Stochastic currents
Let \(D^1\) be the space of \(1\)-forms with compact supports in the Euclidean space \(\mathbb{R}^d\), i.e., if \(\varphi\in D^1\), then \[ \widetilde\varphi= \sum^d_{i=1} \varphi_i dx_i,\tag{1} \] where for \(i= 1,\dots, n\), \(\varphi_i\in C^\infty_K(\mathbb{R}^d)\), the space of smooth functions with compact support. Given a probability space \((\Omega, A,P)\) the authors call ``stochastic current'' (S-current for short) a linear continuous mapping from the space \(D^1\) into \(L^0(\Omega)\) (the space of real-valued (r.v.) in \((\Omega,A,P)\) equipped with the topology of convergence in probability). Let us denote by \(D_1\) (resp. \(D\)) the space of S-currents, and so if \(\widetilde S\in D_1\), then \(\forall\varphi\in D^1\Rightarrow\widetilde S(\varphi)\in L^0(\Omega)\). Definition 4. The S-current \(\widetilde S\in D_1\) is said to have a ``pathwise regularization'' provided that there exists a measurable mapping \(S^r: \Omega\to D_1\), such that \[ [\widetilde S(\varphi)](\omega)= S^r[\omega](\varphi)\quad\text{a.s. \(P\)-\(\omega\)},\quad\forall\varphi\in D^1.\tag{2} \] The authors are interested in the following problem of pathwise regularization for S-currents. Let \[ F:= (\Omega, A,(A_t)_{t\in [0,T]}, P)\tag{3} \] be a filtration and denote by \(X= M+ V\) a given continuous semimartingale over \(F\), in which \(M= (M_t)\) (resp. \(V= (V_t)\)) is a continuous local martingale (resp. a continuous, adapted and of bounded variation process) and define: \[ \widetilde S(\varphi):= \int^T_0 \langle\varphi(X_t), d^s X_t\rangle,\tag{4} \] where \(d^s X_t\) denotes the Stratanovich-stochastic differential generated by \(X\); resp. \[ \widetilde I(\varphi):= \int^T_0 \langle\varphi(X_t), d^1 X_t\rangle,\tag{5} \] where \(d^1 X_t\) denotes the Itô-stochastic differential generated by \(X\). The main result is the following theorem, which gives sufficient conditions for the existence of a pathwide regularization of the S-current (4) (resp. (5)). Theorem 9. Under the same hypotheses as quoted above the S-current (4) (resp. (5)) admits a pathwise regularizations \(\widetilde S^r(\omega)\) (resp. \(\widetilde I^r(\omega)\)) (see (2)) such that for all \(s> d/2\), \[ \widetilde S^r(\omega)\in H^{-s-1}(\mathbb{R}^d, \mathbb{R}^d)\text{ a.s. }P,\tag{6} \] in which \(H^{-s-1}\) denotes the Sobolev space of order \(-(s+1)\). Furthermore, if the local martingale \(M\) is identified with \(d\) independent one-dimensional Brownian motions, then we get a sharper result by replacing \(H^{-s-1}\) by \(H^{-s}\) in (6): \[ \widetilde I^r(\omega)\in H^{-s}(\mathbb{R}^d, \mathbb{R}^d)\text{ a.s. }P,\tag{7} \] in which \(H^{-s}\) denotes the Sobolev space of order \(-s\). Comments of the reviewer: In the introduction the authors mention the following ``optimal regularity criterion'' for the case in which the given semimartingale \(M\) (see (3) above!) is identified with the standard Brownian motion in \(\mathbb{R}^d\). Theorem 1. With probability one, a Brownian motion path defines a 1-current of Sobolev class \(H^{-s}(\mathbb{R}^d, \mathbb{R}^d)\) for every \(s> d/2\). On the contrary, for any \(s\leq d/2\), the current does not belong to \(H^{-s}(\mathbb{R}^d, \mathbb{R}^d)\). Let us investgated the following deterministic 1-current: \[ I(\varphi):= \int^T_0 \langle\varphi(x_t), dx_t\rangle,\tag{8} \] where \((x_t)_{t\in [0,t]}\) is a given regular curve in \(\mathbb{R}^d\), and \(dx_t\) denotes the differential generated by \(x_t\). It is clear that (8) defines a tempered distribution in \(\mathbb{R}^d\) (since it is a Schwartz distribution with compact support!), therefore it follows from the ``lifting theorem'' [given by the reviewer, ``A theory of stochastic Sobolev spaces'' (Preprint Hanoi, 2002/2003), Prop. 2.2] that the above tempered distribution is lifted to an ``S-distribution'' (stochastic distribution) in the sense given by the reviewer (loc. cit., Definition 2.1). In the present situation a ``pathwide regularization'' (in the sense given by the authors of the paper under review) of the above-mentioned ``S-distribution'' is explicitly given by the lifting theorem mentioned above, furthermore if the tempered distribution (8) is locally-\(L^2\), then we could assert that the paths of the associated ``S-distribution'' is a.s. \(P\) locally-\(L^2\) provided that \(s> d/2\). Now if we choose the regular curve \((x_t)_{t\in \mathbb{R}^d}\) in (8) to be \((t,\dots, t)_{t\in [0,T]}\), then the ``lifted S-distribution'' \(I(\widetilde\varphi)\) associated with (8) turns out to be following ``complex Itô-integral'': For each rapidly decreasing function \(\varphi\in \mathbb{R}^d\), let \(\widetilde\varphi\) be its Laplace-Fourier transform, then \[ I(\widetilde\varphi):= \int^T_0 \langle\widetilde\varphi(Z_t), dZ_t\rangle,\tag{9} \] where \((X_t)\) denotes the B.M. in \(\mathbb{R}^d\), and \((Z_t= X_t+ jY_t)_{t\in [0,\infty]}\) the standard complex B.M. in \(C^d\). And so it is possible here to recover the ``Theorem'' mentioned above by ``pull-back'' (9) to \(\mathbb{R}^d\,(\subset \mathbb{C}^d)\). It is also interesting to note that by the ``conformal lifting theorem'' due to \textit{R. K. Getoor} and \textit{M. G. Sharpe} [Invent. Math. 16, 271--308 (1972; Zbl 0268.60048)] it is always possible to adjoint to each given continuous \((A_t)\)-local martingale in \(\mathbb{R}^d\) a ``conjugate local martingale'' defined in an extended filtration \((\widetilde A_t)_{t\in [0,T]}\) in order to obtain an ``\((\widetilde A_t)\)-conformal martingale'' in \(\mathbb{C}^d\). On the other hand since conformal martingales are nothing else than complex processes obtained from the complex Brownian motion via a ``time change'', this indicates that Theorem 9 of the article under review could also be obtained from results given by Getoor and Sharpe (loc. cit.).
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