Peter Imkeller

Ito's formula for continuous (N,d)-processes

On cherche a etudier les temps locaux de processus multi-indices continus en utilisant plusieurs notions appropriees du calcul stochastique

A class of two-parameter stochastic integrators

Let M be a continuous square integrable two-parameter martingale. Then the quadratic i-variations [M]i appear as integrators of terms of the second differential order in Itos formula, whereas terms of the third differential order are described by mixed variations Ni which behave like [M]i in parameter direction i and like M in the complementary direction. We prove that both [M]sup:i

On the perturbation problem for occupation densities

esup and Nsup:i

Local times for a class of multi-parameter processes

esup,=l,2, are stochastic integrators the integrals of which are defined on some vector space of 1- resp. 2-previsible processes. On one hand, this result shows that non-continuous previsible processes are integrable and is therefore basic for an Ito formula for non-continuous two-parameter martingales. On the other hand, the way it is derived may give a hint what multi-parameter semimartingales (martingale-like processes) are

Occupation densities of stratonovitch stochastic differential equations with boundary conditions

Let X be a semimartingale, perturbed by a process V of bounded variation, but with completely arbitrary measurability properties. We prove that if V is twice continuously differentiable such that its second derivative is Holder continuous of order , then the perturbed process X + V possesses occupation densities which are continuous under usual circumstances.

Stochastic integrals of point processes and the decomposition of two-parameter martingales

Given p≧1, let X be a real-valued, continuous N-parameter process, submitted to a “domination condition” which ensures the existence of its kth variations μX(k) 1≦k≦k0 p-stochastic measures which are the multi-parameter analogues of the “variation” dX and the “quadratic variation” d[X] of one-parameter semimartingales. This condition is fulfilled for example for processes of integrable variation, for the (N,1)-Wiener process W,or, more generally in case N=2, for those “representable semimartingales” (Wong Zakai or Guyon, Prum) which have sufficiently well-behaved representing functionals. Furthermore, it allows us to establish a stochastic calculus in Lp sense for X with a simple version of Itos formula in terms of the integrals of μX(k),1≦k≦k0. We slightly generalize the latter to a larger class of functions to obtain Tankas formula for X. Imposing the condition that μx=0 defines an increasing process. We show that Tanaks formula yields local times for X with occupation times measured in μX(ko) scale—...

Adaptedness and Existence of Occupation Densities for Stochastic Integral Processes in the Second Wiener Chaos

To solve diffusion type stochastic differential equations in which the Stratonovitch integral describes the coupling to the driving Wiener process, and with boundary conditions at the endpoints of the parameter space, the unit interval, one can consider the “flow” of diffusions corresponding to the initial condition , and look for a random variable X 0 such that X 0 and satisfy tne given boundary condition. Then solves the equation. By using semimartingale inequalities, we prove that the corresponding “flow” L(x) of occupation densities of is continuous in x. This way we are able to identify occupation densities of solutions of the diffusion type equations with coupled boundary data, and give conditions under which they are continuous.

The transformation theorem for two-parameter pure jump martingales

Let M be a square integrable martingale indexed by [0, 1]2 with respect to a filtration which possesses the property of conditional independence. Assume that M has trajectories which are continuous for approach from the right upper quadrant and possess limits for the remaining three. M can have three kinds of jumps. A point t is a 0-jump if [Delta]tM = lims[short up arrow]t[Mt - M(t1,s2) - M(s1,t2) + Ms] [not equal to] 0, a 1-jump if [Delta]tM = 0 and lims1[short up arrow]t1[Mt - M(s1,t2)] [not equal to] 0. Analogously, 2-jumps are defined. With the 0-jumps associate the two-parameter point process [mu]M which assigns unit point mass to nontrivial (t, [Delta]tM), with the 1-jumps the one-parameter point process [mu]1M which puts unit mass to nontrivial (t1, [Delta]t1M(+, 1)), and with the 2-jumps a corresponding [mu]2M. We define stochastic integrals with respect to the compensated [mu]M, [mu]iM, i = 1, 2, with the help of which we can describe the jump components associated with the respective jumps in the orthogonal decomposition of M by discontinuous and continuous parts.

The continuity of the quadratic variation of two-parameter martingales

Let U be a Skorohod integral process in the second Wiener chaos associated with a square integrable function f on the unit square. We derive a necessary and sufficient integral criterion for the existence of a square integrable occupation density of U in terms of the Hilbert-Schmidt operator associated with f. This criterion trivializes in case U is adapted and thus exhibits the special role played by adaptedness in this area.

On some sample path properties of Skorohod integral processes

SummaryLetM be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps ofM in the plane.M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which forC4-functions f the processf(M) is described by integrals off(k) (M),k=1, 2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula.