A Math Circle in Third Grade
Problems developed for a mathematics circle conducted throughout the school year in a primary school class, involving all pupils without any selection, thus working with a heterogeneous group of children.
math.HO↗
arXiv subjects
Evgeny Lakshtanov
Publications and source records attributed to Evgeny Lakshtanov.
Unbiased Monte Carlo Greeks for Discontinuous Payoffs
Pathwise differentiation of Monte Carlo estimators fails at payoff discontinuities, producing zero or biased sensitivities for barriers, autocallables, and digital options. The industry workaround --- smoothing the indicator functions --- introduces bias and requires per-product calibration. We derive a correction formula that restores unbiased Greeks without smoothing. For a payoff $F(Z,θ)$ that is piecewise smooth with discontinuities on surfaces $\{g_i = 0\}$, we show that the sensitivity decomposes into a pathwise term (computed by standard AAD) plus a sum of boundary corrections, each involving the payoff jump, the Gaussian density at the boundary, and the sensitivity of the boundary to the parameter. The correction is computed by Newton root-finding in the normal-random space, with the jump evaluated by two forward replays of the pricing kernel. The implementation uses AADC (\texttt{pip install aadc}), whose tape replay and automatic discontinuity tracking make the method fully automatic --- the quant writes standard pricing code, and the correction driver identifies and handles all discontinuities. We prove the formula for arbitrary compositions of smooth functions and indicator functions (not just outer products), covering real autocallable payoff structures with recursive alive/dead logic. Benchmarks on QuantLib models (GBM, Heston, Hull-White) show all Greeks within 0.1--4\% of analytic or bump-and-revalue references.
q-fin.CP↗
A C++ implementation of the discrete adjoint sensitivity analysis method for explicit adaptive Runge-Kutta methods enabled by automatic adjoint differentiation and SIMD vectorization
A C++ library for sensitivity analysis of optimisation problems involving ordinary differential equations (ODEs) enabled by automatic differentiation (AD) and SIMD (Single Instruction, Multiple data) vectorization is presented. The discrete adjoint sensitivity analysis method is implemented for adaptive explicit Runge-Kutta (ERK) methods. Automatic adjoint differentiation (AAD) is employed for efficient evaluations of products of vectors and the Jacobian matrix of the right hand side of the ODE system. This approach avoids the low-level drawbacks of the black box approach of employing AAD on the entire ODE solver and opens the possibility to leverage parallelization. SIMD vectorization is employed to compute the vector-Jacobian products concurrently. We study the performance of other methods and implementations of sensitivity analysis and we find that our algorithm presents a small advantage compared to equivalent existing software.
math.NA↗
Denoised Monte Carlo for option pricing and Greeks estimation
We present a novel technique of Monte Carlo error reduction that finds direct application in option pricing and Greeks estimation. The method is applicable to any LSV modelling framework and concerns a broad class of payoffs, including path-dependent and multi-asset cases. Most importantly, it allows to reduce the Monte Carlo error even by an order of magnitude, which is shown in several numerical examples.
q-fin.PR↗
Adjoint Differentiation for generic matrix functions
We derive a formula for the adjoint $\overline{A}$ of a square-matrix operation of the form $C=f(A)$, where $f$ is holomorphic in the neighborhood of each eigenvalue. We then apply the formula to derive closed-form expressions in particular cases of interest such as the case when we have a spectral decomposition $A=UDU^{-1}$, the spectrum cut-off $C=A_+$ and the Nearest Correlation Matrix routine. Finally, we explain how to simplify the computation of adjoints for regularized linear regression coefficients.
q-fin.CP↗
Remarks on stochastic automatic adjoint differentiation and financial models calibration
In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form $G=\frac{1}{2}\sum_1^m (Ey_i-C_i)^2$, which often appear in the calibration of stochastic models. { We demonstrate that it allows a perfect SIMD\footnote{Single Input Multiple Data} parallelization and provide its relative computational cost. In addition we demonstrate that this theoretical result is in concordance with numeric experiments.}
q-fin.CP↗
Designing lasing and perfectly absorbing potentials
Existence of a spectral singularity (SS) in the spectrum of {a Schrödinger operator with} a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, $\PT-$symmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, $\PT-$symmetric potential. We illustrate all these opportunities with examples. We also describe splitting of a second-order SSs under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.
physics.optics↗
Global solution of the initial value problem for the focusing Davey-Stewartson II system
We consider the two dimensional focusing Davey-Stewartson II system and construct the global solution of the Cauchy problem for a dense in $L^2(\mathbb C)$ set of initial data. We do not assume that the initial data is small. So, the solutions may have singularities. We show that the blow-up may occur only on a real analytic variety and the variety is bounded in each strip $t \leq T$.
math-ph↗
Boris R. Vainberg (on his 80th birthday)
Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.
math.HO↗
On reconstruction of complex-valued once differentiable conductivities
The classical $\overline \partial$-method has been generalized recently [lnv], [lnv2] to be used in the presence of exceptional points. We apply this generalization to solve Dirac inverse scattering problem with weak assumptions on smoothness of potentials. As a consequence, this provides an effective method of reconstruction of complex-valued one time differentiable conductivities in the inverse impedance tomography problem.
math-ph↗
Solution of the initial value problem for the focusing Davey-Stewartson II system
We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves).
nlin.SI↗
Uniqueness in the inverse conductivity problem for complex-valued Lipschitz conductivities in the plane
We consider the impedance tomography problem in the plane. Using Bukhgeim's scattering data for the Dirac problem, we prove that the conductivity is uniquely determined by the Dirichlet-to-Neuman map
math-ph↗
Recovery of $L^p$-potential in the plane
An inverse problem for the two-dimensional Schrodinger equation with $L^p_{com}$-potential, $p>1$, is considered. Using the $\overline{\partial}$-method, the potential is recovered from the Dirichlet-to-Neumann map on the boundary of a domain containing the support of the potential. We do not assume that the potential is small or that the Faddeev scattering problem does not have exceptional points. The paper contains a new estimate on the Faddeev Green function that immediately implies the absence of exceptional points near the origin and infinity when $v\in L^p_{com}$.
math-ph↗
Uniqueness in potential scattering with reduced near field data
We consider inverse potential scattering problems where the source of the incident waves is located on a smooth closed surface outside of the inhomogeneity of the media. The scattered waves are measured on the same surface at a fixed value of the energy. We show that this data determines the bounded potential uniquely.
math-ph↗
A Factorization Method and Monotonicity Bounds in Inverse Medium Scattering for Contrasts with Fixed Sign on the Boundary
We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. Whilst the factorization method been used so far mainly to identify the shape of a scatterer's support, we show that factorizations based on Dirichlet-to-Neumann operators can be used to compute bounds for numerical values of the medium on the boundary of its support. To this end, we generalize ideas from inside-outside duality to obtain a monotonicity principle that allows for alternative uniqueness proofs for particular inverse scattering problems (e.g., when obstacles are present inside the medium). This monotonicity principle indeed is our most important technical tool: It further directly shows that the boundary values of the medium's contrast function are uniquely determined by the corresponding far field operator. Our particular factorization of far field operators additionally implies that the factorization method rigorously characterizes the support of an inhomogeneous medium if the contrast function takes merely positive or negative values on the boundary of its support, independent of the contrast's values inside its support. Finally, the monotonicity principle yields a simple algorithm to compute upper and lower bounds for these boundary values, assuming the support of the contrast is known. Numerical experiments show feasibility of a resulting numerical algorithm.
math.AP↗
Recovery of interior eigenvalues from reduced near field data
We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can be arbitrarily small but fixed.The scattered waves are measured on the same surface. An effective procedure is suggested for recovery of interior eigenvalues by these data.
math-ph↗
Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues
We consider the interior transmission eigenvalue (ITE) problem, which arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (when the latter approach {\bf$z=1$)}. We obtain a Weyl type formula for the counting function of positive ITEs, which are taken together with ascribed signs.
math-ph↗
Exceptional points in Faddeev scattering problem
Exceptional points are values of the spectral parameter for which the homogeneous Faddeev scattering problem has a non-trivial solution. We study the existence/absence of exceptional points for small perturbations of conductive potentials of arbitrary shape and show that problems with absorbing potentials do not have exceptional points in a neighborhood of the origin. A criterion for existence of exceptional points is given.
math-ph↗