120009511
On semimartingale diffusions and stochastic differential equations
Given a regular diffusion X on the real axis which is a semimartingale we describe the semimartingale decomposition of X. We then give necessary and sufficient conditions in terms of the scale and the speed measure for X being a solution of an Ito type stochastic differential equation driven by a Wiener process and with classical drift or a drift term involving the local time of X. A regular diffusion is also characterized as unique solution of a certain martingale problem. Finally we discuss an example related to skew Brownian motion
120957526
On Stochastic Differential Equations with Reflecting Barriers
121309269
Stochastic Integration for Some Rough Non‐adapted Processes
For the Wiener process W on the unit interval and 0 ⩽ θ ⩽ 1/4 consider the non-adapted process ut =sgn(Wt−θ)sgn(W1−WT+θ), θ⩽t⩽1−θ. By giving explicit descriptions of their expansions in terms of generalized Hermite polynomials we show that they possess Skorohod integral processes. In the more interesting case θ = 0, a component due to the interaction of forwards and backwards Wiener process seems to dominate the non-interaction part, whereas for θ > 0 interaction drops geometrically with respect to non-interaction. We translate these purely analytic properties into more familiar terms of samples of stochastic processes. Firstly, we construct simple approximations of the integral process which are of the type of Riemann sums. We then show that their quadratic variations are the same as for the Wiener process on [θ, 1 − θ]. Finally, using the fact that W remains a semimartingale if its filtration is enlarged by W1, we compare the resulting Ito integral process to the Skorohod integral process of u in case θ = 0. We show that they differ by a local time. Consequently the Skorohod integral process of u is a semimartingale.
121422615
Flächenapproximation beim Jacobialgorithmus
119634267
On Thue's equation
and more precisely for the number of primitive solutions to (1.1), that is, solutions in coprime integers x, y. The first important results on the number of solutions of Thue's equation were obtained by C.L. Siegel, in the case r = 3 and in the binomial case F ( x , y ) = a x r b y r. In view of his results and comments, we should attribute to Siegel the problem to decide whether the number of primitive solutions to (1.1), for irreducible F of degree r > 3, could be bounded by a function depending only on r and h, but otherwise independent of F. In 1983, Siegel's question was answered in the affirmative by J.-H. Evertse [2]. As a special case of his results (he also treats equations in number fields), Evertse obtains the bound
118410662
The joint distribution of the digits of certain integer s-tuples
Let B(n) denote the number of digits 1 in the representation of a natural number n in the binary scale. For example, since 5 in the binary scale is 101, we have B(5) = 2. It is well known that for most integers n, the number B(n) is about half the total number of digits in the binary representation of n, so that B(n) is roughly equal to \(\frac{1} {2}v\), where 
 
$$v = v(n) = _2 \log n,$$ 
 
(1.1) 
 
and where 2log is the logarithm to the base 2. In fact it follows from the Central Limit Theorem of probability theory that the numbers n with 
 
$$\frac{{B(n) - \frac{1} {2}v}} {{\sqrt v }} \leqq \xi$$ 
 
(1.2) 
 
have density 
 
$$\phi (\xi ) = \sqrt {\frac{2} {\pi }\int\limits_{ - \infty }^\xi {e^{ - 2t^2 } dt:} }$$ 
 
(1.3) 
 
Here and always we say that a set S of natural numbers has density ψ if the number N(S, x) of numbers n ∈ S, n ≦ x satisfies the asymptotic relation N(S, x) ∼ ψx as x → ∞.
123201234
Badly approximable systems of linear forms
A number α is called badly approximable if there is a constant c = c(α)>0 such that 
|q|αq − p| > c(α) 
for any integers p, q ≠ 0. The existence of continuum—many badly approximable numbers follows easily from the theory of continued fractions. In the present paper the notion of a badly approximable number is generalized to badly approximable systems of m linear forms in n variables. Among other results we prove for every given m and n the existence of continuum many badly approximable systems of linear forms.