## 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

### Citations:

Zbl 0704.60045; Zbl 0655.60038
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