We clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes $\delta$. They reduce to the better-known Markov properties if $\delta$ is the empty set, or, with the exception of the local property, if $\delta$ is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes $\delta_0$, all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set $\delta\setminus\delta_0$. We note that many of the above results are new even for reciprocal chains.

We use multivariate total positivity theory to exhibit new families of peacocks. As theauthors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associatedmartingales vol. 3. Bocconi-Springer (2011)], our guiding example is the resultof Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao,Finance Res. Lett. 5 (2008) 162–171]. We shall introducethe notion of strong conditional monotonicity. This concept is strictly more restrictivethan the conditional monotonicity as defined in [F. Hirsch, C. Profeta, B. Roynette and M.Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011)] (see also [R.H.Berk, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1978) 303–307], [A.M. Bogso, C. Profeta and B. Roynette, Lect. Notes Math.Springer, Berlin (2012) 281–315.] and [M. Shaked and J.G. Shanthikumar,Probab. Math. Statistics. Academic Press, Boston (1994)].). There aremany random vectors which are strongly conditionally monotone (SCM). Indeed, we shallprove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As aconsequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This familyincludes processes with independent and log-concave increments, and one-dimensionaldiffusions which have absolutely continuous transition kernels.

We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(Xn →∞) is positive or 0. It turns out that essentially this depends on the magnitude of E(Xn+1 | Xn = x) compared to that of E(X2n+1 | Xn = x) for large x. The assumptions are chosen such that for example population-dependent branching processes can be treated by our results.

It is shown that many properties of Markov processes can be extended to the case where the time parameter is multidimensional (a partially ordered set). This includes Kolmogorov's backward and forward equations, and semi-group treatment of homogeneous processes.

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.