On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.


Introduction
Financial analysis and decision making rely on quantification and modelling of future risk exposures. A systematic approach for the latter was put forward in [3], laying the foundations of an axiomatic framework for coherent measurement of risk. A subsequent breakthrough was the development and application of the notion of backward stochastic differential equations (BSDEs) in the context of risk analysis, which gave rise to the (strongly) time-consistent extension of coherent risk-measures to continuous-time dynamic settings [39,42]. Building on these advances, we consider in this article a new class of such continuous-time dynamic coherent risk measures, which we propose to call dynamic spectral risk measures (DSRs).
Quantile-based coherent risk measures, such as Expected Shortfall, belong to the most widely used risk-measures in risk analysis, and are also known as spectral risk measures, Choquet expectations (based on probability distortions) and Weighted VaR; see [1,12,34,48]. In order to carry out, for instance, an analysis of portfolios involving dynamic rebalancing, one is lead to consider the (strongly) time-consistent extension of such coherent risk-measures to given time-grids, which are defined by iterative application of the spectral risk-measure along these particular grids. Due to its continuoustime domain of definition a DSR is, in contrast, independent of a grid structure. While the latter holds for any continuous-time risk measure we show that DSRs emerge as the limits of such iterated spectral risk measures when the time-step vanishes and under appropriate scaling of the parameters, by establishing a functional limit theorem.
To explore its use in financial decision problems, we consider subsequently a dynamic portfolio optimisation problem under DSR, which we analyse in terms of its associated Hamilton-Jacobi-Bellman (HJB) equation. In the case of a long-only investor (who is allowed neither to borrow nor to short-sell stocks) we identify explicitly dynamic optimal allocation strategies.
DSR, like any dynamic risk-measure obtained from a BSDE, is (strongly) time-consistent in the sense that if the value of a random variable X is not larger than Y under DSR at time t almost surely, then the same relation holds at earlier times s, s < t. For dynamic risk-measures the property of strong time-consistency is well known to be equivalent to recursiveness, a tower-type property which is referred to as filtration-consistency in [16] Such concepts have been investigated extensively in the literature; among others we mention [4,11,15,17,25,30,31,40]. For studies on weaker forms of time-consistency we refer to [41,47,49].
The notion of strong time-consistency in economics goes back at least as far as [46] and has been standard in the economics literature ever since; see for instance [10,20,23,24,27,32,33].
Due to their recursive structure financial optimisation problems, such as utility opimisation under the entropic risk-measure and related robust portfolio optimisation problems satisfy the Dynamic Programming Principle and admit time-consistent dynamically optimal strategies (see for instance [6,36] and references therein). In Section 6 we demonstrate that this also holds for the optimal portfolio allocation problem phrased in terms of the minimisation under a DSR, and phrase and solve this problem via the associated HJB equation.
For a given DSR, the functional limit theorem that we obtain (see Theorem 5.2) shows how to construct an approximating sequence of iterated spectral risk-measures driven by lattice random walks, suggesting an effective method to evaluate functionals under a given DSR and solutions to associated PIDEs, by recursively applying (distorted) Choquet expectations. The functional limit theorem involves a certain non-standard scaling of the parameters of the iterated spectral risk measures, which is given in Definition 5.1. The advantage of this approximation method is that it sidesteps the (typically non-trivial) task of computing the Malliavin derivatives. A numerical study is beyond the scope of the current paper, and is left for future research.
While one may prove the functional limit theorem directly through duality arguments, we present in the interest of brevity a proof that draws on the convergence results obtained in [37] for weak approximation of BSDEs. In the literature various related convergence results are available, of which we next mention a number (refer to [37] for additional references). The construction of continuoustime dynamic risk-measures arising as limits of discrete-time ones was studied in [45] in a Brownian setting. In a more general setting including in addition finitely many Poisson processes, [35] presents a limit theorem for recursive coherent quantile-based risk measures, which is proved via an associated non-linear partial differential equation. In [19] a Donsker-type theorem is established under a Gexpectation.
Contents. The remainder of the paper is organised as follows. In Section 2 we collect preliminary results concerning dynamic coherent risk measures and related BSDEs, adopting a pure jump setting driven by a Poisson random measure. In Section 3 we are concerned with the Choquet-type integrals which appear in the definitions of dynamic and iterated spectral risk measures. With these results in hand, we phrase the definition of a DSR in Section 4 and identify its dual representation. In Section 5 we present the functional limit theorem for iterated spectral risk measures. Finally, in Section 6 we turn to the study of a dynamic portfolio allocation problems under a DSR.

Preliminaries
In this section we collect elements of the theory of time-consistent dynamic coherent risk measures and associated BSDEs, in both continuous-time and discrete-time settings. To avoid repetition we state some results and definitions in terms of the index set I, which is taken to be either I = [0, T ] or I = π ∆ := {t i = i∆, i = 0, . . . , N }, with ∆ = T /N for some N ∈ N and T > 0.

Time-consistent dynamic coherent risk measures
On some filtered probability space (Ω, F, F, P) with F = (F t ) t∈I , we consider risks described by random variables X ∈ L p = L p (F T ), p > 0, the set of F T -measurable random variables X with E[|X| p ] = Ω |X| p dP < ∞. We denote by L p t = L p (F t ) and L p (G) the elements X in L p (F) that are measurable with respect to the sigma-algebras F t and G ⊂ F, respectively, and by L ∞ , For a given measure µ on a measurable space (U, U) we denote by L p (µ), p > 0, the set of Borel , and by L p + (µ) the set of non-negative elements in L p (µ). Dynamic coherent risk-measures and (strong) time-consistency, we recall, are defined as follows in an L 2 -setting: Definition 2.1 A dynamic coherent risk measure ρ = (ρ t ) t∈I is a map ρ : L 2 → S 2 (I) that satisfies the following properties: (iii) (positive homogeneity) for X ∈ L 2 and λ ∈ L ∞ t , ρ t (|λ|X) = |λ|ρ t (X); Definition 2.2 A dynamic coherent risk measure ρ is called (strongly) time-consistent if either of the following holds: (vi) (recursiveness) for X ∈ L 2 and s, t with s ≤ t, ρ s (ρ t (X)) = ρ s (X).
More generally, a map ρ : L 2 → S 2 (I) is called a time-consistent dynamic risk measure if ρ satisfies conditions (i) and (v). For a proof of the equivalence of items (v) and (vi) we refer to [26,Lemma 11.11]; for a discussion of (the unconditional version of) the properties (i)-(iv) see [3,4]. One way to construct time-consistent dynamic risk measures is as solution to an associated backward stochastic differential equation (BSDE) or backward stochastic difference equation (BS∆E). To ensure that such dynamic risk measures are positively homogeneous and subadditive, the corresponding driver functions are to be positively homogeneous, subadditive and should not dependent on the value of the risk-measure (see Proposition 11 in [42] and Lemma 2.1 in [16]). For background on the notion of strong time-consistency and its relation to g-expectations we refer to [7,8,39,42]. Specifically, in our setting such driver functions are defined as follows: 3 For a given Borel measure µ on R k \{0} we call a function g : I × L 2 (µ) → R a driver function if for any z ∈ L 2 (µ) t → g(t, z) is continuous (in case I = [0, T ]) and the following holds: (i) (Lipschitz-continuity) for some K ∈ R + \{0} and any t ∈ I and z 1 , z 2 ∈ L 2 (µ) A driver function g is called coherent if the following hold: (ii) (positive homogeneity) for any r ∈ R + , t ∈ I and z ∈ L 2 (µ), we have g(t, rz) = rg(t, z); (iii) (subadditivity) for any t ∈ I and z 1 , z 2 ∈ L 2 (µ), we have g(t, z 1 + z 2 ) ≤ g(t, z 1 ) + g(t, z 2 ).
We describe next the dynamic (coherent) risk-measure defined via the BSDEs (if I = [0, T ]) or BS∆Es (if I is a finite partition of [0, T ]).

Discrete-time lattice setting
We turn first to the discrete-time lattice setting, fixing a uniform partition π = π ∆ of [0, T ] with as before ∆ = T /N for some N ∈ N. Let L (π) = (L (π) t ) t∈π denote a square-integrable zero-mean random walk starting at zero and taking values in ( √ ∆Z) k , and let F (π) = (F (π) t ) t∈π denote the filtration generated by L (π) . Furthermore, we let g (π) be a coherent driver function as in Definition 2.3 with I = π and µ(dx) equal to the scaled law ν (π) (dx) of ∆L t , t ∈ π\{T }, given by The BS∆E for (Y (π) , Z (π) ) corresponding to final value −X (π) ∈ L 2 (F (π) T ) and driver function g (π) takes the following form (in view of [37, Proposition 3.2]), which is analogous to the one in continuoustime case given in (2.6) below: for t ∈ π\{T } and with Y (π) T = −X (π) , where I A denotes the indicator of a set A. In difference notation the BS∆E (2.2) is for t ∈ π\{T } given by t ). If the Lipschitz-constant K = K (π) of the driver function g (π) is strictly smaller than the reciprocal 1/∆ of the mesh-size then it follows from [37, Propositions 3.1 and 3.2] (and using that, in the notation of [37], F (pi) is independent of W (pi) ) that there exists a unique solution (Y (π) , Z (π) ) to the BS∆E which satisfies the following relations for t ∈ π: = x}) denotes the smallest sigmaalgebra containing F (π) t as well as the sigma-algebra σ({∆L In analogy with the continuous-time case (reviewed below), the dynamic coherent risk-measure associated to the solution to the BS∆E is defined as follows: Definition 2.4 For a driver function g (π) as in Definition 2.3 with I = π and µ(dx) = ν (π) and the solution (Y (π) , Z (π) ) of the corresponding BS∆E (2.2), ρ g (π) ,(π) = (ρ g (π) ,(π) t ) t∈π denotes the timeconsistent dynamic risk measure given by ρ

Continuous-time setting
In the continuous-time case (I = [0, T ]) we consider risky positions described by random variables X that are measurable with respect to F T , where F = {F t } t∈[0,T ] denotes the right-continuous and completed filtration generated by a Poisson random measure N on [0, T ] × R k \{0} for some k ∈ N. We suppose throughout that the associated Lévy measure ν satisfies the following condition: The Lévy measure ν associated to the Poisson random measure N has no atoms and, for some ε 0 > 0, ν 2+ε 0 ∈ R + \{0} where for p ≥ 0 We denote byÑ (dt × dx) = N (dt × dx) − ν(dx)dt the compensated Poisson random measure and by L = (L t ) t∈[0,T ] the (column-vector) Lévy process given by [44,Theorem 25.3] LetH 2 denote the set ofP-measurable square-integrable processes, where, with P denoting the predictable sigma-algebra,P = P ⊗ B(R k \{0}), and let U denote the Borel sigma-algebra induced by the L 2 (ν(dx))-norm. In particular, U ∈H 2 is such that U H2 < ∞, where Moreover, let M 2 denote the set of probability measures Q = Q ξ on (Ω, F T ) that are absolutely continuous with respect to P with square-integrable Radon-Nikodym derivatives ξ ∈ L 2 + (F T ), and write Let us next consider a coherent driver function g as in Definition 2.3 with µ = ν and I = [0, T ]. and fix a final condition X ∈ L 2 . The associated BSDE for the pair (Y, Z) ∈ S 2 ×H 2 is given by . This BSDE, we recall from [5], admits a unique solution. By combining [38,43,42], we have that the BSDE (2.6) gives rise to a dynamic coherent risk-measure as follows: Definition 2.6 For a given coherent driver function g, the corresponding dynamic coherent risk mea- where (Y, Z) ∈ S 2 ×H 2 solves (2.6).
for some function f : R k → R the dynamic coherent risk-measure ρ g (X) is related to the following semi-linear PIDE (denotingv = ∂v ∂t ): where Dv t,x : R k → R and Gv(t, x) are given by Dv t,x (y) = v(t, x + y) − v(t, x) and This non-linear Feynman-Kac result is shown by an application of Itô's lemma. (ii) The risk measure ρ g admits a dual representation for a certain representing subset S g of the set M 1 of probability measures that are absolutely continuous with respect to P. The set S g is convex and closed (see [26,Theorem 11.22]).
We describe next a representation result for dynamic risk-measure ρ g in terms of the representing processes (H ξ ) of the stochastic logarithms of the Radon-Nikodym derivatives ξ ∈ L 2 + (F T ) of the measure Q ξ ∈ M 2 , which are given by where E(·) denotes the Doléans-Dade stochastic exponential. We call a B( Theorem 2.8 Let g be a coherent driver function. Then for some P ⊗ U-measurable set C g that is closed, convex, contains 0 and is bounded, we have for any t ∈ [0, T ] that ρ g t (X) satisfies (2.11) with 14) The proof of Theorem 2.8 follows by adaptation of the arguments given in [36,Theorem A.25], and is omitted.
Remark 2.9 (i) Note that two driver functions g 1 and g 2 are equal if and only if the corresponding sets C g 1 and C g 2 in the representation (2.14) are equal.
then the corresponding driver function is given by g(t, z) =ḡ(z) wherē (2.15)

Convergence
We next turn to the question of the convergence of a sequence (ρ g (π) ,(π) ) π of dynamic coherent risk measures as in Definition 2.4 when the mesh size ∆ = ∆ π tends to zero. Let us suppose that (ρ g (π) ,(π) ) π are driven by the random walks (L (π) ) π that are defined as follows: for some probability distribution (p ∆ j , j ∈ Z k ) on Z k that is given as follows in terms of a constant c ≥ 1 (that will be specified shortly) and a partition ( where, as before, ν 2 = R k \{0} |x| 2 ν(dx). When ∆ 0, we have by the dominated convergence theorem that T and x ∈ R k . Moreover, we have by functional weak convergence theory (see e.g. [29,Theorem VII.3.7]) On a suitably chosen probability space L (π) T converges to L T in probability as ∆ 0. The latter convergence also holds in a stronger sense thanks to moment-conditions satisfied by L (π) T that we show next. We define the value of c as follows in terms of ε 0 > 0 given in Assumption 2.5: Lemma 2.10 The collection (L (π) ) π of random walks defined in (2.16) and (2.17)-(2. 19) is such that we have, for any uniform partition π and t ∈ π\{T }, E ∆L where ε 0 > 0 and ν 2+ε 0 are as in Assumption 2.5, and c ∆ is given in (2.20) and (2.23). Furthermore, we have

Remark 2.11
Under the bound in the right-hand side of (2.24) we have numerical stability of the solutions to sequence of BS∆Es driven by (L (π) ) (see [37,Theorem 3.4]).
Proof. Letting π = π ∆ denote the partition with mesh ∆ ∈ R + \{0} and ε = ε 0 , a first observation is that, for any t ∈ π\{T }, a ∈ R + \{0} and p ∈ [2, 2 + ε], we have by Chebyshev's inequality where, as before, ν p = R k \{0} |x| p ν(dx). By multiplying (2.26) by p a p−1 and integrating we have the estimate (2.28) Taking in (2.27) p = 2 and a = b √ c ∆ ν 2 ∆ and (a) setting b = 1 shows that which yields the bound in the right-hand side of (2.24), while (b) integrating over b ≥ 1 shows that t |]/ √ ∆ ≤ ν 2 /c ∆ , which tends to zero as ∆ 0 in view of the form of c ∆ . To establish (2.25) the proof next proceeds analogously as that of the moment result for Lévy processes (see [44,Theorem 25.3]). The key step is to transfer the uniform estimate of moments of the increments to a uniform estimate of moments of the random walk at T is the following estimate for a sub-multiplicative functions g (a function g : R k → R is called sub-multiplicative, we recall, if for some b g ∈ R + and any x, y ∈ R k we have g(x + y) ≤ b g g(x)g(y)): where we used that the increments ∆L (π) t , t ∈ π\{T }, are independent. For any a ∈ R + the function g a given by g a (x) := |x| 2+ε ∨ a, we recall from [44,Proposition 25.4] is sub-multiplicative. From (2.28) and (2.29) we have that E g 1 ∆L (π) t is bounded above by Combining the bounds (2.30) and (2.31) with the facts that c defined in 2.23 is such that As the right-hand side of (2.32) is bounded above by c −1 exp(c ν 2+ε T ) we have (2.25), and the proof is complete. 2 The moment-conditions in Lemma 2.10 carry over to those of path-functionals as follows:

Choquet-type integrals
We describe next the Choquet-type integrals that feature in the definition of dynamic spectral riskmeasures given in the next section. We refer to [18] for a treatment of the theory of non-linear integration. The Choquet-type integrals that we consider are given in terms of measure distortions that we define next.  On a given measure space (U, U) a set A ∈ U with µ(A) > 0 is called an atom, we recall, if C ⊂ A implies µ(C) ∈ {0, µ(A)}. We assume throughout that the measure distortions and associated measure spaces are of the following type: The measure µ on (U, U) is sigma-finite and has no atoms, and the measure distortion Γ : [0, µ(U)) → R + is bounded and such that The Choquet-type integrals that we consider are defined as follows: Definition 3.3 Let (U, U, µ) be a measure space and let Γ + and Γ − be associated measure distortions which satisfy Assumption 3.2.

Remark 3.4 (i)
To see that C Γ•µ + (f ) ∈ R + for f ∈ L 2 + (µ) and µ and Γ satisfying Assumption 3.2 we note that by Chebyshev's inequality, monotonicty of Γ and a change of variables, we have if µ(U) = ∞. If µ(U) < ∞ we find by a similar line of reasoning that (ii) Taking in Definition 3.3, (U, U, µ) = (Ω, F T , P), and taking the measure distortions Γ + and Γ − equal to a continuous probability distortion Ψ and the function Ψ given by Ψ( 1], it is straightforward to check that Ψ • P is a capacity and the Choquet-type integral of X ∈ L 2 in (3.2) coincides with the classical Choquet expectation corresponding to Ψ • P: We record next a robust representation result for Choquet-type integrals that plays an important role in the sequel. Let M p,µ , p ≥ 1, denote the set of measures m on (U, U) that are absolutely continuous with respect to a given measure µ on this space with Radon-Nikodym derivatives such that dm dµ ∈ L p + (µ). Then we have that C Γ•µ + : L 2 + (µ) → R + is K Γ -Lipschitz-continuous and In particular, C Γ•µ + is positively homogeneous and subadditive, that is, for any λ ∈ R + and f, g ∈ L 2 Proof of Proposition 3.5.. The representation in (3.6), we recall, is known to hold true when (a) Γ(1) = 1 and (b) µ has unit mass and (c) M Γ 1,µ is replaced by the set of m ∈ M Γ 1,µ with m(U) = 1 (see [9] and [26,Corollary 4.80]). We note that, by positive homogeneity and (a) and (b), (c) is not needed for the representation in (3.6) to hold true. Let ε > 0, let µ be as given and let m ∈ M Γ 1,µ , and denote by O ε , ε > 0, a collection of sets with finite non-zero µ-measure and such that O ε U. Denoting we thus have for any f ∈ L 2 + (µ) that Since, as is readily verified by an application of the monotone convergence theorem, C Γε•µε and m ε (f ) m(f ) as ε ↓ 0, and Γ ε (1) ∈ R + \{0}, we obtain (3.6) by taking ε 0 in (3.8). The positive homogeneity and convexity of C Γ•µ + (f ) as stated in (3.7) follow as direct consequences of the robust representation in (3.6).

Conditional and iterated Choquet integrals
Analogously, we define F t -conditional Choquet-type integrals as follows: Definition 3.6 For any t ∈ [0, T ] and probability distortions Ψ andΨ satisfying Assumption 3.2 (relative to the measure P restricted to (Ω, F t )), the conditional Choquet-type integral C Ψ•P,Ψ•P ( · |F t ) : L 2 → L 2 t is given by Remark 3.7 (i) Reasoning similarly as in Remark 3.4(i) and as in the proof of Lemma 3.5, we have that (a) for any X ∈ L 2 , C Ψ•P,Ψ•P (X|F t ) is square-integrable; and (b) the map C Ψ•P,Ψ•P ( · |F t ) is Lipschitz-continuous on L 2 with Lipschitz-constant K Ψ + KΨ (which are given by the constant K Γ in (3.1) with µ(U) = 1 and Γ equal to Ψ andΨ, respectively).
(ii) The conditional Choquet expectation in (3.2) of X ∈ L 2 withΨ = Ψ may equivalently be expressed as weighted integral of the conditional Expected Shortfall of X at different levels. Specifically, associated to any concave probability distortion Ψ is a unique Borel measure µ on [0, 1] defined by µ({0}) = 0 and by µ(ds) = sF (ds) for s ∈ (0, 1], where F is the locally finite positive measure given in terms of the right-derivative Ψ + of Ψ by F ((s, 1]) = Ψ + (s) (see [ The conditional Choquet expectation in Definition 3.6 can then be expressed in terms of the measure µ and the F t -conditional Expected Shortfall, as follows: where the F t -conditional Expected Shortfall ES λ (X|F t ) of X ∈ L 2 at level λ ∈ (0, 1] is given in terms of the F t -conditional Value-at-Risk VaR λ (X|F t ) = inf{z ∈ R : P(X < −z|F t ) < λ} at level λ by The proof of (3.11) follows by a straightforward adaptation to the conditional setting of the proof for the static setting given in Föllmer and Schied (2011).
(iii) It follows from the representation in (3.11) that the collection of the conditional Choquet expec- One way to define a sequence of conditional spectral risk-measures that is adapted to the filtration F (π) = (F (π) t ) t∈π is recursive in terms of conditional Choquet-integrals, as follows : Definition 3.8 Given a concave probability distortion Ψ satisfying Assumption 3.2 and a filtration F (π) = (F (π) t ) t∈π the corresponding iterated spectral risk measure S = (S t ) t∈π , S t : The class of iterated spectral risk-measures defined as such contains in particular the Iterated Tail Conditional Expectation proposed in [28] and is closely related to the Dynamic Weighted V@R that is defined in [13] for adapted processes via its robust representation. As already noted in the proof of Proposition 3.5, in the static case such a representation was derived in [9] for bounded random variables; see also [26,Theorems 4.79 and 4.94] , and see [12] for the extension to the set of measurable random variables (we refer to [22] for families of dynamic risk measure defined via stochastic distortion probabilties in a binomial tree setting; see [14] for a general theory of finite state BSDEs). We show next that iterated spectral risk measures are discrete-time time-consistent dynamic coherent risk measures and identify the driver function of the associated BS∆E. Proposition 3.9 The iterated spectral risk measure S = (S t ) t∈π given in Definition 3.8 is a discretetime coherent risk measure ρḡ ∆ ,π with driver functionḡ ∆ given bȳ where ν (π) is defined in (2.1).
Proof. It follows from Proposition 3.5 that the functionḡ ∆ defined in (3.13) is a coherent driver function in the sense of Definition 2.3 with I = π and µ = ν (π) . Let X ∈ L 2 (F (π) ) be arbitrary and denote by (Y (π) , Z (π) ) the solution of the BS∆E with driver functionḡ ∆ . To show that the dynamic coherent risk measure corresponding toḡ ∆ coincides with the spectral risk measure S = (S t ) t∈π it suffices to verify thatḡ (3.14) Letting t ∈ π\{T } and denoting ∆L = ∆L (π) t , we note from Definition 3.8 and where we used that, due to stationarity of the increments of L (π) , ∆L (π) t (which has law ν (π) ∆) is independent of t. Thus we have (3.14) and the proof is complete. 2

Dynamic spectral risk measures
With the previous results in hand we move to the definition of dynamic spectral risk-measures in continuous time. Let us fix in the sequel a pair of concave measure distortions functions Γ + and Γ − that satisfy Assumption 3.2 and are such that Γ − (x) ≤ x for x ∈ R + . We define dynamic spectral risk measures to be those coherent spectral risk measures ρ g for which the driver functions g are given in terms of Choquet integrals, as follows: The spectral driver functionḡ : L 2 (ν) → R + is given bȳ for u ∈ L 2 (ν).
By Lemma 3.5 we have thatḡ is Lipschitz-continuous, positively homogeneous and convex, so that g is a coherent driver function in the sense of Definition 2.3. The corresponding dynamic coherent risk-measure ρḡ is the object of study for the remainder of the paper, which we label as follows: The dynamic coherent risk-measure ρḡ with spectral driver functionḡ given in Definition 4.1 is called the (continuous-time) dynamic spectral risk-measure corresponding to measure distortions Γ + and Γ − .
We next show that dynamic spectral risk measure admit a dual representation of the form (2.11) and (2.13) with a representing set that is explicitly expressed in terms of the measure distortions Γ + and Γ − , as follows: and letḡ be a spectral driver function. The dynamic spectral risk-measure ρḡ satisfies the dual representation in (2.11), (2.13) with representing set Cḡ given by These expressions follow by deploying the dual representation in Theorem 4.3 and Girsanov's theorem (e.g., Theorems III.3.24 and III.5.19 in Jacod and Shiryaev (1987)): we have that ρḡ 0 (I(a)) is equal to (a,∞) (1 + H ξ t (y))ν(dy)dt = − exp(−T ν(a)) exp (−T Γ + (ν(a))) , while the expression for ρḡ 0 (−I(a)) follows in a similar manner.
Proof of Theorem 4.3.. In view of Theorem 2.8 and Remark 2.9(i)-(ii) it suffices to verify that for any h ∈ L 2 (ν) we have where h k dν = R k \{0} h(x)k(x)ν(dx). Our next observation is that the set Cḡ in (4.1) admits the following equivalent representation: To see that this is the case, we note that, for any U ∈ L 2 (ν), we have −U − ≤ U ≤ U + , while U + = U 1 and −U − = U 2 for U 1 = U I {U ≥0} and U 2 = U I {U <0} .
To see that (4.2) holds we note from (4.3), Proposition 3.5 and the identity for any h, k 1 , k 2 ∈ L 2 (ν), thatḡ(h) = sup k∈Cḡ h k dν is bounded below by which is by Proposition 3.5 equal to C Given this lower bound and the fact thatḡ(h) is bounded above by we conclude that (4.2) holds true. 2

Limit theorem
We next turn to the functional limit theorem which shows that dynamic spectral risk measures arise as a limit of iterated spectral risk measures, under a suitable scaling of the corresponding probability distortions. We suppose that, uniformly in p ∈ [0, 1], Ψ ∆ (p) − p scales in the mesh size ∆ and the measure distortions Γ + and Γ − as follows: Specifically, the condition that we require is phrased as follows: Definition 5.1 We denote by (Ψ ∆ ) ∆∈(0,1] a sequence of probability distortions that is such that Ψ ∆ and Ψ ∆ given by Ψ ∆ (p) = 1 − Ψ ∆ (1 − p) satisfy Assumption 3.2 with respect to the measure µ(dx) ≡ P(∆L (π) t 1 ∈ dx) and we have where for ∆ ∈ (0, 1] and x ∈ [0, 1] Here, we recall, Γ + and Γ − denote the given concave measure distortions which are such that Γ − (x) ≤ x for x ∈ R + and Assumption 3.2 holds with µ(dx) ≡ ν(dx) and Γ ≡ Γ + or Γ − .
Proof of Theorem 5.2.. We note first that, as L (π) d → L when ∆ 0, F (L (π) ) converges in distribution to F (L), which is element of L 2 . Furthermore, by Corollary 2.12, the collection {F (L (π) ) 2 } π is uniformly integrable. Thus, in view of Theorem 2.14 it suffices next to verify that the sequence of driver functions (ḡ ∆ ) ∆∈(0,1] of the iterated spectral risk measures S ∆ given in Proposition 3.9 satisfies Condition 2.13, which we proceed to do.

Dynamically optimal portfolio allocation
We next consider dynamic portfolio problems concerning balancing gain and risk as quantified by the DSR. We suppose the investment horizon is equal to T > 0 and consider the DSR associated to the spectral driver functionḡ. In this section we impose the following restriction on the Lévy measure ν: The support of ν is included in the set (−1, ∞) k .
We suppose that the financial market consists of a risk-free bond and n risky stocks with discounted pricesŜ = (Ŝ 1 , . . . ,Ŝ n ) evolving according to the following system of SDEs: where d i ∈ R is the excess log-return and R i ∈ R k is the (row) vector of jump-coefficients with nonnegative coordinates that are such that ((R i ) 1 ≤ 1 (where 1 ∈ R k denotes the k-column vector of ones and where, for any vector v, v denotes its transposition). Given the form of the model we havê S i t ∈ L 2 t andŜ i t > 0 for any i = 1, . . . , k and t ∈ [0, T ]. Let us consider the case of a small investor whose trades have a negligible impact on the price and let us adopt the classical frictionless and self-financing setting (no transaction cost, infinitely divisible assets, continuous-time trading, no funds are infused into or withdrawn from the portfolio at intermediate times, etc.). At any time t ∈ [0, T ] the investor decides to allocate part θ i t of the current wealth for investment into the stockŜ i , i = 1, . . . , n, so that, if X θ t− denotes the discounted wealth just before time t, we have that θ i t X θ t− /Ŝ i t− is the number of stocks i held in the portfolio at time t. We suppose that certain limits are placed on the leverage ratio of the portfolio and on the size of the short-holdings in the various stocks, and that this restriction is phrased in terms of a bounded and closed set B ⊂ R n as the requirement that θ t (ω) ∈ B for any (t, ω) ∈ [0, T ] × Ω.
(6.1) Example 6.2 To impose constraints on the fractions of the current wealth invested in the bond account and the stock accounts we take x i ≤ 1 + L 0 for some L 0 , . . . , L n ∈ R + . In particular, by taking L i > 0 we impose a limit on the borrowing (i = 0) or the number of stock i that may be shorted (i = 0). The case of a "long only" investor that has no shortsales and only invests own wealth (no borrowing) corresponds to taking in L 0 = L 1 = · · · = L n = 0.
We call an allocation strategy θ = (θ t ) t∈[0,T ] admissible if θ is predictable and (6.1) holds. We denote by A the collection of admissible allocation strategies. Denoting by R = (R i ) i=1,...,k the R n×kmatrix with ith row equal to R i , we have that the discounted value X θ = (X θ t ) t∈[0,T ] of a portfolio corresponding to θ ∈ A evolves according to the SDE t ∈ (0, τ θ ∧ T ], where τ θ = inf{t ∈ [0, T ] : X θ t < 0} (with inf ∅ = +∞) is the first time that the value of the portfolio becomes negative, when the investor has to stop trading.

Portfolio optimisation under dynamic spectral risk measures
We consider next the stochastic optimisation problem given in terms of DSR by the following criterion that is to be minimised for t ∈ [0, T ]:J θ t = ρḡ t (X θ T ∧τ θ ), (6. 2) The investor's problem is to identify a stochastic processJ * = (J * t ) t∈[0,T ] and an allocation strategy θ * ∈ A such thatJ by (denoting f = ∂f ∂x ) x) x R θ y ν(dy).
The non-linear Feynman-Kac formula (see Remark 2.7) implies that if the following semi-linear PIDE has a sufficienly regular solution it is equal to Vθ: Standard arguments suggest then that if the optimal allocation strategy θ * is of feedback-type and the corresponding value-function V is sufficiently regular, then V satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:  be such thatθ ∈Θ. Then the feedback strategyθ * = (θ * t ) t∈[0,T ] with feedback functionθ is optimal for (6.3) and we haveJ * t =Jθ * t = w(t, Xθ t∧τθ ), where Xθ solves the SDE in (6.4) and (6.5) withθ replaced byθ.
Note that by the HJB equation (6.7) the first term on the right-hand side of (6.10) is non-positive. Hence by taking conditional expectation in (6.10) and using (6.8)-(6.9) we have that w(t, X θ t ) ≤ E −X θ T ∧τ θ +