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Canonical RDEs and general semimartingales as rough paths. (English) Zbl 07036341
In verbatim from the abstract of the authors: “In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle’s BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased a uniformly-controlled-variations (UCV) condition (Kurtz-Protter, Jakubowski-Mémin-Pagès). A number of examples illustrate the scope of our results.”
Let us beforehand try to understand what the authors mean by the term canonical stochastic differential equation (SDE) in the sense of Marcus (resp. the term general semimartingle) in the above mentioned abstract. From the original paper of Marcus (see [S. I. Marcus, Stochastics 4, 223–245 (1981; Zbl 0456.60064)]) it follows that the canonical SDE driven by a multidimensional semimartingale \(z=(z^1,\dots,z^k)\) can be written as follows: \[ \begin{split} x_t=x_0+\int_0^t f(x_s)ds + \sum_{i=1}^k \int_0^t g_i(x_{s^-})dz_s^i + \sum_{i=1}^k \int_0^t g_i(x_{s^-})dz_s^i + \frac12 \sum_{i,j=1}^k \int_0^t D_i g_j(x_{s^-}d\langle z_s^{ic},z_s^{jc}\rangle_s \\ +\sum_{s\leq t} \left[ \phi(x_{s^-,})-\Delta z_s-x_{s^-}-\sum_{i=1}^k\int_0^t g_i(x_{s^-})\Delta z_s^i \right] \end{split} \tag{1} \] where \(z=(z^1,\dots,z^k)\), \(\Delta z_s=(\Delta z_s^1,\dots,\Delta z_s^k)'\). Each component \(z^i=(z^i_s)\) of \(z\) is a semimartingale with a unique decomposition of the form \(z^i=z^i_0+z^{ic}+z^{id}+a^i\), in which \(z^{ic}\) (resp. \(z^{id}\)) is a continuous (resp. purely discontinuous) local martingale, and \(a^i\) is a process with paths a.s. of bounded variation. The following assumptions are assumed throughout the work of Marcus:
(B1) all noise processes \(z=(z^1,\dots,z^k)\) are vectors with semimartingale components such that, with probability one, each \(z^i\) has a finite number of jumps on each bounded interval and no two components jump simultaneously.
It is essential to note that the first assumption in (B1) permits Marcus to investigate individually each path \(t\to z_t(\omega)\) of \(z\) as càdlàg curve in \(\mathbb R^k\), so that he could “glue any jump with a piecewise linear approximation” in order to obtain continuous path curve in \(\mathbb R^k\). The second assumption allows only one jump at a time, this fact makes the equation (1) “resembles” a single-input system in an infinitesimal time. The proof of the existence and uniqueness theorems for a “path solution” of the stochastic equation (1) is purely real analysis and easy to understand, however if the input driven semimartingale is a continuous local martingale then one recovers the known results of Lyons (see [T. J. Lyons, Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)]), where the Holder’s continuity assumptions prevalent in the rough path method is dropped, and provided further a proper solution of Lyons’ main result in rough path analysis: continuity of the solution map as a function of the driving rough path. These new results pose the following interesting problem: Is the rough path method really essential for the modeling and approximation of SDE driven by semimartingales where both jumps and continuous components are considered?
There is no explicit explanations of the paper under review for the term “general semimartingales” but I guess that they used the same subclass of semimartingales introduced in the Marcus’ work, because the authors write in page 421 the following: “In fact, we reserve the prefix “Marcus” to situations in which jumps only arise in the \(d\)-dimensional driving signal, and are handled (in the spirit of Marcus) by connecting \(X_{t^-}\) and \(X_t\) by a straight line. (As a straight line has no area, this creates no jump in the area)...”.
To read the work under review the readers have to be familiar with the “preparatory materials” introduced in the Section 2, that ate unfortunately too technical to present in a review. As the authors explain in the beginning of the same 421: “Loosely speaking, given a multidimensional continuous semimartingale, \(X\) the Stratonovich integral \(\int f(X)\circ dX\) can be given a robust (pathwise) meaning in term of \(\mathbf{X}=(X,\int X\otimes \circ dX)\), a.e. realization of which constitutes a geometric rough path of finite p-variation for any \(p>2\).”
One of the main theorems of the work under review that shows the link between the (pathwise) solution of the RDES given by the authors with that given by Marcus [loc. cit.] is Theorem 5.18. Loosely speaking, the above mentioned result asserts that if \(X, (X_n)_{n\geq1}\) are \(R^d\)-valued semimartingales such that \((X_n)_{n\geq1}\) satisfies the uniformly controlled variation of semimartingale sequences and \(X_n\to X\) in law (resp. in probability) for the Skorokhod topology, then the solutions to Marcus SDEs driven by \(X_n\) (along fixed vector fields) converge in la law (resp. in probability) to the Marcus SDE driven by \(X\). The proof of Theorem 5.18 given by the authors involves rough paths method, a detour that integrates the problem into its solution, the reviewer thinks this is only a taste of modeling choice, there may exist a less sophisticated proof for the result!
The following is extracted from the introduction of the paper under review: “In the second part of the paper, we show how general (càdlàg) semimartingales fit into the theory. In particular, we show that the canonical lift of a semimartingale indeed is a.s. a (geometric) rough path of finite \(p\)-variation for any \(p>2\) (several special case, including Lévy processes, were in [P. K. Friz and A. Shekhar, Ann. Probab. 45, No. 4, 2707–2765 (2017; Zbl 1412.60103); D. R. E. Williams, Rev. Mat. Iberoam. 17, No. 2, 295–329 (2001; Zbl 1002.60060)] but the general case remain open).”
Comments of the reviewer. In the sequel the symbols in the equation (1) are used, the reviewer uses also some results from [P. A. Meyer, Lect. Notes Math. None, 245–400 (1976; Zbl 0374.60070)].

60L20 Rough paths
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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