Neural network regression for Bermudan option pricing

The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques on a finite-dimensional vector space. In this work, we study neural networks approximations of conditional expectations. We prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation, assuming an efficient algorithm to compute this approximation. We illustrate the numerical efficiency of neural networks as an alternative to standard regression methods for approximating conditional expectations on several numerical examples.

Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

Binary Hamming codes and Boolean designs

In this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of $${\mathcal {P}}$$ P , respectively of $${\mathcal {P}}^*$$ P ∗ , that induce permutations of $${\mathcal {B}}_k$$ B k , respectively of $${\mathcal {B}}_k^*.$$ B k ∗ . In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

With Wronskian through the Looking Glass

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plücker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.

Permutations of zero-sumsets in a finite vector space

In this paper, we consider a finite-dimensional vector space {{\mathcal{P}}} over the Galois field {\operatorname{GF}(p)}, with p being an odd prime, and the family {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for {x=0}, of the permutations of {\mathcal{P}} inducing permutations of {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space {\mathcal{P}} if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.

Existence of continuous right inverses to linear mappings in finite-dimensional geometry

A linear mapping of a compact convex subset of a finite-dimensional vector space always possesses a right inverse, but may lack a continuous right inverse, even if the set is smoothly bounded. Examples showing this are given, as well as conditions guaranteeing the existence of a continuous right inverse.

Eccentric topological properties of a graph associated to a finite dimensional vector space

A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

Covariants, Invariant Subsets, and First Integrals

Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be a finite-dimensional vector space. Let $End(V)$ be the semigroup of all polynomial endomorphisms of $V$. Let $E$ be a subset of $End(V)$ which is a linear subspace and also a semi-subgroup. Both $End(V)$ and $E$ are ind-varieties which act on $V$ in the obvious way. In this paper, we study important aspects of such actions. We assign to $E$ a linear subspace $D_{E}$ of the vector fields on $V$. A subvariety $X$ of $V$ is said to $D_{E}$ -invariant if $h(x)$ is in the tangent space of $x$ for all $h$ in $D_{E}$ and $x$ in $X$. We show that $X$ is $D_{E}$ -invariant if and only if it is the union of $E$-orbits. For such $X$, we define first integrals and construct a quotient space for the $E$-action. An important case occurs when $G$ is an algebraic subgroup of $GL(V$) and $E$ consists of the $G$-equivariant polynomial endomorphisms. In this case, the associated $D_{E}$ is the space the $G$-invariant vector fields. A significant question here is whether there are non-constant $G$-invariant first integrals on $X$. As examples, we study the adjoint representation, orbit closures of highest weight vectors, and representations of the additive group. We also look at finite-dimensional irreducible representations of SL2 and its nullcone.

Response and Sensitivity Using Markov Chains

<p>Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study</p><p>how this affects their statistical properties. A prominent real-life example of this class of problems is the investigation</p><p>of climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system is</p><p>as a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study the</p><p>response and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto a</p><p>suitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markov</p><p>processes. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinear</p><p>response of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on the</p><p>evolution law of densities (the Liouville/Fokker-Planck equation), allowing to effectively calculate the sensitivity and</p><p>response of two representative dynamical systems.</p>