Abstract
There is now an increasingly large number of proposed concordance measures available to capture, measure and quantify different notions of dependence in stochastic processes. However, evaluation of concordance measures to quantify such types of dependence for different copula models can be challenging. In this work, we propose a class of new methods that involves a highly accurate and computationally efficient procedure to evaluate concordance measures for a given copula, applicable even when sampling from the copula is not easily achieved. In addition, this then allows us to reconstruct maps of concordance measures locally in all regions of the state space for any range of copula parameters. We believe this technique will be a valuable tool for practitioners to understand better the behaviour of copula models and associated concordance measures expressed in terms of these copula models.
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Appendices
Appendix A: Background Results on Reconstruction of a Copula via a Tensor Decomposition
This appendix provides a summary of the key results and algorithms from Dalessandro and Peters (2017), which include:
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1.
Construction of a process mimicking a diffusion through local moment matching. At this purpose the mimicking process is a birth-death process (BDP), and we show how to calculate its birth and death rate in a way to exactly replicate some target moments. The BDP is represented through a continuous time Markov chain (CTMC).
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2.
Construction of a process mimicking a conditional diffusion. At this purpose we introduce the concept of conditional infinitesimal generator matrix, which is a sequence of generator matrices. We also introduce the concept of orthogonal projections in a tensor space.
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3.
Approximation of the weak solution of a diffusion through the matrix exponentiation of the infinitesimal generator matrix.
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4.
Approximation of the weak solution of a multidimensional diffusion through the matrix exponentiation of an infinitesimal generator matrix which spans a tensor Kronecker product space. The weak solution is approximated only over orthogonal dimensions and this means that the multivariate weak solution is not directly constructed rather calculated from elementary one dimensional conditional weak solutions.
Let Xt, t ≥ 0 a d dimensional Markov process with \((X_{t}^1, \ldots , X_{t}^d) \in \mathbb {R}^d\) denoting its components vector at time t, with \(\mathbb {R}^d\) denoting the d-dimensional Euclidean space. Our modelling framework deals with the local approximation of Xt by birth-death processes (BDP) and their associated continuous time Markov chains (CTMC).
It suffices, in the methods required for this manuscript, to restrict our attention to the analysis of continuous Markov processes with stochastic differential equation (SDE) of the form,
where b(x) = (bi(x)), i = 1,…,d is a drift vector function with \(\mathbf {b}(\mathbf {x}): \mathbb {R}^d \rightarrow \mathbb {R}^d\) and \({\Sigma } = {\Psi } {\Psi }^{\prime }\) is a covariance matrix continuous function, with Σ = (aij(x)), i,j = 1,…,d, \({\Psi }(\mathbf {x}): \mathbb {R}^d \rightarrow \mathbb {R}^{d \times d}\) and \(\mathbf {W}_t \in \mathbb {R}^d\), and aij(x) = σi(x)σj(x)ρij(x), where σi(x) for all i is the diffusion positive function for the i-th process, and ρij(x) ∈ [− 1, 1] is the pairwise correlation function between process \(X^{i}_t\) and \(X^{j}_t\).
In addition we specify an initial condition to Eq. A.1 of the form Xt = x0, with \(x_0 \in \mathbb {R}^d, t \geq 0\).
We can write the infinitesimal generator associated to Eq. A.1 as,
To understand this mathematically we recall the following results, we begin with some important notations. We denote a countably set by \(\mathbf {\mathcal {X}}^{k} := \{ x^{(k)}_0, \ldots , x^{(k)}_{n_k} \} \in \mathbb {R}\), k = 1,…,d as the state space for the Markov chain \(X^{(n_k)}_{t}\) approximating \({X^{k}_{t}}\), with \(X^{(n_k)}_t \rightarrow X^{k}_t\) in the weak sense as \(n \rightarrow \infty \), being nk the number of discretization points for the k-dimension state space. The continuous time Markov chain represents the mimicking BDP. We denote by \(h_k = (x^{(k)}_{n_k} - x^{(k)}_{0})/ n_k\), the positive interval of the discretization grid \(\mathbf {\mathcal {X}}^{k}\).
The infinitesimal generator matrix entries, approximating the operator in Eq. A.2, are the local birth rate and death rate of the mimicking BDP and are calculated by matching the first and second instantaneous moments of the Markov chain \(X_{t}^{(n_k)}\) that have to coincide with those of \(X^{k}_t\) on the set \( \mathbf {\mathcal {X}}^{k,o} := \mathbf {\mathcal {X}}^k\backslash \partial \mathbf {\mathcal {X}}^k\). This is achieved by local moment matching using the following equations,
where the boundary \(\partial \mathbf {\mathcal {X}}\) consists of the smallest (i.e. \(x^{(k)}_0\)) and largest (i.e. \(x^{(k)}_{n_k}\)) elements in \(\mathbf {\mathcal {X}}^k\) and the interior \(\mathbf {\mathcal {X}}^{k,o}\) is the complement of the boundary.
In the following definition we illustrate the construction of the approximated infinitesimal generator matrix.
Definition 15 (Approximated Infinitesimal Generator matrix.)
Let us denote by \(A^{(n_k)} = A^{(n_k)}_{X^k} = (a(x^{(k)}_i,x^{(k)}_{j}))\) for i,j = 1,…,nk the approximated infinitesimal generator of the chain \(X_{t}^{(n_k)}\) mimicking the Markov process \(X_{t}^k\) with local drift parameters bk(⋅) and diffusion parameter σk(⋅). \(A^{(n_k)} \in \mathbb {R}^{n_k \times n_k}\) is the infinitesimal generator of the mimicking BDP \(X_{t}^{(n_k)}\) and is a tridiagonal matrix, which entries are computed using Eq. A.3 yielding to:
with \(-\frac {\sigma _{k}^2(x^{(k)}_i)}{h_k} \leq b_k(x_i) \leq \frac {\sigma _{k}^2(x_i)}{h}\) for all k, and a(xi,xj) ≥ 0 for all i,j.
Remark A.1 1
The continuous drift and covariance functions introduced above are approximated as piecewise functions over the discretized support of mutidimensional CTMC, which means
The link between the approximated terminal distribution \(P^{(n_k)}_t\) of a Markov process at time t and the infinitesimal generator matrix \(A^{(n_k)}\) is given by the solution of the following Cauchy problem
From a numerical point of view the calculation of \(P^{(n_k)}_t\) in Eq. A.6 requires the computation of the matrix exponential \(\exp (t A^{(n_k)}) \). One way to then tackle the curse of dimensionality that will arise when trying to evaluate such a matrix exponential in practice for large numbers of states, is to exploit the orthogonality across the Markov process dimensions.
In case of d independent Markov processes, the infinitesimal generator
is approximated by the following infinitesimal generator matrix:
where the symbol ⊕ denotes the Kroneker sum. We approximate (A.7) on a discrete multidimensional state space \(\mathcal {X} = \bigoplus _{i = 1}^d \mathbf {\mathcal {X}}^i\), namely only over the orthogonal spaces. In this case the solution to Eq. A.6 is given by
where the symbol ⊗ denotes the Kroneker product. We set the notation \(P^{(n)}_t := P^{(n) \perp }_t\), indicating the approximated density when the marginals are all orthogonal.
Remark A.2 1
Approximating over orthogonal dimension has a considerable numerical advantage. The problem of exponentiating the matrix A(n) becomes the exponential of single dimension (1-D) matrices which are the marginal generator matrices \(A^{(n_k)}\), for all k.
We would like to exploit the approximation over orthogonal dimension also for the case of the generator in Eq. A.2, where there is a correlation structure across the Markov process’ dimensions.
This is possible using a projected mimicking process constructed as a conditional BDP and which infinitesimal generator computation is proposed below.
Definition 16 (Conditional Local Infinitesimal Generator Matrix)
Let Xt be a d-dimensional Markov process and \(\mathbf {X}^{(n)}_t\) its corresponding approximating Markov chain. Let \(X^{(n)}_t = (X^{(n_k)}_t, \mathbf {Y}^{(n^{\prime })}_t)\) denote a partition of the chain, formed by the BDP process \(X^{(n_k)}_t\), with k ∈ (1,…,d), and by d − 1 BDP processes \(\mathbf {Y}^{(n^{\prime })}_t = (X^{(n_j)}_t)\) for j = 1,…,dk≠j. Then, if we consider the conditional Markov chain \((X^{(n_k)}_t| \mathbf {Y}^{(n^{\prime })}_t = \mathbf {y})\), where \( \mathbf {y} \in \mathcal {Y} := \bigotimes _{\scriptstyle i = 1\atop \scriptstyle i \ne k }^d \mathcal {X}^k\) is a state of the Markov chain partition \(\mathbf {Y}^{(n^{\prime })}_t\), we can calculate its infinitesimal generator matrix \(A^{(n_k)}_{X_k|\mathbf {Y} }\) entries by matching the the first two instantaneous moments of \((X^{k}_t| \mathbf {Y}_t)\) and obtaining,
where the instantaneous conditional moments \((b^{(c)}_k, \sigma _{k}^{2,(c)})\) are given by
for all \(x^{(k)}_i \in \mathbf {\mathcal {X}}^k\), and with moments partitions \( \mathbf {b}(\mathbf {x}) = \left [\begin {array}{llll} b_k \\ b_y \end {array}\right ] \) for the drift function and \({\Sigma }(\mathbf {x}) = \left [\begin {array}{llll} {\Sigma }_{kk} & {\Sigma }_{ky} \\ {\Sigma }_{yk} & {\Sigma }_{yy} \end {array}\right ] \) for the covariance matrix function respectively.
Example 1 (Two dimensional CTMC approximation)
For example in a two dimensional case the conditional Markov chain \((X^{(n_1)}_t| X^{(n_2)}_t = x^{(2)}_j )\), \(x^{(2)}_j \in \mathbf {\mathcal {X}}^2\), j = 1,…,n2, has infinitesimal generator matrix
where the instantaneous conditional moments \((b^{(c)}_k, \sigma _{k}^{2,(c)})\) are given by
Then one of the useful results derived in Dalessandro and Peters (2016) which we will utilise in this paper is the following approximation of the matrix exponential of the Kronecker sum of a sequence of matrices,
where the notation {Bj}, j = 1, 2,… denotes a sequence of matrices {B1,B2,…}. In Eq. A.14 we utilise the definition that if A is an \(\mathbb {R}^{m \times m}\) matrix and {Bj},j = 1,…,m is a sequence of m matrices in \(\mathbb {R}^{n \times n }\), then the Kronecker product of a matrix sequence A ⊗S{Bj} is the nm × nm block matrix:
In general, the construction of the infinitesimal generator matrix by means of the conditional operators is done using the following tensor equation:
where ⊕S denotes the Kronecker sum over a matrix sequence.
In this way, using the approximation in Eq. A.14, it is possible to compute the transition density for the multidimensional Markov chain, which is given by
1.1 Convergence Discussion for Discrete Approximation
For completeness we report some convergence results of the proposed approximation scheme from Dalessandro and Peters (2017), which are in its Section 4 entitled ‘Convergence of the approximated Generator’. In our methodology we construct a continuous Markov chain X(n) with dynamics as close as possible to the corresponding approximated process (Xt)t≥ 0. At this purpose we define and study the following error
which, according to Theorem 13 in Dalessandro and Peters (2017) implies that also the Markov chain transition density at time t converges to the process transition density at the same time, namely \(P_{t}^{(n)} \pi _n f \rightarrow P_t f\), for all f in the core of A, where πn is an auxiliary mapping function. The theoretical result of convergence states that the error is at most some constant c times |(Δx)2| when Δx is close to zero.
We illustrate this theoretical result in Fig. 17, where we consider a d = 5 dimensional diffusion with drift and diffusion functions equal to bk(x) = 0,σk(x) = 1 for all k respectively and we closely study the cases of correlation ρ(x) = 0 and ρ(x)≠ 0.
The plot reports the convergence rate error of the proposed CTMC approximation scheme, as a function of n, the number of grid points in each dimension. The chain approximates a d = 5 with drift and diffusion functions equal to bk(x) = 0,σk(x) = 1 for all k respectively. The cases of correlation ρ(x) = 0 and ρ(x)≠ 0, (ρ(x) = ± 0.7) are reported. There is consistent agreement of the schemes convergence rate when correlation is zero. In fact, the blue circles display the convergence of a full generator matrix with zero correlations. Identical convergence rates are displayed for the full generator matrix with non-zero correlation and the rank-1 generator matrix with zero correlation. The rate of convergence is lower for the rank-1 generator matrix with non-zero correlation equal to (ρ(x) = ± 0.7). Note that the convergence error functions are benchmarked against the function 1/n2, which is the reference rate for the theoretical results
The plots report the error function \( \epsilon (n) = \| P_t - P^{(n)}_t \|_2 \), which is the L2-norm of the difference between the Markov chain transition density at time t and the original process transition density at the same time, as the number of states in each dimension nk = n for all k, goes towards infinity.
The empirical study of the scheme convergence meets and satisfies the theoretical findings. In particular we can observe the consistent agreement of the schemes convergence rate when correlation is zero. We can furthermore notice how non-zero correlation (ρ(x) = ± 0.7) makes convergence lower.
In all the examples and empirical studies and in the concordance measure analysis we use n = 200 grid points across all chain dimensions when constructing an approximating Markov chain. This is a number of grid points which guarantees a convergence error ≤ 10− 4 according to Fig. 17.
Appendix B: Construction of the Mimicking Local Gaussian Copula
This appendix provides a practical summary of the key results and algorithms from Dalessandro and Peters (2016) about the approximated local Gaussian copula construction, which include:
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1.
Construction of the approximated local Gaussian copula density c(n) as the ratio of a joint terminal distribution \(P^{(n)}_t\) and the corresponding independent terminal distribution \(P^{(n)\perp }_t\), see Eq. 1.7. This is done within the discretized hypercube \(\mathbf {U} = \bigotimes _{k = 1}^d \mathbf {U}^k\) support of the multidimensional CTMC.
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2.
Mapping of the approximated local Gaussian copula C(n) into any target copula distributions evaluated pointwise on the hypercube U.
The construction of the mimicking local Gaussian copula function c(n) can be obtained as a decomposition of the generator matrix into the approximated operator matrix of a generic copula function and the marginal approximated densities’ operators Kronecker product. This important result from Dalessandro and Peters (2016), which we report below, clearly show how the approximated local Gaussian copula density c(n) is the ratio of a joint terminal distribution \(P^{(n)}_t\) and the corresponding independent terminal distribution \(P^{(n)\perp }_t\).
Proposition (CTMC Functional Copula Infinitesimal Operator)
Let \(\mathbf {X}_{t}^{(n)} := (X^{(n_1)}_t, X^{(n_2)}_t)\) be a bivariate continuous time Markov chain with approximate weak solution \(P^{(n)}_t\), mimicking the correlated diffusion \(\mathbf {X}_t := (X^{1}_t, X^{2}_t)\). Then the CTMC mimicking process can be characterized by the bivariate functional copula density operator representation given by
where \(I^{(n_1)} \in \mathbb {R}^{n_1 \times n_1}\) is the unitary matrix, and \(\{\hat {A}^{(n_2)}_{X^2|X^1} \}\) is the operator matrix sequence of the linear least square estimator of the conditional Markov chain process \((X^{(n_1)}_t|X^{(n_2)}_t)\).
Proof
In order to calculate the approximated expression for the mimicking copula density c(n), we first look at how we can rewrite the approximate solution P(n). We have:
Then we can rewrite
□
The copula operator in the d-dimensional case was shown to be of the form:
In Section 3 of Dalessandro and Peters (2016) they then show how to utilise this decomposition of the generalized diffusions generator to locally reconstruct what they term a generalized Gaussian copula reconstruction of any target copula model. In fact this is possible through the mapping
for all u ∈U, where
where the support \(\mathbf {\mathcal {X}}\) is represented as union of disjoint subsets {Bj}, i.e. \(\mathbf {\mathcal {X}} = \bigotimes _{i = 1}^d \mathcal {X}^i = \bigcup _{j = 1}^n B_j \), with each set Bj being a coordinate vector point x in the space \(\mathbf {\mathcal {X}}\) and \(B_0 = \{\mathbf {x} = (x^{(1)}_0, \ldots , x^{(d)}_0)\}, B_1 = \{\mathbf {x} = (x^{(1)}_1, \ldots , x^{(d)}_1)\}, \ldots , B_n = \{\mathbf {x} = (x^{(1)}_{n_1}, \ldots , x^{(d)}_{n_d})\}\) being a countable and ordered sequence of sets spanning the whole discretized support \(\mathbf {\mathcal {X}}\).
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Dalessandro, A., Peters, G.W. Efficient and Accurate Evaluation Methods for Concordance Measures via Functional Tensor Characterizations of Copulas. Methodol Comput Appl Probab 22, 1089–1124 (2020). https://doi.org/10.1007/s11009-019-09752-2
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DOI: https://doi.org/10.1007/s11009-019-09752-2
Keywords
- Concordance measures
- Copula functions
- Copula infinitesimal generators
- Martingale problem
- Multidimensional semimartingales decomposition approximations
- Semimartingales decomposition
- Tensor algebra