  Given a standard Brownian motion $W$ and a stationary Poisson point process
$p$ with values in ${\mathbb R}^d$, we first consider the following SDE of
It\^o's type \begin{align*}
  &dX_{t}=b\left(r, X_{r}\right) dr + \alpha\left(r, X_{r} \right) dW_r+ \int_{
|z| \le 1}g\left(X_{r-},r,z\right)\widetilde{N}_p\left(dr,dz\right) + \int_{
|z| >1 } f\left(X_{r-},r,z\right){N}_p\left(dr,dz\right), \end{align*} where $
X_s=x\in\mathbb{R}^d,\,0\le s \le t \le T.$ Here $N_p$ [resp.,
$\widetilde{N}_p$] is the Poisson [resp., compensated Poisson] random measure
associated with $p$. The coefficients $b$, $\alpha$, and $g$ satisfy
Lipschitz--type conditions in the $x-$variable. We prove that there exists a
sharp version of the pathwise unique strong solution $X = (X^{s,x}_t)_{t\ge
s}$, i.e., there exists an almost sure event $\Omega'$ such that, for every
$\omega \in \Omega'$, the map $(s,x,t)\mapsto X^{s,x}_t(\omega)$ is c\`adl\`ag
in $s$ (for $t$ and $x$ fixed), c\`adl\`ag in $t$ (for $s$ and $x$ fixed) and
continuous in $x$ (for $s$ and $t$ fixed). This result also solves a problem
indicated in H. Kunita (Stochastic Flows and Jump-Diffusions, 2019). We then
extend our approach to encompass controlled SDEs where the coefficients may
additionally depend on a random control function $(c_r)_{r \in [0,T]}$. Using
our sharp stochastic flow we establish rigorously a dynamic programming
principle (DPP) for controlled SDEs. The DPP is a fundamental tool in
stochastic control with several applications in physics and mathematical
finance. Such result has a proof of independent interest and seems to be new
even in the case where there is no large-jumps component.
