arXiv subjects

Chao Zhou

Publications and source records attributed to Chao Zhou.

At least 145 records · Page 8Linked to original sources

arXiv2022

The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation

Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development. In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the $\ell^2$ grid error is bounded by the sum of $O(h^p)$ and the network approximation error, where $h$ is the time step size and $p$ is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.

math.NA↗

arXiv2022

Robust control problems of BSDEs coupled with value functions

A robust control problem is considered in this paper, where the controlled stochastic differential equations (SDEs) include ambiguity parameters and their coefficients satisfy non-Lipschitz continuous and non-linear growth conditions, the objective function is expressed as a backward stochastic differential equation (BSDE) with the generator depending on the value function. We establish the existence and uniqueness of the value function in a proper space and provide a verification theorem. Moreover, we apply the results to solve two typical optimal investment problems in the market with ambiguity, one of which is with Heston stochastic volatility model. In particular, we establish some sharp estimations for Heston model with ambiguity parameters.

q-fin.MF↗

arXiv2022

Backward Stochastic Differential Equations and Backward Stochastic Volterra Integral Equations with Anticipating Generators

For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might not admit adapted solutions, shown by an example. However, for backward stochastic Volterra integral equations (BSVIEs, for short), the generators are allowed to be anticipating. This gives, among other things, an essential difference between BSDEs and BSVIEs. Under some proper conditions, the well-posedness of such kinds of BSVIEs is established. Further, the results are extended to path-dependent BSVIEs, in which the generators can depend on the future paths of unknown processes. An additional finding is that for path-dependent BSVIEs, in general, the situation of anticipating generators is not avoidable and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng--Yang [22] is not necessary.

math.PR↗

arXiv2022

Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding $N$-player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from LQ mean field games of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the $N$-player games directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the $N$-player game to the mean field game and the propagation of chaos property for the related optimal trajectories.

math.OC↗

arXiv2022

Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features

In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.

math.NA↗

arXiv2022

Mean Field Portfolio Games

We study mean field portfolio games with random market parameters, where each player is concerned with not only her own wealth but also relative performance to her competitors. We use the martingale optimality principle approach to characterize the unique Nash equilibrium in terms of a mean field FBSDE with quadratic growth, which is solvable under a weak interaction assumption. Motivated by the weak interaction assumption, we establish an asymptotic expansion result in powers of the competition parameter. When the market parameters do not depend on the Brownian paths, we obtain the Nash equilibrium in closed form.

q-fin.MF↗

arXiv2022

Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE, for short) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a Type-II backward SVIE (BSVIE, for short). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a Type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding Type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback.

math.OC↗

arXiv2022

A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. Unlike the previous related approaches in simpler settings, our results do not only rely on a simple application of Itô's formula and classical estimates, but use crucially appropriate generalizations of deep estimates for supermartingales obtained by Meyer. As an example of application, we prove that reflected BSDEs are well-posed in a general framework which has not been covered so far in the existing literature.

math.PR↗

arXiv2022

Power Forward Performance in Semimartingale Markets with Stochastic Integrated Factors

We study the forward investment performance process (FIPP) in an incomplete semimartingale market model with closed and convex portfolio constraints, when the investor's risk preferences are of the power form. We provide necessary and sufficient conditions for the construction of such a performance process, and show that it can be recovered as the unique solution of an infinite horizon quadratic backward stochastic differential equation (BSDE) with a nonmonotone driver. In an integrated stochastic factor model, we relate the factor representation of the BSDE solution to the smooth solution of an ill-posed partial integro-differential Hamilton-Jacobi-Bellman (HJB) equation. We provide an explicit construction of the BSDE solution for the class of time-monotone FIPPs, generalizing existing results from Brownian to semimartingale market models.

math.PR↗

arXiv2022

Power Forward Performance in Semimartingale Markets with Stochastic Integrated Factors

We study the forward investment performance process (FIPP) in an incomplete semimartingale market model with closed and convex portfolio constraints, when the investor's risk preferences are of the power form. We provide necessary and sufficient conditions for the existence of such FIPP. In a semimartingale factor model, we show that the FIPP can be recovered as a triplet of processes which admit an integral representation with respect to semimartingales. Using an integrated stochastic factor model, we relate the factor representation of the triplet of processes to the smooth solution of an ill-posed partial integro-differential Hamilton-Jacobi-Bellman (HJB) equation. We develop explicit constructions for the class of time-monotone FIPPs, generalizing existing results from Brownian to semimartingale market models.

q-fin.PM↗

arXiv2022

Scalable production of single 2D van der Waals layers through atomic layer deposition: Bilayer silica on metal foils and films

The self-limiting nature of atomic layer deposition (ALD) makes it an appealing option for growing single layers of two-dimensional van der Waals (2D-VDW) materials. In this paper it is demonstrated that a single layer of a 2D-VDW form of SiO2 can be grown by ALD on Au and Pd polycrystalline foils and epitaxial films. The silica was deposited by two cycles of bis (diethylamino) silane and oxygen plasma exposure at 525 K. Initial deposition produced a three-dimensionally disordered silica layer; however, subsequent annealing above 950 K drove a structural rearrangement resulting in 2D-VDW; this annealing could be performed at ambient pressure. Surface spectra recorded after annealing indicated that the two ALD cycles yielded close to the silica coverage obtained for 2D-VDW silica prepared by precision SiO deposition in ultra-high vacuum. Analysis of ALD-grown 2D-VDW silica on a Pd(111) film revealed the co-existence of amorphous and incommensurate crystalline 2D phases. In contrast, ALD growth on Au(111) films produced predominantly the amorphous phase while SiO deposition in UHV led to only the crystalline phase, suggesting that the choice of Si source can enable phase control.

cond-mat.mtrl-sci↗

arXiv2021

Portfolio diversification and model uncertainty: a robust dynamic mean-variance approach

This paper focuses on a dynamic multi-asset mean-variance portfolio selection problem under model uncertainty. We develop a continuous time framework for taking into account ambiguity aversion about both expected return rates and correlation matrix of the assets, and for studying the join effects on portfolio diversification. The dynamic setting allows us to consider time varying ambiguity sets, which include the cases where the drift and correlation are estimated on a rolling window of historical data or when the investor takes into account learning on the ambiguity. In this context, we prove a general separation principle for the associated robust control problem, which allows us to reduce the determination of the optimal dynamic strategy to the parametric computation of the minimal risk premium function. Our results provide a justification for under-diversification, as documented in empirical studies and in the static models [16], [34]. Furthermore, we explicitly quantify the degree of under-diversification in termsof correlation bounds and Sharpe ratios proximities, and emphasize the different features induced by drift and correlation ambiguity. In particular, we show that an investor with a poor confidence in the expected return estimation does not hold any risky asset, and on the other hand, trades only one risky asset when the level of ambiguity on correlation matrix is large. We also provide a complete picture of the diversification for the optimal robust portfolio in the three-asset case JEL Classification: G11, C61 MSC Classification: 91G10, 91G80, 60H30

q-fin.PM↗

arXiv2021

Robust lEarned Shrinkage-Thresholding (REST): Robust unrolling for sparse recover

In this paper, we consider deep neural networks for solving inverse problems that are robust to forward model mis-specifications. Specifically, we treat sensing problems with model mismatch where one wishes to recover a sparse high-dimensional vector from low-dimensional observations subject to uncertainty in the measurement operator. We then design a new robust deep neural network architecture by applying algorithm unfolding techniques to a robust version of the underlying recovery problem. Our proposed network - named Robust lEarned Shrinkage-Thresholding (REST) - exhibits an additional normalization processing compared to Learned Iterative Shrinkage-Thresholding Algorithm (LISTA), leading to reliable recovery of the signal under sample-wise varying model mismatch. The proposed REST network is shown to outperform state-of-the-art model-based and data-driven algorithms in both compressive sensing and radar imaging problems wherein model mismatch is taken into consideration.

cs.LG↗

arXiv2021

Robust exploratory mean-variance problem with drift uncertainty

We solve a min-max problem in a robust exploratory mean-variance problem with drift uncertainty in this paper. It is verified that robust investors choose the Sharpe ratio with minimal $L^2$ norm in an admissible set. A reinforcement learning framework in the mean-variance problem provides an exploration-exploitation trade-off mechanism; if we additionally consider model uncertainty, the robust strategy essentially weights more on exploitation rather than exploration and thus reflects a more conservative optimization scheme. Finally, we use financial data to backtest the performance of the robust exploratory investment and find that the robust strategy can outperform the purely exploratory strategy and resist the downside risk in a bear market.

math.OC↗

arXiv2021

A plasmonic nanorod that walks on DNA origami

In nano-optics, a formidable challenge remains in precise transport of a single optical nano-object along a programmed and routed path toward a predefined destination. Molecular motors in living cells that can walk directionally along microtubules have been the inspiration for realizing artificial molecular walkers. Here we demonstrate an active plasmonic system, in which a plasmonic nanorod can execute directional, progressive and reverse nanoscale walking on two or three-dimensional DNA origami.Such a walker comprises an anisotropic gold nanorod as its body and discrete DNA strands as its feet. Specifically, our walker carries optical information and can in situ optically report its own walking directions and consecutive steps at nanometer accuracy, through dynamic coupling to a plasmonic stator immobilized along its walking track. Our concept will enable a variety of smart nanophotonic platforms for studying dynamic light-matter interaction, which requires controlled motion at the nanoscale well below the optical diffraction limit.

physics.bio-ph↗

arXiv2021

A rotary plasmonic nanoclock

One of the fundamental challenges in nanophotonics is to gain full control over nanoscale optical elements. The precise spatiotemporal arrangement determines their interactions and collective behavior. To this end, DNA nanotechnology is employed as an unprecedented tool to create nanophotonic devices with excellent spatial addressability and temporal programmability. However, most of the current DNA-assembled nanophotonic devices can only reconfigure among random or very few defined states. Here, we demonstrate a DNA-assembled rotary plasmonic nanoclock. In this system, a rotor gold nanorod can carry out directional and reversible 360 degree rotation with respect to a stator gold nanorod, transitioning among 16 well-defined configurations powered by DNA fuels. The full-turn rotation process is monitored by optical spectroscopy in real time. We further demonstrate autonomous rotation of the plasmonic nanoclock powered by DNAzyme-RNA interactions. Such assembly approaches pave a viable route towards advanced nanophotonic systems entirely from the bottom-up.

physics.app-ph↗

arXiv2021

Dynamic Plasmonic System That Responds to Thermal and Aptamer-Target Regulations

The DNA origami technique has empowered a new paradigm in plasmonics for manipulating light and matter at the nanoscale. This interdisciplinary field has witnessed vigorous growth, outlining a viable route to construct advanced plasmonic architectures with tailored optical properties. However, so far plasmonic systems templated by DNA origami have been restricted to respond to only single stimuli. Despite broad interest and scientific importance, thermal and aptamer-target regulations have not yet been widely utilized to reconfigure three-dimensional plasmonic architectures. In this Letter, we demonstrate a chiral plasmonic nanosystem integrated with split aptamers, which can respond to both thermal and aptamer-target regulations. We show that our dual-responsive system can be noninvasively tuned in a wide range of temperatures, readily correlating thermal control with optical signal changes. Meanwhile, our system can detect specific targets including adenosine triphosphate and cocaine molecules, which further enhance the optical response modulations and in turn influence the thermal tunability.

physics.app-ph↗

arXiv2021

Watching a Single Fluorophore Molecule Walk into a Plasmonic Hotspot

Plasmonic nanoantennas allow for enhancing the spontaneous emission, altering the emission polarization, and shaping the radiation pattern of quantum emitters. A critical challenge for the experimental realizations is positioning a single emitter into the hotspot of a plasmonic antenna with nanoscale accuracy. We demonstrate a dynamic light-matter interaction nanosystem enabled by the DNA origami technique. A single fluorophore molecule can autonomously and unidirectionally walk into the hotspot of a plasmonic nanoantenna along a designated origami track. Successive fluorescence intensity increase and lifetime reduction are in situ monitored using single-molecule fluorescence spectroscopy, while the fluorophore walker gradually approaches and eventually enters the plasmonic hotspot. Our scheme offers a dynamic platform, which can be used to develop functional materials, investigate intriguing light-matter interaction phenomena, and serve as prototype system for examining theoretical models.

physics.optics↗