It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also allows us to characterize Riccati chain equations geometrically in terms of the pro…
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48 results for “Riccati equation”
Paper derives Riccati equation for static spaces and proves its applications.
problem Deriving Riccati equation for static spaces.
method Proving splitting theorem and connectivity of conformal boundary.
result Establishes compactness of universal covering for static triples.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We study algebraic solutions of the Riccati equation over the field of rational functions , and over the elliptic function field .
New techniques solve Riccati equations on 3D manifolds, finding 4th order metric obstructions.
problem Solving Riccati-type equations with algebraic constraints on 3D Riemannian manifolds.
method Real algebraic geometry techniques, focusing on connection coefficients and Hessian equations.
result Obstruction to solving Riccati equations has order 4 in metric coefficients.
New theory extends LQ control to non-exponential discount scenarios.
problem Time-inconsistent deterministic LQ control problems.
method Extended equivalent relationship to non-exponential discount functions, studied Riccati equation solvability.
result Existence and uniqueness of linear equilibrium for time-inconsistent LQ problem.
In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical perspective. The nonlinear superposition principle also arises in a simple way.
Investigates mean-variance portfolio selection in non-Markovian markets.
problem Continuous-time Markowitz mean-variance portfolio selection in fake stationary affine Volterra models.
method Stochastic factor solution to a Riccati BSDE, deriving explicit solutions as multi-dimensional Riccati-Volterra equations.
result Analytical closed-form expressions for optimal portfolio policies and mean-variance efficient frontier.
A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategiesq-fin.MF
Study optimizes portfolio liquidation strategies with complex market impacts.
problem Optimizing portfolio liquidation with transient market impacts and self-exciting order flow.
method Mean-field control problem with semimartingale strategies, passing to continuous-time limit, and solving Riccati equations.
result Existence of optimal strategy with jumps only at start and end of trading period.
A new tontine design aims to protect longevity risk with non-indexed investments.
problem Pooling longevity risk with traditional methods.
method Non-indexed investments with negatively correlated returns to mortality.
result Mathematical proof of recovery schedule using a Riccati equation.
Mapping Among Line Elementsmath.DG
Paper maps Hamiltonians and line elements in manifolds.
problem Mapping among generalized Hamiltonians and line elements.
method Constructing Calabi's Riemannian Line Elements and solving matrix Riccati equations.
result Analytical and exact solutions of mapping between manifolds.
This paper optimizes portfolio selection for multivariate affine and quadratic Volterra models with rough volatilities.
problem Optimizing portfolio selection for multivariate models with rough volatilities and stochastic correlations.
method Investigates continuous-time Markowitz mean-variance problem for multivariate affine and quadratic Volterra models using Riccati backward stochastic differential equations (BSDEs).
result Derives explicit solutions for BSDEs in affine Volterra models and new analytic formulae for quadratic models.
We propose two methods to obtain exact solutions for the Almgren-Chriss model about optimal execution of portfolio transactions. In the first method we rewrite the Almgren-Chriss equation and find two exact solutions. In the second method, employing a general reparametrized time, we show that the Almgren-Chriss equatio…
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Br…
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
Investigates optimal investment strategies in financial markets with jumps.
problem Optimal portfolio selection for investors in multi-asset financial markets with jumps.
method Uses martingale optimality principle and Riccati backward stochastic differential equations with jumps.
result Derives semi-closed form optimal strategies and value function for Merton's problem.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to -dimensional Lorentz--Finsler manifolds.
result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
We solve a family of fractional Riccati differential equations with constant (possibly complex) coefficients. These equations arise, e.g., in fractional Heston stochastic volatility models, that have received great attention in the recent financial literature thanks to their ability to reproduce a rough volatility beha…
Paper solves Merton's portfolio problem in a non-Markovian, non-semimartingale model.
problem Merton's portfolio optimization in a fake stationary Volterra-Heston model.
method Stochastic factor solution to a Riccati BSDE, combined with martingale optimality principle.
result Derives semi-closed form optimal strategies and value function.
Solvability of Differential Riccati Equations and Applications to Algorithmic Trading with Signalsq-fin.TR
Study solves DREs for trading strategies using signals and past prices.
problem Solving DREs for optimal trading strategies.
method Analyzes DREs with indefinite matrix coefficients and applies to trading problems.
result Derives optimal trading strategies using signals and past prices.
Expanding the rough Heston model in q-fin.MF
Expanding the rough Heston model in
problem Analyzing the dependence of the fractional Riccati equation on the Hurst parameter
method Deriving a Taylor expansion of the Riccati solution in
result Local uniform convergence and analyticity of the fractional Riccati solution
We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to simila…
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
Affine forward variance modelsq-fin.MF
We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equat…
Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.
Some partial differential equations and conformal surfaces of the 4-dimensional Minkowski spacemath.DG
This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space , based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of , the representation then allows us to obtain interesting part…
In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called Riccati met…
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
Study of J-Hermitian matrices and geometric mean definition.
problem Understanding the cone of J-Hermitian matrices and its geometric mean.
method Analysis of the cone structure, Riemannian structure, and definition of J-geometric mean.
result Uniquely characterized J-geometric mean defined as a solution to a Riccati-type equation.
The paper solves a complex financial optimization problem using a novel mathematical technique.
problem Optimizing portfolio selection in financial markets.
method Maximal monotone operator method and Riccati transformation.
result Existence and uniqueness of a solution to the transformed parabolic equation in a Sobolev space.
Study bond market making with hit-ratio target using optimal control and HJB equations.
problem Optimizing bond market making with hit-ratio target in OTC markets.
method Stochastic optimal control approach, dualizing hit-ratio target, HJB equation, Riccati equation, linearization.
result Explicit quote decompositions into riskless spread, inventory-risk correction, and hit-ratio correction.
Model liquidity premia using a risk-sharing economy with quadratic costs.
problem Understanding the cross-section of liquidity premia earned by assets with different trading costs.
method Developed a risk-sharing economy model with quadratic transaction costs, leading to matrix-valued Riccati equations for equilibrium.
result Calibrated model to time series data, revealing liquidity premia across assets with varying trading costs.
Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
Affine connections linked to Riccati distributions on compact surfaces.
problem Understanding affine structures on complex compact surfaces.
method Established a correspondence between affine connections and Riccati distributions.
result One-to-one correspondence between affine structures and Riccati foliations on compact surfaces.
Investigates optimal portfolio selection with regime-switching-induced stock price shocks.
problem Mean-variance portfolio selection with regime-switching and stock price jumps.
method Modeling regime-switching and stock price jumps, deriving optimal portfolio strategy and efficient frontier using ODEs.
result Added complexity due to regime-switching-induced stock price shocks, leading to nonlinear ODEs.
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, char…
The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…
Construct geometric interpretation of Heston model using group quantization.
problem Geometric interpretation of Heston model
method Lifted local Lie groupoid formulation
result Geometric interpretation of Heston pricing operator and Riccati equations
We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenböck formula and Laplacian comparison theorem, and study the h…
This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
problem Optimizing trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
method Relative entropy-regularized robust optimal control problem, modeled as a stochastic differential game.
result Analytical expressions for optimal strategy and trajectory are derived under specific assumptions.
New method solves stochastic control problems with delays using deep learning.
problem Stochastic control problems with delayed control in drift and diffusion.
method Characterization via Riccati PDEs and deep learning scheme.
result Illustrates effect of delay on Markowitz portfolio allocation problem.
Novel signature approach for pricing and hedging path-dependent options with market frictions.
problem Pricing and hedging path-dependent options with market frictions.
method Signature approach, mean-quadratic variation criterion, non-standard infinite-dimensional Riccati equations, time-augmented signature, non-Markovian stochastic control problem.
result Effective hedging strategies in frictional markets with low-truncated signature approximations.
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…
Paper solves complex game theory problems with new equations.
problem Zero-sum stochastic games with non-Markovian switching.
method New multidimensional SRE and BSDE solutions.
result Existence and uniqueness of SRE solutions.
We determine the homogeneous Kähler diffeomorphism which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…
In the continuous time mean-variance model, we want to minimize the variance (risk) of the investment portfolio with a given mean at terminal time. However, the investor can stop the investment plan at any time before the terminal time. To solve this kind of problem, we consider to minimize the variances of the investm…
This paper refines the Gaussian Sinkhorn algorithm for general multivariate models.
problem Finite-dimensional solutions for general Gaussian multivariate models.
method Recursive formulation of the Sinkhorn algorithm for Gaussian models, including closed form expressions of entropic transport maps and Schrödinger bridges.
result Refined convergence analysis of Gaussian Sinkhorn algorithms.
Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…