Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Professor of Mathematics



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Publication Type

67 results found

jialiang L, Zheng H, 2022, Deep neural network solution for finite state mean field game with error estimation, Dynamic Games and Applications, ISSN: 2153-0785

We discuss the numerical solution to a class of continuous time finite state mean field games. We apply the deep neural network (DNN) approach to solving the fully-coupled forward and backward ordinary differential equation (ODE) system that characterizes the equilibrium value function and probability measure of the finite state mean field game. We prove that the error between the true solution and the approximate solution is linear to the square root of DNN loss function. We give an example of applying the DNN method to solve the optimal market making problem with terminal rank based trading volume reward.

Journal article

Zhu D, Zheng H, 2022, Effective approximation methods for constrained utility maximization with drift uncertainty, Journal of Optimization Theory and Applications, Vol: 194, Pages: 191-219, ISSN: 0022-3239

In this paper we propose a novel and effective approximation method for finding the value function for general utility maximization with closed convex control constraints and partial information. Using the separation principle and the weak duality relation, we transform the stochastic maximum principle of the fully-observable dual control problem into an equivalenterror minimization stochastic control problem and find the tight lower and upper bounds of the value function and its approximate value. Numerical examples show the goodness and usefulness of the proposed method.

Journal article

Zheng H, Jang HJ, Xu Z, 2022, Optimal investment, heterogeneous consumption and best time for retirement, Operations Research, ISSN: 0030-364X

This paper studies an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of timefor retirement. The utility for luxury goods is not necessarily a concave function.The optimal heterogeneous consumption strategies for a class of non-homotheticutility maximizer are shown to consume only basic goods when the wealth is small,to consume basic goods and make savings when the wealth is intermediate, and toconsume almost all in luxury goods when the wealth is large. The optimal retirement policy is shown to be both universal, in the sense that all individuals shouldretire at the same level of marginal utility that is determined only by income, labor cost, discount factor as well as market parameters, and not universal, in thesense that all individuals can achieve the same marginal utility with different utilityand wealth. It is also shown that individuals prefer to retire as time goes by ifthe marginal labor cost increases faster than that of income. The main tools usedin analyzing the problem are from PDE and stochastic control theory includingvariational inequality and dual transformation. We finally conduct the simulationanalysis for the featured model parameters to investigate practical and economicimplications by providing their figures.

Journal article

Cesari R, Zheng H, 2022, Stochastic maximum principle for optimal liquidation with control-dependent terminal time, Applied Mathematics and Optimization, Vol: 85, Pages: 1-32, ISSN: 0095-4616

In this paper we study a general optimal liquidation problem with a control-dependent stopping time which is the first time the stock holding becomes zero or a fixed terminal time, whichever comes first. We prove a stochastic maximum principle (SMP) which is markedly different in its Hamiltonian condition from that of the standard SMP with fixed terminal time. We present a simple example in which the optimal solution satisfies the SMP in this paper but fails the standard SMP in the literature.

Journal article

davey A, Zheng H, 2021, Deep learning for constrained utility maximisation, Methodology and Computing in Applied Probability, Vol: 24, Pages: 661-692, ISSN: 1387-5841

This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

Journal article

Davey A, Monoyios M, Zheng H, 2021, Duality for optimal consumption with randomly terminating income, Mathematical Finance, Vol: 31, Pages: 1275-1314, ISSN: 0960-1627

We establish a rigorous duality theory, under No Unbounded Profit with Bounded Risk, for an infinite horizon problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered in the Davis-Vellekoop example in a version of this problem in a Black-Scholes market. Many of the classical tenets of duality theory hold, with the notable exception that marginal utility at zero initial wealth is finite. We use as dual variables a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption in excess of income is a supermartingale. We show that the space of discounted local martingale deflators is dense in our dual domain, so that the dual problem can also be expressed as an infimum over the discounted local martingale deflators. We characterize the optimal wealth process, showing that optimal deflated wealth is a potential decaying to zero, while deflated wealth plus cumulative deflated consumption over income is a uniformly integrable martingale at the optimum. We apply the analysis to the Davis-Vellekoop example and give a numerical solution.

Journal article

Wang T, Zheng H, 2021, Closed-loop equilibrium strategies for general time-inconsistent optimal control problems, SIAM Journal on Control and Optimization, Vol: 59, Pages: 3152-3178, ISSN: 0363-0129

In this paper we introduce a general framework for time-inconsistent optimal control problems. We characterize the closed-loop equilibrium strategy in both the integral and point wise forms with the newly developed methodology. We recover and improve the results of some well-known models, including the classical optimal control, Bjork et al. (2017), He and Jiang(2020), and Yong (2012) models, and reveal some interesting aspects that appear for the first time in the literature. We illustrate the usefulness of the model and the results by a number of examples in dynamic portfolio selection, including mean-variance with state-dependent risk aversion, investment/consumption with non-exponential discounting, and utility-deviation-risk with coupled terminal state and expected terminal state.

Journal article

Choi SE, Jang HJ, Lee K, Zheng Het al., 2021, Optimal market-Making strategies under synchronised order arrivals with deep neural networks, Journal of Economic Dynamics and Control, Vol: 125, Pages: 1-25, ISSN: 0165-1889

This study investigates the optimal execution strategy of market-making for market and limit order arrival dynamics under a novel framework that includes a synchronised factor between buy and sell order arrivals. Using statistical tests, we empirically confirm that a synchrony propensity appears in the market, where a buy order arrival tends to follow the sell order’s long-term mean level and vice versa. This is presumably closely related to the drastic increase in the influence of high-frequency trading activities in markets. To solve the high-dimensional Hamilton–Jacobi–Bellman equation, we propose a deep neural network approximation and theoretically verify the existence of a network structure that guarantees a sufficiently small loss function. Finally, we implement the terminal profit and loss profile of market-making using the estimated optimal strategy and compare its performance distribution with that of other feasible strategies. We find that our estimation of the optimal market-making placement allows significantly stable and steady profit accumulation over time through the implementation of strict inventory management.

Journal article

Ching W, Gu J, Zheng H, 2021, On correlated defaults and incomplete information, Journal of Industrial and Management Optimization, Vol: 17, Pages: 889-908, ISSN: 1547-5816

In this paper, we study a continuous time structural asset value model for two correlatedfirms using a two-dimensional Brownian motion. We consider the situation of incompleteinformation, where the information set available to the market participants includes the defaulttime of each firm and the periodic asset value reports. In this situation, the default timeof each firm becomes a totally inaccessible stopping time to the market participants. Theoriginal structural model is first transformed to a reduced-form model. Then the conditionaldistribution of the default time together with the asset value of each name are derived. Weprove the existence of the intensity processes of default times and also give the explicit formof the intensity processes. Numerical studies on the intensities of the two correlated names areconducted for some special cases.

Journal article

Gu J-W, Steffensen M, Zheng H, 2021, A note on P- vs. Q-expected loss portfolio constraints, Quantitative Finance, Vol: 21, Pages: 263-270, ISSN: 1469-7688

We consider portfolio optimization problems with expected loss constraints un-der the physical measure P and the risk neutral measure Q, respectively. Using Merton’s portfolio as a benchmark portfolio, the optimal terminal wealth of the Q-risk constraint problem can be easily replicated with the standard delta hedg-ing strategy. Motivated by this, we consider the Q-strategy fulfilling the P-risk constraint and compare its solution with the true optimal solution of theP-riskconstraint problem. We show the existence and uniqueness of the optimal solution to theQ-strategy fulfilling theP-risk constraint, and provide a tractable evalua-tion method. The Q-strategy fulfilling the P-risk constraint is not only easier toimplement with standard forwards and puts on a benchmark portfolio than the P-risk constraint problem, but also easier to solve than either of theQ- or P-riskconstraint problem. The numerical test shows that the difference of the values ofthe two strategies (the Q-strategy fulfilling the P-risk constraint and the optimal strategy solving the P-risk constraint problem) is reasonably small.

Journal article

Luo J, Zheng H, 2020, Dynamic equilibrium of market making with price competition, Dynamic Games and Applications, Vol: 11, Pages: 556-579, ISSN: 2153-0785

In this paper, we discuss the dynamic equilibrium of market making with price competition and incomplete information. The arrival of market sell/buy orders follows a pure jump process with intensity depending on bid/ask spreads among market makers and having a looping countermonotonic structure. We solve the problem with the nonzero-sum stochastic differential game approach and characterize the equilibrium value function with a coupled system of Hamilton–Jacobi nonlinear ordinary differential equations. We prove, do not assume a priori, that the generalized Issac’s condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spreads than those derived using a benchmark model without price competition, which indicates the market liquidity would be enhanced in the presence of price competition of market makers.

Journal article

Jang HJ, Jia L, Zheng H, 2020, Why should we invest in CoCos than stocks? An optimal growth portfolio approach, The European Journal of Finance, Vol: 26, Pages: 1606-1622, ISSN: 1351-847X

We investigate an optimal growth portfolio problem with contingent convertible bonds (CoCos). As the conversion risk in CoCos is closely associated with the issuer's capital structure and the stock price at conversion, we model both equity and credit risk to frame this optimisation problem. This study aims to answer two questions that (i) how investors should optimally allocate their financial wealth between a CoCo and a risk-free bond; and (ii) which approach – investing in a CoCo or in a stock issued by the same bank – could result in higher expected returns. First, we derive the dynamic of a coupon-paying CoCo price under a reduced-form approach. We then decompose the problem into pre- and post-conversion regimes to obtain closed-form optimal strategies. A comparative simulation leads us to conclude that, under various market conditions, investing in a CoCo with a risk-free bond provides a higher expected growth than investing in stock.

Journal article

Zheng H, gu J, Si S, 2020, Constrained utility deviation-risk optimization and time-consistent HJB equation, SIAM Journal on Control and Optimization, Vol: 58, Pages: 866-894, ISSN: 0363-0129

In this paper we propose a unified utility deviation-risk model which covers both utilitymaximization and mean-variance analysis as special cases. We derive the time-consistentHamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and significantlyreduce the number of state variables, which makes the HJB equation derived in this papermuch easier to solve than the extended HJB equation in the literature. We illustrate theusefulness of the time-consistent HJB equation with several examples which recover theknown results in the literature and go beyond, including mean-variance model with stochasticvolatility dependent risk aversion, utility deviation-risk model with state dependent riskaversion and control constraint, and constrained portfolio selection model. The numericaland statistical tests show that the utility and deviation-risk have significant impact on theequilibrium control strategy and the distribution of the terminal wealth.

Journal article

Dong Y, Zheng H, 2020, Optimal investment with S-shaped utility and trading and value at risk constraints: An application to defined contribution pension plan, European Journal of Operational Research, Vol: 281, Pages: 341-356, ISSN: 0377-2217

In this paper we investigate an optimal investment problem under loss aversion (S-shaped utility) and with trading and Value-at-Risk (VaR) constraints faced by a defined contribution (DC) pension fund manager. We apply the concavification and dual control method to solve the problem and derive the closed-form representation of the optimal terminal wealth in terms of a controlled dual state variable. We propose a simple and effective algorithm for computing the initial dual state value, the Lagrange multiplier and the optimal terminal wealth. Theoretical and numerical results show that the VaR constraint can significantly impact the distribution of the optimal terminal wealth and may greatly reduce the risk of losses in bad economic states due to loss aversion.

Journal article

Ma J, Li W, Zheng H, 2020, Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model, European Journal of Operational Research, Vol: 280, Pages: 428-440, ISSN: 0377-2217

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution to the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte-Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yaari utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.

Journal article

ma J, Xing J, Zheng H, 2019, Global closed-form approximation of free boundary for optimal investment stopping problems, SIAM Journal on Control and Optimization, Vol: 57, Pages: 2092-2121, ISSN: 0363-0129

In this paper we study a utility maximization problem with both optimal control and opti-mal stopping in a finite time horizon. The value function can be characterized by a variationalequation that involves a free boundary problem of a fully nonlinear partial differential equation.Using the dual control method, we derive the asymptotic properties of the dual value functionand the associated dual free boundary for a class of utility functions, including power and non-HARA utilities. We construct a global closed-form approximation to the dual free boundary,which greatly reduces the computational cost. Using the duality relation, we find the approx-imate formulas for the optimal value function, trading strategy, and exercise boundary for theoptimal investment stopping problem. Numerical examples show the approximation is robust,accurate and fast.

Journal article

Dong Y, Zheng H, 2019, Optimal investment of DC pension plan under short-selling constraints and portfolio insurance, Insurance: Mathematics and Economics, Vol: 85, Pages: 47-59, ISSN: 0167-6687

In this paper we investigate an optimal investment problem under short-selling and portfolio insurance constraints faced by a defined contribution pension fund manager who is loss averse. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the S-shaped utility, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.

Journal article

Bian B, Zheng H, 2019, Turnpike property and convergence rate for an investment and consumption model, Mathematics and Financial Economics, Vol: 13, Pages: 227-251, ISSN: 1862-9660

We discuss the turnpike property for optimal investment and consumption problems. We find there exists a threshold value that determines the turnpike property for investment policy. The threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate. We show that if utilities behave asymptotically like power utilities and satisfy some simple relations with the threshold value, then the turnpike property for investment holds. There is in general no turnpike property for consumption policy. We also provide the rate of convergence and illustrate the main results with examples of power and non-HARA utilities and numerical tests.

Journal article

Jia L, Pistorius M, Zheng H, 2019, Dynamic portfolio optimization with looping contagion risk, SIAM Journal on Financial Mathematics, Vol: 10, Pages: 1-36, ISSN: 1945-497X

In this paper we consider a utility maximization problem with defaultable stocks and looping contagion risk.We assume that the default intensity of one company depends on the stock prices of itself and other companies,and the default of the company induces immediate drops in the stock prices of the surviving companies. Weprove that the value function is the unique viscosity solution of the HJB equation. We also perform somenumerical tests to compare and analyse the statistical distributions of the terminal wealth of log utility andpower utility based on two strategies, one using the full information of intensity process and the other a proxyconstant intensity process.

Journal article

Jang HJ, Na YH, Zheng H, 2018, Contingent convertible bonds with the default risk premium, International Review of Financial Analysis, Vol: 59, Pages: 77-93, ISSN: 1057-5219

Contingent convertible bonds (CoCos) are hybrid instruments characterized by both debt and equity. CoCos are automatically converted into equity or written down when a predefined trigger event occurs. The present study quantifies the issuing bank's default risk that only manifests in the post-conversion period for pricing CoCos depending on a loss-absorbing method. This work aims to reflect the distinct features of equity-conversion CoCos - in contrast to a write-down CoCos - in a valuation framework. Accordingly, we propose a model to compute the ratio of common equity Tier 1 (CET1), which is composed of core capital and risky assets, by employing a geometric Brownian motion and a random variable. Then, we formulate the post-conversion risk premium by measuring the probability with which the bank's CET1 ratio breaches a regulatory default threshold after conversion. Finally, we empirically examine a positive value of the post-conversion risk premium embedded in the market prices of equity-conversion CoCos.

Journal article

li Y, Zheng H, 2018, Dynamic convex duality in constrained utility maximization, Stochastics: An International Journal of Probability and Stochastic Processes, Vol: 90, Pages: 1145-1169, ISSN: 1744-2508

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.

Journal article

Gu J-W, Steffensen M, Zheng H, 2018, Optimal dividend strategies of two collaborating businesses in the diffusion approximation model, Mathematics of Operations Research, Vol: 43, Pages: 377-398, ISSN: 0364-765X

In this paper, we consider the optimal dividend payment strategy for an insurance company that has two collaborating business lines. The surpluses of the business lines are modeled by diffusion processes. The collaboration between the two business lines permits that money can be transferred from one line to another with or without proportional transaction costs, while money must be transferred from one line to another to help both business lines keep running before simultaneous ruin of the two lines eventually occurs.

Journal article

Li Y, Zheng H, 2018, Constrained quadratic risk minimization via forward and backward stochastic differential equations, SIAM Journal on Control and Optimization, Vol: 56, Pages: 1130-1153, ISSN: 0363-0129

In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach, we show that the necessary and sufficient optimality conditions for both the primal and dual problems can be written in terms of processes satisfying a system of forward and backward stochastic differential equations (FBSDEs) together with other conditions. We characterize explicitly the optimal wealth and portfolio processes as functions of adjoint processes from the dual FBSDEs in a dynamic fashion, and vice versa. We apply the results to solve quadratic risk minimization problems with cone constraints and derive the explicit representations of solutions to the extended stochastic Riccati equations for such problems.

Journal article

Ma J, Li W, Zheng H, 2017, Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization, European Journal of Operational Research, Vol: 262, Pages: 851-862, ISSN: 0377-2217

In this paper, we study the dual control approach for the optimal asset allocation problem in a continuous-time regime-switching market. We find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations. These bounds can be tightened with additional controls to the dual process. We suggest a Monte-Carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions, including power, non-HARA and Yaari utilities. The latter two utilities are beyond the scope of any current methods available in finding the value function.

Journal article

Wong KC, Yam SCP, Zheng H, 2017, Utility-deviation-risk portfolio selection, SIAM Journal on Control and Optimization, Vol: 55, Pages: 1819-1861, ISSN: 0363-0129

We here provide a comprehensive study of the utility-deviation-risk portfolio selection problem. By considering the first-order condition for the corresponding objective function, we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem, called the nonlinear moment problem. We then illustrate the application of this general necessity result by revisiting the nonexistence of the optimal solution for the mean-semivariance problem. Second, we establish an alternative version of the verification theorem serving as the sufficient condition that the solution, which satisfies another mild condition different from that for necessity, of the nonlinear moment problem is the optimal terminal wealth of the original utility-deviation-risk portfolio selection problem. We then apply this general sufficiency result to revisit the various well-posed mean-risk problems already known in the literature and to also establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems under the assumption that the underlying utility satisfies the Inada condition. To the best of our knowledge, positive answers to the latter two problems have long been absent in the literature. In particular, the existence result in the utility-downside-risk problem is in contrast to the well-known nonexistence of an optimal solution for the mean-downside-risk problem. As a corollary, the existence result in utility-semivariance problem allows us to utilize the semivariance as a proper risk measure in the classical portfolio management paradigm.

Journal article

Huang YT, Song Q, Zheng H, 2017, Weak Convergence of Path-Dependent SDEs in Basket Credit Default Swap Pricing with Contagion Risk, SIAM Journal on Financial Mathematics, Vol: 8, Pages: 1-27, ISSN: 1945-497X

We investigate computational aspects of basket credit default swap pricing with counterparty credit risk under a multiname contagion model. This model enables us to capture systematic volatility increases in the market triggered by particular bankruptcies. A drawback of this model is its analytical intractability due to a combination of path-dependent coefficients and a path-dependent functional, which furthermore causes potential failure of convergence of numerical approximations under standing assumptions. In this paper, we find sufficient conditions for the desired convergence of functionals associated with approximated solution of certain path-dependent stochastic differential equations.

Journal article

Bian B, Wu N, Zheng H, 2016, Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact, Discrete and Continuous Dynamical Systems - Series B, Vol: 21, Pages: 1401-1420, ISSN: 1553-524X

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.

Journal article

Liu C, Zheng H, 2015, Asymptotic analysis for target asset portfolio allocation with small transaction costs, Insurance Mathematics & Economics, Vol: 66, Pages: 59-68, ISSN: 0167-6687

In this paper we discuss the asset allocation in the presence of small proportional transaction costs. The objective is to keep the asset portfolio close to a target portfolio and at the same time to reduce the trading cost in doing so. We derive the variational inequality and prove a verification theorem. Furthermore, we apply the second order asymptotic expansion method to characterize explicitly the optimal no transaction region when the transaction cost is small and show that the boundary points are asymmetric in relation to the target portfolio position, in contrast to the symmetric relation when only the first order asymptotic expansion method is used, and the leading order is a constant proportion of the cubic root of the small transaction cost. In addition, we use the asymptotic results for the boundary points and obtain an expansion for the value function. The results are illustrated in the numerical example.

Journal article

Ma J, Deng D, Zheng H, 2015, Convergence analysis and optimal strike choice for static hedges of general path-independent pay-offs, Quantitative Finance, Vol: 16, Pages: 593-603, ISSN: 1469-7696

In this paper, we propose a new algorithm to find the optimal static replicating portfolios for general path-independent nonlinear pay-off functions and give an estimate for the rate of convergence that is absent in the literature. We choose the static replication by designing an adaptation function arising in the error bound between the nonlinear pay-off function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strikes. The numerical tests for variance swaps, swaptions, static quadratic hedges and also for a jump-diffusion process, allowing for the default of the underlying asset, show that the proposed iterative equidistribution equation algorithm is simple, fast and accurate. The paper generalizes and improves the results on static replication and approximation in the literature.

Journal article


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