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Pairs-trading under path-dependent cointegration
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Abstract
Continuous-time pairs-trading rules are often developed based on the diffusion limit of first-order autoregressive cointegration models. Empirical identification of cointegration effects is generally made according to discrete-time error correction representation of vector autoregressive (VAR(p)) processes. We show that the diffusion limit of a VAR(p) process appears as a stochastic delayed differential equation. Motivated by this, we investigate the dynamic portfolio problem under a class of path-dependent models embracing path-dependent cointegration models as special case. Under certain regular conditions, we prove the existence of the optimal strategy and show that it is related to a system of Riccati partial differential equations. The proof is developed by means of functional It\^{o}'s calculus. When the process satisfies cointegration conditions, our results lead to the optimal dynamic pairs-trading rule. Our numerical study shows that the path-dependent effect has significantly impact on the pairs-trading strategy.
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We consider a simple class of stochastic control problems with a delayed control, in both the drift and the diffusion part of the state stochastic differential equation. We provide a new characterization of the solution in terms of a set of Riccati partial differential equations. Existence and uniqueness of a solution are obtained under a sufficient condition expressed directly as a relation between the time horizon, the drift, the volatility and the delay. Furthermore, a deep learning scheme (The code is available in a IPython notebook.) is designed and used to illustrate the effect of the delay feature on the Markowitz portfolio allocation problem with execution delay.
Stochastic control problems with delay are challenging due to the path-dependent feature of the system and thus its intrinsic high dimensions. In this paper, we propose and systematically study deep neural network-based algorithms to solve stochastic control problems with delay features. Specifically, we employ neural networks for sequence modeling (e.g., recurrent neural networks such as long short-term memory) to parameterize the policy and optimize the objective function. The proposed algorithms are tested on three benchmark examples: a linear-quadratic problem, optimal consumption with fixed finite delay, and portfolio optimization with complete memory. Particularly, we notice that the architecture of recurrent neural networks naturally captures the path-dependent feature with much flexibility and yields better performance with more efficient and stable training of the network compared to feedforward networks. The superiority is even evident in the case of portfolio optimization with complete memory, which features infinite delay.
This paper concerns portfolio selection with multiple assets under rough covariance matrix.
We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models.
In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients,
the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models.
Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.
This paper solves for the robust time-consistent mean–variance portfolio selection problem on multiple risky assets under a principle component stochastic volatility model. The model uncertainty is introduced to the drifts of the risky assets prices and the stochastic eigenvalues of the covariance matrix of asset returns. Using an extended dynamic programming approach, we manage to derive a semi-closed form solution of the desired portfolio via the solution to a coupled matrix Riccati equation. We provide the conditions, under which we prove the existence and the boundedness of the solution to the coupled matrix Riccati equation and derive the value function of the control problem. Moreover, we conduct numerical and empirical studies to perform sensitivity analyses and examine the losses due to ignoring model uncertainty or volatility information.
In this paper we consider the functional Itô calculus framework to find a path-dependent version of the Hamilton-Jacobi-Bellman equation for stochastic control problems that feature dynamics and running costs that depend on the path of the control. We also prove a dynamic programming principle for such problems. We apply our results to path-dependence of the delay type. We further study stochastic differential games in this context.
We consider a stochastic optimal control problem governed by a stochastic differential equation with delay in the control. Using a result of existence and uniqueness of a sufficiently regular mild solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, see the companion paper "Stochastic Optimal Control with Delay in the Control I: solving the HJB equation through partial smoothing ", we solve the control problem by proving a Verification Theorem and the existence of optimal feedback controls.
Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the the ones when the delay appears only in the state. This is particularly true when we look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong for the deterministic case) the HJB equation is an infinite dimensional second order semilinear Partial Differential Equation (PDE) that does not satisfy the so-called "structure condition" which substantially means that the control can act on the system modifying its dynamics at most along the same directions along which the noise acts. The absence of such condition, together with the lack of smoothing properties which is a common feature of problems with delay, prevents the use of the known techniques (based on Backward Stochastic Differential Equations (BSDEs) or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation and so no results on this direction have been proved till now. In this paper we provide a result on existence of regular solutions of such kind of HJB equations. This opens the road to prove existence of optimal feedback controls, a task that will be accomplished in a companion paper. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specific class of equations and data which arises naturally in many applied problems.
We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of N banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is an N-player linear-quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a close-loop Nash equilibrium using an HJB approach.
We analyze the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman equation is a two-dimensional nonlinear parabolic partial differential equation. We produce an explicit solution to this equation by reducing it to a simpler one-dimensional linear parabolic equation. This reduction is achieved through a Cole-Hopf type transformation, recently introduced in portfolio optimization theory by Zariphopoulou [9]. A verification argument is then used to prove that this solution coincides with the value function of the control problem. The optimal investment strategy is also given explicitly. On the technical side, the main problem we are facing here is the necessity to identify conditions on the parameters of the control problem ensuring uniform integrability of a family of random variables that, roughly speaking, are the exponentials of squared Wiener integrals.
This paper studies the problem of determining the optimal cut-off for pairs
trading rules. We consider two correlated assets whose spread is modelled by a
mean-reverting process with stochastic volatility, and the optimal pair trading
rule is formulated as an optimal switching problem between three regimes: flat
position (no holding stocks), long one short the other and short one long the
other. A fixed commission cost is charged with each transaction. We use a
viscosity solutions approach to prove the existence and the explicit
characterization of cut-off points via the resolution of quasi-algebraic
equations. We illustrate our results by numerical simulations.
In this paper, we present an elementary and self-contained proof of the stochastic Fubini theorem, which states that one can interchange a Lebesgue integral and a stochastic integral. The integrability conditions we use are weaker and more natural than the usual conditions in the literature. In particular, we do not need integrability in , and we use -integrability instead of -integrability in the additional parameter.
Very few studies have explored the structure of optimal switching regimes. We extend the existing research on the infinite-horizon multiple-regime switching problem with an arbitrary number of switch options by replacing the linear running reward function with a quadratic function in the objective function. To make our analysis more rigorous, we establish the theoretical basis for the application of the simultaneous multiple-regime switches to the problem with the extended objective function, and provide the sufficient condition under which each certain separated region in the space includes, at most, one single connected optimal switching region, which determines the structure of the optimal switching regions, and we identify the structure of the optimal switching regions for the particular problem.
This paper investigates a class of robust non-zero-sum reinsurance-investment stochastic differential games between two competing insurers under the time-consistent mean–variance criterion. We allow each insurer to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. The surplus processes of both insurers are governed by the classical Cramér-Lundberg model and each insurer is an ambiguity-averse insurer (AAI) who concerns about model uncertainty. The objective of each insurer is to maximize the expected terminal surplus relative to that of his competitor and minimize the variance of this relative terminal surplus under the worst-case scenario of alternative measures. Applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi-Bellman (HJB) equations for both insurers. We establish the robust equilibrium reinsurance-investment strategies and the corresponding equilibrium value functions of both insurers by solving the extended HJB equations under both the compound Poisson risk model and its diffusion-approximated model. Finally, we conduct some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies.
In many applications of mathematical modeling to biology, economics, social sciences and engineering, the objective is to find optimal solutions. Usually we want to minimize an objective function depending on a number of functions subject to constraints given, for example, by systems of differential equations. Two main numerical approaches are used to solve these optimal control problems, depending on whether the problem is optimized first and then discretized, or viceversa. Each of these two approaches has its advantages and disadvantages. In this paper we describe both methods an apply them to a plant virus propagation model, where the virus is propagated through a vector that bites the infected plants. The model includes delays due to the time the virus takes to infect the plant and the vector, and seasonality due to the dependence of the behavior on the seasons. The objective function is the total cost to a farmer of having infected plants, and includes the actual cost of a plant plus the cost of the controls which are insecticides and a predator species that preys on the insects. Numerical simulations are done using both methods and comparisons are made.
In this study, we consider a time-consistent mean–variance asset–liability management portfolio selection problem in which the liability is controllable. The objective is to find an equilibrium investment strategy and an equilibrium debt ratio in the financial market consisting of one risk-free asset and one risky asset. By using forward backward stochastic differential equations (FBSDEs), we derive a sufficient condition and a necessary condition for the open-loop equilibrium strategies. The uniqueness of the strategies is also provided. Furthermore, to illustrate our results, we provide numerical examples to show how the parameters impact on the equilibrium strategies and the corresponding efficient frontier.
This paper investigates the open-loop equilibrium reinsurance-investment (RI) strategy under general stochastic volatility (SV) models. We resolve difficulties arising from the unbounded volatility process and the non-negativity constraint on the reinsurance strategy. The resolution enables us to derive the existence and uniqueness result for the time-consistent mean variance RI policy under both situations of constant and state-dependent risk aversions. We apply the general framework to popular SV models including the Heston, the 3/2 and the Hull–White models. Closed-form solutions are obtained for the aforementioned models under constant risk aversion, and the non-leveraged models under state-dependent risk aversion.
We study the theoretical implications of cointegrated stock prices on the profitability of pairs trading strategies. If stock returns are fairly weakly correlated across time, cointegration implies very high Sharpe ratios. To the extent that the theoretical Sharpe ratios are "too large," this suggests that either (i) cointegration does not exist pairwise among stocks, and pairs trading profits are a result of a weaker or less stable dependency structure among stock pairs, or (ii) the serial correlation in stock returns stretches over considerably longer horizons than is usually assumed. Empirically, there is little evidence of cointegration, favoring the first explanation.
Executing a basket of co‐integrated assets is an important task facing investors. Here, we show how to do this accounting for the informational advantage gained from assets within and outside the basket, as well as for the permanent price impact of market orders (MOs) from all market participants, and the temporary impact that the agent's MOs have on prices. The execution problem is posed as an optimal stochastic control problem and we demonstrate that, under some mild conditions, the value function admits a closed‐form solution, and prove a verification theorem. Furthermore, we use data of five stocks traded in the Nasdaq exchange to estimate the model parameters and use simulations to illustrate the performance of the strategy. As an example, the agent liquidates a portfolio consisting of shares in Intel Corporation and Market Vectors Semiconductor ETF. We show that including the information provided by three additional assets (FARO Technologies, NetApp, Oracle Corporation) considerably improves the strategy's performance; for the portfolio we execute, it outperforms the multiasset version of Almgren–Chriss by approximately 4–4.5 basis points.
In this paper, we investigate an optimal investment and excess-of-loss reinsurance problem with delay and jump–diffusion risk process for an insurer. Specifically, the insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market, where the surplus of insurer is represented by a jump–diffusion model and the financial market consists of one risk-free asset and one risky asset whose price process is governed by a constant elasticity of variance model. In addition, the performance-related capital inflow/outflow is introduced, the wealth process of insurer is modeled by a stochastic differential delay equation. The insurer aims to seek the optimal excess-of-loss reinsurance and investment strategy to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. By solving a Hamilton–Jacobi–Bellman equation, the closed-form expressions for the optimal strategy and the optimal value function are derived. Finally, some special cases of our model and results are presented, and some numerical examples for our results are provided.
We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b). Copyright Blackwell Publishers Inc. 1997.
We solve optimal iterative three-regime switching problems with transaction costs, with investment in a mean-reverting asset that follows an Ornstein–Uhlenbeck process and find the explicit solutions. The investor can take either a long, short or square position and can switch positions during the period. Modeling the short sales position is necessary to study optimal trading strategies such as the pair trading. Few studies provide explicit solutions to problems with multiple (more than two) regimes (states). The value function is proved to be a unique viscosity solution of a Hamilton–Jacobi–Bellman variational inequality (HJB-VI). Multiple-regime switching problems are more difficult to solve than conventional two-regime switching problems, because they need to identify not only when to switch, but also where to switch. Therefore, multiple-regime switching problems need to identify the structure of the continuation/switching regions in the free boundary problem for each regime. If the number of the states is two, only two regions have to be identified, but if , regions have to be detected. We identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach.
This paper concerns the continuous-time, mean-variance portfolio selection problem in a complete market with random interest rate, appreciation rates, and volatility coefficients. The problem is tackled using the results of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), two theories that have been extensively studied and developed in recent years. Specifically, the mean-variance problem is formulated as a linearly constrained stochastic LQ control problem. Solvability of this LQ problem is reduced, in turn, to proving global solvability of a stochastic Riccati equation. The proof of existence and uniqueness of this Riccati equation, which is a fully nonlinear and singular BSDE with random coefficients, is interesting in its own right and relies heavily on the structural properties of the equation. Efficient investment strategies as well as the mean-variance efficient frontier are then analytically derived in terms of the solution of this equation. In particular, it is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random. Finally, a version of the Mutual Fund Theorem is presented.
We study the optimal trading policy of an arbitrageur who can exploit temporary mispricing in a market with two convergent assets. We build on the model of Liu and Timmermann (2013) and include transaction costs, which impose additional limits to the implementation of such convergence trade strategy. We show that the presence of transaction costs could reveal an endogenous stop-loss concern in a certain economy, which affects the optimal policy of the arbitrageur in significant ways. Using pairs of Chinese stock shares that are dual-listed in Shanghai and in Hong Kong, we show that our co-integration strategy is generally superior to the relative-value pairs trading strategy studied in Gatev et al. (2006). Several extensions of our model are also discussed.
This paper concerns a mean–variance portfolio selection problem in a complete market with unbounded random coefficients. In particular, we use adapted processes to model market coefficients, and assume that only the interest rate is bounded, while the appreciation rate, volatility and market price of risk are unbounded. Under an exponential integrability assumption of the market price of risk process, we first prove the uniqueness and existence of solutions to two backward stochastic differential equations with unbounded coefficients. Then we apply the stochastic linear–quadratic control theory and the Lagrangian method to solve the problem. We represent the efficient portfolio and efficient frontier in terms of the unique solutions to the two backward stochastic differential equations. To illustrate our results, we derive explicit expressions for the efficient portfolio and efficient frontier in one example with Markovian models of a bounded interest rate and an unbounded market price of risk.
Cointegration is a useful econometric tool for identifying assets which share a common equilibrium. Cointegrated pairs trading is a trading strategy which attempts to take a profit when cointegrated assets depart from their equilibrium. This paper investigates the optimal dynamic trading of cointe-grated assets using the classical mean-variance portfolio selection criterion. To ensure rational economic decisions, the optimal strategy is obtained over the set of time-consistent policies from which the optimization problem is enforced to obey the dynamic programming principle. We solve the opti-mal dynamic trading strategy in a closed-form explicit solution. This ana-lytical tractability enables us to prove rigorously that cointegration ensures the existence of statistical arbitrage using a dynamic time-consistent mean-variance strategy. This provides the theoretical grounds for the market belief in cointegrated pairs trading. Comparison between time-consistent and pre-commitment trading strategies for cointegrated assets shows the former to be a persistent approach, whereas the latter makes it possible to generate infinite leverage once a cointegrating factor of the assets has a high mean reversion rate.
We give a definition of integration and cointegration in continuous time and study characterizations of those definitions, in particular for CARMA (Continuous time ARMA) processes. We check also that such singularities in CARMA processes may imply unit roots problems in the discretized corresponding process. Then, error correction representations in continuous time are exhibited and discussed, and a general theorem of representation giving a precise description of the singularities is proved. Lastly, we look at other singularities emphasizing the importance of estimating the continuous time model if it is the true one: a one-dimensional noise can generate a two-dimensional regular AR process.
This paper considers a portfolio management problem of Merton's type in which the risky asset return is related to the return history. The problem is modeled by a stochastic system with delay. The investor's goal is to choose the investment control as well as the consumption control to maximize his total expected, discounted utility. Under certain situations, we derive the explicit solutions in a finite dimensional space.
This paper is concerned with an optimal investment and reinsurance problem with delay for an insurer under the mean–variance criterion. A three-stage procedure is employed to solve the insurer’s mean–variance problem. We first use the maximum principle approach to solve a benchmark problem. Then applying the Lagrangian duality method, we derive the optimal solutions for a variance-minimization problem. Based on these solutions, we finally obtain the efficient strategy and the efficient frontier of the insurer’s mean–variance problem. Some numerical examples are also provided to illustrate our results.
This paper considers a portfolio management problem of Merton's type in which the risky asset return is related to the return history. The problem is modeled by a stochastic system with delay. The investor's goal is to choose the investment control as well as the consumption control to maximize his total expected, discounted utility. Under certain situations, we derive the explicit solutions in a finite dimensional space.
Convergence trades exploit temporary mispricing by simultaneously buying relatively underpriced assets and selling short relatively
overpriced assets. This paper studies optimal convergence trades under both recurring and nonrecurring arbitrage opportunities
represented by continuing and “stopped” cointegrated price processes and considers both fixed and stochastic (Poisson) horizons.
Conventional long-short delta neutral strategies are generally suboptimal and it can be optimal to simultaneously go long
(or short) in two mispriced assets. Optimal portfolio holdings critically depend on whether the risky asset position is liquidated
when prices converge. Our theoretical results are illustrated on pairs of Chinese bank shares traded on both the Hong Kong
and China stock exchanges.
This paper is concerned with one kind of forward–backward linear quadratic stochastic control problem whose system is described by a linear anticipated forward–backward stochastic differential delayed equation. The explicit form of the optimal control is derived. Optimal state feedback regulators are studied in two special cases. For the case with delay in just the control variable, the optimal state feedback regulator is obtained by the Riccati equation. For the other case with delay in just the state variable, the optimal state feedback regulator is analyzed by the value function approach.
We propose a model for analyzing dynamic pairs trading strategies using the stochastic control approach. The model is explored in an optimal portfolio setting, where the portfolio consists of a bank account and two co-integrated stocks and the objective is to maximize for a fixed time horizon, the expected terminal utility of wealth. For the exponential utility function, we reduce the problem to a linear parabolic Partial Differential Equation which can be solved in closed form. In particular, we exhibit the optimal positions in the two stocks.
A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.
We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages
A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems
We study whether investors can exploit serial dependence in stock returns to improve out-of-sample portfolio performance. We show that a vector-autoregressive (VAR) model captures stock return serial dependence in a statistically significant manner. Analytically, we demonstrate that, unlike contrarian and momentum portfolios, an arbitrage portfolio based on the VAR model attains positive expected returns regardless of the sign of asset return cross-covariances and autocovariances. Empirically, we show, however, that both the arbitrage and mean-variance portfolios based on the VAR model outperform the traditional unconditional portfolios only for transaction costs below ten basis points. © 2014 © The Author 2014. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: [email protected]
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Itô calculus deals with functions of the current state whilst we deal with functions of the current path to acknowledge the fact that often the impact of randomness is cumulative. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem, providing an alternative to the Clark-Ocone formula from Malliavin Calculus. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense.
This paper considers the continuous-time mean-variance portfolio selection problem in a financial market in which asset prices are cointegrated. The asset price dynamics are then postulated as the diffusion limit of the corresponding discrete-time error-correction model of cointegrated time series. The problem is completely solved in the sense that solutions of the continuous-time portfolio policy and the efficient frontier are obtained as explicit and closed-form formulas. The analytical results are applied to pairs trading using cointegration techniques. Numerical examples show that identifying a cointegrated pair with a high mean-reversion rate can generate significant statistical arbitrage profits once the current state of the economy sufficiently departs from the long-term equilibrium. We propose an index to simultaneously measure the departure level of a cointegrated pair from equilibrium and the mean-reversion speed based on the mean-variance paradigm. An empirical example is given to illustrate the use of the theory in practice.
In this paper we extend the traditional price change hedge ratio estimation method by applying the theory of cointegration to hedging with stock index futures contracts for France (CAC 40), the United Kingdom (FTSE 100), Germany (DAX), and Japan (NIKKEI). Previous studies ignore the last period's equilibrium error and short-run deviations. The findings of this study indicate that the hedge ratios obtained from the error correction method are superior to those obtained from the traditional method as evidenced by the likelihood ratio test and out-of-sample forecasts. Using the procedures developed in this paper, hedgers can control the risk of their portfolios more effectively at a lower cost.
Throughout these notes r is a fixed constant, 0 ≤ r < ∞. We denote by C the Banach space of continuous functions [–r, 0] → Cn
with norm ║ø║ = sup –r≤θ≤0 |ø(θ)|, where | · | is any vector norm in Cn
. If x : [–r, α) → Cn
, α > 0, is a continuous function, then x
t
∈ C, 0 ≤ t < α, is defined by