\right)$ for all $x \in \mathbb {R}$ and $p_0 \ge 4l$,

$$\begin{aligned} \mathbb {E}\left[ \left| \int _{t \wedge \tau _N}^{t} b(X_s) d s\right| ^{2}\right]&\le C \int _{0}^{t} \mathbb {E}\left[ \left( 1+|X_s|^{2 \ell }\right) \mathbb {I}_{\left\{ \tau _N \le s\right\} }\right] \mathrm{d} s \\&\le \kappa C \int _{0}^{T} \mathbb {E}\left[ 1+|X_s|^{2 \ell }\right] ^{2} \mathrm{d} s+\frac{C \mathbb {P}\left[ \tau _N<T\right] }{\kappa } \\&\le \kappa C+\frac{C \mathbb {P}\left[ \tau _N<T\right] }{\kappa } . \end{aligned}$$

Let $N \rightarrow \infty$ and $\kappa \rightarrow 0$, we have

$$\begin{aligned} \int _{0}^{t \wedge \tau _N} b_{N}\left( X^N_s\right) \mathrm{d} s {\mathop {\longrightarrow }\limits ^{L^{2}}} \int _{0}^{t} b(X_s) \mathrm{d} s \quad \text{ as } N \rightarrow \infty . \end{aligned}$$

(51)

In the same manner, we can see that

$$\begin{aligned} \int _{0}^{t \wedge \tau _N} \sigma _{N}\left( X^N_s\right) \mathrm{d} W_{s} {\mathop {\longrightarrow }\limits ^{L^{2}}} \int _{0}^{t} \sigma (X_s) \mathrm{d} W_{s} \quad \text{ as } N \rightarrow \infty , \end{aligned}$$

(52)

and, by $\int _{\mathbb {R}_{0}} z^{2} \nu (\mathrm{d}z)<\infty$,

$$\begin{aligned} \int _0^{t \wedge \tau _N} \int _{\mathbb {R}_0} c_N\left( X^N_{s-}\right) z{\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) {\mathop {\longrightarrow }\limits ^{L^{2}}} \int _0^t\int _{\mathbb {R}_0} c\left( X_{s-}\right) z{\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) \quad \text{ as } N \rightarrow \infty . \end{aligned}$$

(53)

By combining (49), (50), (51), (52) and (53), we get

$$\begin{aligned} X_{t}=x_{0}+\int _{0}^{t} b(X_s) \mathrm{d} s+\int _{0}^{t} \sigma (X_s) \mathrm{d} W_{s}+\int _0^t\int _{\mathbb {R}_0} c\left( X_{s-}\right) z{\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) \end{aligned}$$

almost surely for all $t \in [0, T] .$ This shows that $\left( X_{t}\right) _{t \in [0, T]}$ is a solution of Eq. (1). The proof is complete.

1.2 Pathwise uniqueness

Suppose that Eq. (1) has a solution $(X_t)_{0 \le t \le T}$, and $(X'_{t})_{0 \le t \le T}$ is another solution of Eq. (1). It follows from the proof of Proposition 2.3 that the sample paths of $(X_t)_{0\le t \le T}$ and $(X'_t)_{0\le t \le T}$ do not explode. We are going to show that $\mathbb {E}\left[ |X_{t}-X'_{t}|\right] =0$ for all $t \in [0, T]$, which implies the uniqueness of solution. For each $N>0$, let $\tau _N= T \wedge \inf \left\{ t \ge 0:\left| X_{t}\right| \vee \left| X_{t}^{\prime }\right| \ge N\right\}$.

Firstly, applying Itô’s formula for $X^2_t$ and Condition C2 with $p_0=2$, we have

$$\begin{aligned} X_{t}^{2}&=x_{0}^{2}+\int _{0}^{t}\left( 2 X_{s} b\left( X_{s}\right) +\sigma ^{2}\left( X_{s}\right) \right) \mathrm{d} s+\int _{0}^{t} 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s} \\&\quad +\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( \left( X_{s}+c\left( X_{s}\right) z\right) ^{2}-X_{s}^{2}-2 X_{s} c\left( X_{s}\right) z\right) \nu (\mathrm{d} z) \mathrm{d} s \\&\quad +\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( \left( X_{s-}+c\left( X_{s-}\right) z\right) ^{2}-X_{s-}^{2}\right) {\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) \\&=x_{0}^{2}+\int _{0}^{t}\left( 2 X_{s} b\left( X_{s}\right) +\sigma ^{2}\left( X_{s}\right) \right) \mathrm{d} s+\int _{0}^{t} 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s} \\&\quad +\int _{0}^{t} \int _{\mathbb {R}_0} c^{2}\left( X_{s}\right) z^{2} \nu \left( \mathrm{d}z\right) \mathrm{d} s+\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( 2 X_{s-} c\left( X_{s-}\right) z+c^{2}\left( X_{s-}\right) z^{2}\right) {\widetilde{N}}\left( \mathrm{d} s, \mathrm{d} z\right) \\&=x_{0}^{2}+\int _{0}^{t}\left( 2 X_{s} b\left( X_{s}\right) +\sigma ^{2}\left( X_{s}\right) +c^{2}\left( X_{s}\right) \int _{\mathbb {R}_{0}} z^{2} \nu (\mathrm{d} z)\right) \mathrm{d} s \\&\quad +\int _{0}^{t} 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s}+\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( 2 X_{s-} c\left( X_{s-}\right) {z}+c^{2}\left( X_{s-}\right) z^{2}\right) {\widetilde{N}}(\mathrm{d} s, \mathrm{d} z) \\&\le x_{0}^{2}+2\int _{0}^t \left( \gamma \left| X_{s}\right| ^{2}+\eta \right) \mathrm{d} s +\int _{0}^t 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s} \\&\quad +\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( 2 X_{s-} c\left( X_{s-}\right) z+c^{2}\left( X_{s-}\right) z^{2}\right) {\widetilde{N}}(\mathrm{d} s, \mathrm{d} z), \end{aligned}$$

which implies

$$\begin{aligned} X_{t \wedge \tau _N}^{2}&\le x_{0}^{2}+2\int _{0}^{t \wedge \tau _N} \left( \gamma \left| X_{s}\right| ^{2}+\eta \right) \mathrm{d} s +\int _{0}^{t \wedge \tau _N} 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s} \\&\quad +\int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left( 2 X_{s-} c\left( X_{s-}\right) z+c^{2}\left( X_{s-}\right) z^{2}\right) {\widetilde{N}}(\mathrm{d} s, \mathrm{d} z). \end{aligned}$$

For any stopping time $\tau \le T$, we get

$$\begin{aligned} {X_{\tau \wedge \tau _N}^2}&\le x_{0}^{2}+2\int _{0}^{ {\tau \wedge \tau _N}} \left( \gamma \left| X_{s}\right| ^{2}+\eta \right) \mathrm{d} s +\int _{0}^{{\tau \wedge \tau _N}} 2 X_{s} \sigma \left( X_{s}\right) \mathrm{d} W_{s} \nonumber \\&\quad +\int _{0}^{{\tau \wedge \tau _N}} \int _{\mathbb {R}_{0}}\left( 2 X_{s-} c\left( X_{s-}\right) z+c^{2}\left( X_{s-}\right) z^{2}\right) {\widetilde{N}}(\mathrm{d} s, \mathrm{d} z). \end{aligned}$$

(54)

Taking expectation on both sides (54) and using Proposition 2.3 to obtain

$$\begin{aligned} \mathbb {E}\left[ {X_{\tau \wedge \tau _N}^2}\right]&\le x_{0}^{2}+2\mathbb {E}\left[ \int _{0}^{ {\tau \wedge \tau _N}} \left( \gamma \left| X_{s}\right| ^{2}+\eta \right) \mathrm{d} s \right] \\&\le x_{0}^{2}+2\int _{0}^{T} \left( \gamma \mathbb {E}\left[ \left| X_{s}\right| ^{2}\right] +\eta \right) \mathrm{d} s \\&\le C(x_0,\gamma ,\eta ,T). \end{aligned}$$

For any $p \in (0;2)$, thanks to Proposition IV.4.7 in Revuz and Yor (1999), we have

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le T} \left| {X_{t \wedge \tau _N}}\right| ^{p} \right] \le C(x_0,\gamma ,\eta ,p,T). \end{aligned}$$

Let $N \rightarrow \infty$ and apply Fatou’s lemma, we get

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le T} \left| X_t \right| ^{p} \right] \le C(x_0,\gamma ,\eta ,p,T). \end{aligned}$$

(55)

Similarly, since $(X'_{t})_{0 \le t \le T}$ is another solution of Eq. (1), as in (55), for any $p\in (0;2)$, we have

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le T}\left| X'_{t} \right| ^{p}\right] \le C\left( x_{0}, \gamma , \eta , p, T \right) . \end{aligned}$$

(56)

Since $X_t$ and $X'_t$ are solutions of (1), we write

$$\begin{aligned} X_{t}-X'_{t}&=\int _{0}^{t}\left[ b\left( X_{s}\right) -b\left( X'_{s} \right) \right] \mathrm{d} s+\int _{0}^{t}\left[ \sigma \left( X_{s}\right) -\sigma \left( X'_{s} \right) \right] \mathrm{d} W_{s} \\&\quad +\int _{0}^{t} \int _{\mathbb {R}_{0}}\left( c\left( X_{s-}\right) -c\left( X'_{s-} \right) \right) z {\widetilde{N}}\left( \mathrm{d}s, \mathrm{d} z\right) . \end{aligned}$$

Applying Itô’s formula for $\phi _{\delta \varepsilon }\left( X_{t}-X'_{t} \right)$ and using the mean value theorem, YW2 and YW5, we get

$$\begin{aligned}&\left| X_{t \wedge \tau _N}-X'_{t \wedge \tau _N} \right| \\&\quad \le \ \varepsilon +\phi _{\delta \varepsilon }\left( X_{t \wedge \tau _N}-X'_{t \wedge \tau _N} \right) \\&\quad = \varepsilon +\int _{0}^{t \wedge \tau _N} \phi '_{\delta \varepsilon } \left( X_{s}-X'_{s} \right) \left( b\left( X_{s}\right) -b\left( X'_{s} \right) \right) \mathrm{d} s \\&\qquad + \int _{0}^{t \wedge \tau _N} \frac{1}{2} \phi ''_{\delta \varepsilon } \left( X_{s}-X'_{s}\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s}\right) \right) ^{2} \mathrm{d} s\\&\qquad +\int _{0}^{t \wedge \tau _N} \phi '_{\delta \varepsilon } \left( X_{s}-X'_{s}\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s} \right) \right) \mathrm{d} W_{s} \\&\qquad + \int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left[ \phi _{\delta \varepsilon }\left( X_{s}-X'_{s}+\left( c\left( X_{s}\right) -c\left( X'_{s}\right) \right) z\right) \right. \\&\left. \qquad -\phi _{\delta \varepsilon }\left( X_{s}-X'_{s} \right) -\phi '_{\delta \varepsilon }\left( X_{s}-X'_{s} \right) \left( c\left( X_{s}\right) -c\left( X'_{s}\right) z \right) \right] \nu (\mathrm{d}z) \mathrm{d} s \\&\qquad + \int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left[ \phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-}+\left( c\left( X_{s-}\right) -c\left( X'_{s-}\right) \right) z\right) -\phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-} \right) \right] {\widetilde{N}}\left( \mathrm{d} s, \mathrm{d}z\right) \\&\quad \le \varepsilon +\int _{0}^{t \wedge \tau _N} \left| b\left( X_{s}\right) -b\left( X'_{s} \right) \right| \mathrm{d} s \\&\qquad + \int _{0}^{t \wedge \tau _N} \frac{1}{\left| X_{s}-X'_{s}\right| \log \delta } \mathbb {I}_{\left[ \frac{\varepsilon }{\delta }, \varepsilon \right] }\left( | X_{s}-X'_{s} |\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s}\right) \right) ^{2} \mathrm{d} s \\&\qquad + \int _{0}^{t \wedge \tau _N} \phi '_{\delta \varepsilon } \left( X_{s}-X'_{s}\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s} \right) \right) \mathrm{d} W_{s} +\int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}} 2\left| c\left( X_{s}\right) -c\left( X'_{s}\right) \right| |z| \nu \left( \mathrm{d}z\right) \mathrm{d} s \\&\qquad + \int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left[ \phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-}+\left( c\left( X_{s-}\right) -c\left( X'_{s-}\right) \right) z\right) -\phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-} \right) \right] {\widetilde{N}}\left( \mathrm{d} s, \mathrm{d}z\right) . \end{aligned}$$

Then, using Conditions C3, C4, C5, we get

$$\begin{aligned}&\left| X_{t \wedge \tau _N}-X'_{t \wedge \tau _N} \right| \nonumber \\&\quad \le \varepsilon +\int _{0}^{t \wedge \tau _N} L_N\left| X_{s}-X'_{s}\right| \mathrm{d} s+\int _{0}^{t \wedge \tau _N} \frac{\varepsilon ^{2 \alpha } L_N^{2}}{\log \delta } \mathrm{d} s \nonumber \\&\qquad +\int _{0}^{t \wedge \tau _N} \phi '_{\delta \varepsilon } \left( X_{s}-X'_{s}\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s} \right) \right) \mathrm{d} W_{s}+\int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}} 2 L_N\left| X_{s}- X'_s\right| |z| \nu (\mathrm{d} z) \mathrm{d} s \nonumber \\&\qquad + \int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left[ \phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-}+\left( c\left( X_{s-}\right) -c\left( X'_{s-}\right) \right) z\right) -\phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-} \right) \right] {\widetilde{N}}\left( \mathrm{d} s, \mathrm{d}z\right) \nonumber \\&\quad \le \varepsilon +\int _{0}^{t \wedge \tau _N} L_N\left( 1+2\mu \right) \left| X_{s}-X'_{s}\right| \mathrm{d} s+\int _{0}^{t \wedge \tau _N} \frac{\varepsilon ^{2 \alpha } L_N^{2}}{\log \delta } \mathrm{d} s \nonumber \\&\qquad +\int _{0}^{t \wedge \tau _N} \phi '_{\delta \varepsilon } \left( X_{s}-X'_{s}\right) \left( \sigma \left( X_{s}\right) -\sigma \left( X'_{s} \right) \right) \mathrm{d} W_{s} \nonumber \\&\qquad + \int _{0}^{t \wedge \tau _N} \int _{\mathbb {R}_{0}}\left[ \phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-}+\left( c\left( X_{s-}\right) -c\left( X'_{s-}\right) \right) z\right) -\phi _{\delta \varepsilon }\left( X_{s-}-X'_{s-} \right) \right] {\widetilde{N}}\left( \mathrm{d} s, \mathrm{d}z\right) . \end{aligned}$$

(57)

By taking expectation on both sides (57) and using Proposition 2.3, we obtain

$$\begin{aligned} \mathbb {E}\left[ |X_{t \wedge \tau _N} -X'_{t \wedge \tau _N}\right]&\le \varepsilon +L_N\left( 1+2 \mu \right) \int _{0}^{t} \mathbb {E}\left[ \left| X_{s \wedge \tau _N}-X'_{s \wedge \tau _N}\right| \right] \mathrm{d} s +\frac{\varepsilon ^{2 \alpha } L_N^{2}T}{\log \delta }. \end{aligned}$$

By choosing $\delta =2$ and letting $\varepsilon \rightarrow 0$, we get

$$\begin{aligned} \mathbb {E}\left[ \left| X_{t \wedge \tau _N}-X'_{t \wedge \tau _N}\right| \right] \le L_N(1+2 \mu ) \int _{0}^{t} \mathbb {E}\left[ \left| X_{s \wedge \tau _N}-X'_{s \wedge \tau _N}\right| \right] \mathrm{d} s. \end{aligned}$$

Thanks to Gronwall’s inequality, $\mathbb {E}\left[ \left| X_{t \wedge \tau _N}-X'_{t \wedge \tau _N}\right| \right] =0 .$ It means $X_{t \wedge \tau _N}=X'_{t \wedge \tau _N}$ almost surely. This leads to $\mathbb {E}\left[ \left| X_{t}-X'_{t}\right| \right] =\mathbb {E}\left[ \left| X_{t}-X'_{t}\right| \mathbb {I}_{\left[ \tau _N \le t\right] }\right]$. By applying Cauchy’s inequality, (55) and (56) for any $p \in (1;2)$, we obtain

$$\begin{aligned} \mathbb {E}\left[ \left| X_{t}-X'_{t}\right| \right]&\le \frac{1}{2N} \mathbb {E}\left[ \left| X_{t}-X'_{t}\right| ^2\right] +\frac{N}{2} \mathbb {P}\left[ \tau _N \le T\right] \\&\le \frac{1}{2 N} \mathbb {E}\left[ \left| X_{t}-X'_{t}\right| ^{2}\right] +\frac{1}{2N^{p-1}}\left( \mathbb {E}\left[ \sup _{0 \le t \le T}\left| X_{t}\right| ^{p}\right] +\mathbb {E}\left[ \sup _{0 \le t \le T}\left| X'_{t}\right| ^{p}\right] \right) \\&\le \frac{1}{2 N} \mathbb {E}\left[ \left| X_{t}-X'_{t}\right| ^{2}\right] +\frac{C}{2N^{p-1}}. \end{aligned}$$

Let $N \rightarrow \infty$ we obtain $\mathbb {E}\left[ \left| X_{t}-X'_{t}\right| \right] =0 .$ It means $X_{t}=X'_{t}$ almost surely for any $t \in [0;T]$. Since X and $X'$ are càdlàg, they are indistinguishable on [0, T]. The proof is complete.