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Martingale laws, densities and decomposition of Föllmer-Schweizer. (Lois de martingale, densités et décomposition de Föllmer-Schweizer.) (French) Zbl 0772.60033
The paper continues topics in [second author, ibid. 26, No. 3, 451-460 (1990; Zbl 0704.60045)]. Consider a probability space \((\Omega,{\mathcal F}_ 1,P)\), a filtration \(({\mathcal F}_ t)_{t\in[0,1]}\) on it etc. Consider first a continuous \(R^ d\)-valued process \(X=M+B\), where \(M\) is a continuous local martingale and \(B\) is previsible with finite variation. Let \(N^ i\) be orthogonal \(R\)-valued continuous local martingales with \(M=\sigma N\), \(\sigma\) previsible, let \(A\) be continuous ascending adapted with \(d\langle N^ i\), \(N^ i\rangle=\zeta^ iA\). If there exists a probability \(Q\), equivalent to \(P\), such that \(X\) is a \(Q\)-martingale, then one may choose previsible \(\nu,b\) such that \(b=\sigma\nu\), \(X=\sigma N+bA\) and \(\theta^ i=(\nu^ i/\zeta^ i)1_{(\zeta^ i\neq 0)}\) are \(N^ i\)-integrable. Moreover, all possible \(Q\) are \({\mathcal E}(-\theta N+L)P\), where \(L\) is a local martingale orthogonal to all \(N^ i\) (and \({\mathcal E}\) is the “stochastic exponential”). \(Q\) is unique iff \(N\) has the previsible representation property and \(\sigma\) may be chosen invertible.
The authors consider then discontinuous \(R\)-valued processes. They first give a correct proof of a result of the second author [Stochastic differential systems, Proc. IFIP-WG 7/1 Work. Conf., Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 373-380 (1987; Zbl 0655.60038)], namely: if \(D\) is ascending right continuous previsible, \(A\) is right continuous previsible with finite variation, \(p\in(1,\infty)\), \(p^{- 1}+q^{-1}=1\) and \(K=\{\int HdA\); \(H\) previsible bounded, \(\int| H|^ pdD\leq 1\}\) is bounded in probability, then there exists a previsible \(\alpha\) with \(dA=\alpha dD\), \(\int|\alpha|^ qdD<\infty\). Then they prove, via four lemmas etc., that, if \(X\) is adapted, cadlag, \(X_ 0=0\), \(\sup_ s| X_ s|\in L^ 2\) and for the closure \(\overline K\) of \(K\) in \(L^ 1\) we have \(\overline K\cap L^ 1_ +=\{0\}\), then \(X\) is a special semimartingale, \(X=M+\alpha\langle M,M\rangle\) with \(M\) a locally square integrable local martingale, \(\int\alpha^ 2d\langle M,M\rangle<\infty\). A “pathological” example is given.
Let now \(X=M+A\), \(M\) a local martingale, \(A\) previsible with finite variation (the canonical decomposition of the special semimartingale \(X)\). Let \(Q\) be equivalent to \(P\). It is named a “minimal martingale law” for \(X\) if \(X\) is a \(Q\)-martingale and every local \(P\)-martingale \(P\)-orthogonal to \(M\) is a \(Q\)-local martingale. The authors show that if such a \(Q\) exists for \(X\) in the previous theorem, then \({\mathcal E}(-\alpha M)\) is a strictly positive martingale and \(Q={\mathcal E}(-\alpha M)_ 1 P\).
Finally, consider the case when \(M\) is locally square integrable, \(A=\alpha\langle M,M\rangle\), \(\alpha\) previsible, \(\int\alpha^ 2d\langle M,M\rangle<\infty\), \(\alpha\Delta M<1\). Let \(H=\lambda+\int\zeta dX+L_ 1\), where \(\lambda\in R\), \(\zeta\) be previsible, \(L\) be a semimartingale with \(L_ 0=0\), \({\mathcal E}(-\alpha M)\) \((\lambda+\zeta X+L)\) be a martingale and \([X,L]{\mathcal E}(-\alpha M)\) be a local martingale. Then \(\lambda,\zeta,L\) are uniquely determined by \(H\). If \(X\) is continuous and \(H\) is given, then \(H\) is representable as above iff \(H{\mathcal E}(-\alpha M)_ 1\in L^ 1\). If \(X\) is locally bounded, \(H{\mathcal E}(-\alpha M)_ 1\in L^ 1\) and \(E(H{\mathcal E}(-\alpha M)_ 1;{\mathcal F}_ t){\mathcal E}(-\alpha M)^{-1/2}_ t\) is locally square integrable, then \(H\) is representable as above. The paper contains several remarks, showing the “limits” of some of its results.

MSC:
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
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