We consider the stochastic equation $$X_t=x_0+{\int }_0^{t}b(u,X_u)\mathrm{d}{B}_u,t\ge 0,$$ where $B$ is a one-dimensional Brownian motion, $x_0\in ℝ$ is the initial value, and $b[0,\infty )\times ℝ\to ℝ$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution.

The $\sigma$-finite measure ${𝒫}_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$-transform processes with respect to these functions are utilized for the construction of ${𝒫}_{sup}$.

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

Assume that $X$, $Y$ are continuous-path martingales taking values in ${ℝ}^{\nu }$, $\nu \ge 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$\parallel \underset{t\ge 0}{sup}|Y_t{|\parallel }_1\le 2\parallel \underset{t\ge 0}{sup}|X_t{|\parallel }_1.$$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $\mathbb{P}\left(su{p}_{t\ge 0}\right|Y_t\left|\ge 1\right)\le 3.375...\parallel X\parallel ₁$. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $\parallel Y{\parallel }_W\le \left(2(\alpha +1)²\right)/(2\alpha +1)\parallel X{\parallel }_{L^{\infty }}$. Here W is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

In this paper we solve the basic fractional analogue of the classical linear-quadratic gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.