On a statistical information measure for a generalized Samuelson-Black-Scholes model (Q2757198)

For the one-dimensional SDE \(dX_t = b(t,X_t) dt + \sigma(t) X_t dW_t \) on \([0,T]\), with a measurable square-integrable function \(\sigma\geq 0\), this paper examines the \(I_\alpha\)-divergence between the laws of the solution \(X\) and the solution of the corresponding driftless SDE. The main results are sufficient conditions, both stochastic and purely analytic, for the finiteness of all \(I_\alpha\)-divergences with \(\alpha\in\mathbb{R}\). This is done for both time-homogeneous and time-inhomogeneous \(b\). [The closely related paper ibid. 17, No. 4, 359-376 (1999; see the preceding review, Zbl 0985.62007), treats the same question in the multidimensional case with unit volatility matrix.] An application uses these divergences in a testing problem for the drift, and examples illustrate that one can deal with fairly pathological drifts.NEWLINENEWLINENEWLINEThe main tool in the proofs is a space-time version of Khasminskii's lemma. See also the author's paper, Probab. Theory Relat. Fields 97, No. 4, 515-542 (1993; Zbl 0794.60055), and the joint paper of the author and \textit{K.-T. Sturm}, Stochastic Processes Appl. 85, No. 1, 45-60 (2000).