This question is motivated by the desire to build mathematical models that forecast vector-valued discrete time series while guaranteeing a kind of "continuity" via uniform bounds on the magnitude of coordinate-wise first derivatives.
In what follows, let $h,f,k$ be integers, where $h\geqslant 0$, and $f,k\geqslant 1$.
Consider a $k$-dimensional vector time series $v_0, v_1, \ldots, v_t, \ldots$ where the element $v_t \in {\mathbb R}^{k}$ at time $t$ is expressed as $$ v_t = (x_t^1, x_t^2, \ldots, x_t^{k-1}, x_t^k). $$ Fix integer $T\gg 0$. We seek a flexible family of mathematical forecasting models that can be fitted to the sequence $v_0, v_1, \ldots, v_T$, such that the fitted model ${\cal M}$ generates forecasts $f$ time steps into the future. ${\cal M}$ is to satisfy the two conditions given below, of which the first is "standard".
${\cal M}$ can be applied at any time $t\in \{h, h+1, \ldots, T-1,T\}$ and computes its forecast $\tilde v_{t+f}$ as a function of the $h+1$ most recent series elements up to and including time step $t$, i.e. $$ \tilde v_{t+f} \stackrel{def}{=} {\cal M}(t) \stackrel{def}{=} {\cal M}(v_t, v_{t-1}, v_{t-2}, \ldots, v_{t-h}). $$ Here the forecasted vector is of the form $$ \tilde v_{t+f} = (\tilde x_{t+f}^1, \tilde x_{t+f}^2, \ldots, \tilde x_{t+f}^{k-1}, \tilde x_{t+f}^k) $$ The fitted model can thus be seen as a function ${\cal M}: {\mathbb R}^{(h+1)k} \rightarrow {\mathbb R}^{k}$.
Choose $L\in {\mathbb R}^{+}$ such that the coordinates of the vector series upto time $T$ satisfies Lipschitz conditions $$ |x_t^j - x_{t-1}^j| \leqslant L $$ for $0 \lt t \leqslant T$ and $1 \leqslant j \leqslant k$. Then if we apply our model ${\cal M}$ at the $f$ times points $t = T-f+1,\; T-f+2,\; \ldots,\; T-1,\; T,$ the model will generate $f$ corresponding forecasts. Prepending these $f$ forecasts with the last known true value $v_{T}$, we obtain the sequence $$ v_T, \tilde v_{T+1}, \tilde v_{T+2}, \ldots \tilde v_{T+f}. $$ The modeling technique should guarantee that this sequence of $f+1$ vectors also satisfies the same Lipshitz conditions, for the same value of $L$. (Or, if this statement (2) is too severe a requirement, I will settle for any justifiable definition of "continuity")
Condition (1) is easy to achieve via, e.g. lagged vector autoregression. Can such modeling techniques be extended (or are there other techniques) by which one can ensure that the flexible model ${\cal M}$ also satisfies requirement (2)?
If a reader feels they follow the requirements, and knows that the standard techniques (e.g. VARMAX or other variants of VAR) cannot be specialized to achieve condition (2), I would greatly appreciate a comment to this effect.