My question is about a step taken in the proof of Lemma 1.3 in Stochastic Numerics for Mathematical Physics
We have a system of SDEs denoted by: \begin{equation} dX(t) = a(t, X(t)) \text{d}t + \sum_{r=1}^q \sigma_r(t, x) \text{d}W_r(t) \end{equation}
Now, the step in the proof that I do not follow is this:
\begin{equation}\mathbb{E}\lvert X_{t,x}(t+\theta) - X_{t,y}(t+\theta) \rvert^2 \leq \lvert x - y \rvert^2 e^{Kh}, \end{equation} for $0 \leq \theta \leq h$ implies
\begin{equation}\mathbb{E}\lvert X_{t,x}(t+h) - X_{t,y}(t+h) \rvert^2 \leq \lvert x - y \rvert^2 (1 + Kh). \end{equation}
Here, the subscript notation $X_{t, x}(s)$ denotes the initial condition, i.e. $X(t) = x$. Also, the positive constant $K$ can change value but is always independent of $h$.
How can bounding the expectation by something that grows exponentially imply that it is bounded by something growing linearly?
Milstein, Grigori N.; Tretyakov, Michael V., Stochastic numerics for mathematical physics, Scientific Computation. Berlin: Springer (ISBN 3-540-21110-1/hbk; 978-3-642-05930-8/pbk). ixx, 594 p. (2004). ZBL1085.60004.
EDIT: I have attached a picture of the complete proof below:
(1.2) is a globally Lipshcitz condition: \begin{equation} \lvert a(t, x) - a(t, y) \rvert + \sum_{r=1}^q \lvert \sigma_r(t, x) - \sigma_r(t, y) \rvert \leq K \lvert x-y \rvert \end{equation}