The valuation theory for American Contingent Claims, due to Bensoussan (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice, including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price u0; this is expressed a...

In this supplementary material, we will present more details on learning a Random Forest saliency regressor. More evaluation results with state-of-the-art algorithms are also presented. F 1 LEARNING 1.1 Learning a Similarity Score between Two Adjacent Superpixels To learn the similarity score of two adjacent superpixels si and sj , they are described by a 222dimensional feature vector, includin...

Reflected Brownian motion (RBM) in a wedge is a 2-dimensional stochastic process Z whose state space in R is given in polar coordinates by S = {(r, θ) : r ≥ 0, 0 ≤ θ ≤ ξ} for some 0 < ξ < 2π. Let α = (θ1+θ2)/ξ, where −π/2 < θ1, θ2 < π/2 are the directions of reflection of Z off each of the two edges of the wedge as measured from the corresponding inward facing normal. We prove that in the case ...

The theory of noncommutative martingale inequalities has been rapidly developed since the establishment of the noncommutative Burkholder-Gundy inequalities in [12]. Many of the classical martingale inequalities has been transferred to the noncommutative setting. These include, in particular, the Doob maximal inequality in [3], the Burkholder/Rosenthal inequality in [5], [8], several weak type (...

We show that the function h(x) = ∏ i<j(xj − xi) is harmonic for any random walk in Rk with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = {x : x1 < x2 < · · · < xk} onto a point where h vanishes, we define the corresponding Doob h-transform. For certain special cases, we show that the marginal distribution ...

Pioneering work of Doob, Kac, and Kakutani showed that there was a beautiful and deep connection between certain problems in the study of Brownian motion and those of classical potential theory. This work stimulated much research on the theory of Markov processes. In spite of all this work, however, there doesn't appear anywhere in the literature any reasonably complete treatment of the connect...

Stochastic or random integral equations are extremely important in the study of many physical phenomena in life sciences and engineering [3, 14, 16]. There are currently two basic versions of stochastic integral equations being studied by probabilists and mathematical statisticians, namely, those integral equations involving Ito-Doob type of stochastic integrals and those which can be formed as...

The paper deals with the theory of potentials respect to α-Riesz kernel |x − y|α−n order α ∈ (0,2] on $\mathbb R^{n}$ , $n\geqslant 3$ . Focusing first inner α-harmonic measure ${\varepsilon _{y}^{A}}$ (εy being unit Dirac at $y\in \mathbb and μA balayage a Radon μ $A\subset arbitrary), we describe its Euclidean support, provide formula for evaluation total mass, establish vague continuity map ...