The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps

Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markovian property of the process.Comment: 29 pages, to appear in Journal of Theoretical Probabilit

Balance, growth and diversity of financial markets

A financial market comprising of a certain number of distinct companies is considered, and the following statement is proved: either a specific agent will surely beat the whole market unconditionally in the long run, or (and this "or" is not exclusive) all the capital of the market will accumulate in one company. Thus, absence of any "free unbounded lunches relative to the total capital" opportunities lead to the most dramatic failure of diversity in the market: one company takes over all other until the end of time. In order to prove this, we introduce the notion of perfectly balanced markets, which is an equilibrium state in which the relative capitalization of each company is a martingale under the physical probability. Then, the weaker notion of balanced markets is discussed where the martingale property of the relative capitalizations holds only approximately, we show how these concepts relate to growth-optimality and efficiency of the market, as well as how we can infer a shadow interest rate that is implied in the economy in the absence of a bank.Comment: 25 page

Global Solution to the Relativistic Enskog Equation With Near-Vacuum Data

We give two hypotheses of the relativistic collision kernal and show the existence and uniqueness of the global mild solution to the relativistic Enskog equation with the initial data near the vacuum for a hard sphere gas.Comment: 6 page

Stringy Robinson-Trautman Solutions

A class of solutions of the low energy string theory in four dimensions is studied. This class admits a geodesic, shear-free null congruence which is non-twisting but in general diverging and the corresponding solutions in Einstein's theory form the Robinson-Trautman family together with a subset of the Kundt's class. The Robinson-Trautman conditions are found to be frame invariant in string theory. The Lorentz Chern-Simons three form of the stringy Robinson-Trautman solutions is shown to be always closed. The stringy generalizations of the vacuum Robinson-Trautman equation are obtained and three subclasses of solutions are identified. One of these subclasses exists, among all the dilatonic theories, only in Einstein's theory and in string theory. Several known solutions including the dilatonic black holes, the pp- waves, the stringy C-metric and certain solutions which correspond to exact conformal field theories are shown to be particular members of the stringy Robinson-Trautman family. Some new solutions which are static or asymptotically flat and radiating are also presented. The radiating solutions have a positive Bondi mass. One of these radiating solutions has the property that it settles down smoothly to a black hole state at late retarded times.Comment: Latex, 30 Pages, 1 Figure; to appear in Phys. Rev.

Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space \HH, where $L$ is a \RR valued L\'evy process, $A:H\to H$ an infinitesimal generator of a strongly continuous semigroup, \sigma:H\to \RR bounded from below and Lipschitz continuous, and B:\RR\to H a possible unbounded operator. A typical example of such an equation is a stochastic Partial differential equation with boundary L\'evy noise. Let \CP=(\CP_t)_{t\ge 0} %{\CP_t:0\le t<\infty}$the corresponding Markovian semigroup. We show that, if the system (2) du(t) = A u(t)\: dt + B v(t),\quad t>0 u(0) = x is approximate controllable in time$T>0$, then under some additional conditions on$B$and$A$, for any$x\in H$the probability measure$\CP_T^\star \delta_x$is positive on open sets of$H$. Secondly, as an application, we investigate under which condition on$\HH$and on the L\'evy process$L$and on the operator$A$and$B$ the solution of Equation [1] is asymptotically strong Feller, respective, has a unique invariant measure. We apply these results to the damped wave equation driven by L\'evy boundary noise