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

On some filtered probability space \((\varOmega,\mathcal{F}, \mathbf{F}, {\mathbb{P}})\) with \(\mathbf{F}=(\mathcal{F}_{t})_{t\in I}\), we consider risks described by random variables \(X\in\mathcal{L}^{p} = \mathcal{L}^{p}(\mathcal{F}_{T})\), \(p>0\), the set of \(\mathcal{F}_{T}\)-measurable \(X\) with \({\mathbb{E}}[|X|^{p}] = \int_{\varOmega}|X|^{p}\,\mathrm{d}P<\infty\). We denote by \(\mathcal{L} ^{p}_{t} = \mathcal{L}^{p}(\mathcal{F}_{t})\) and \(\mathcal{L}^{p}( \mathcal{G})\) the elements \(X\) in \(\mathcal{L}^{p}(\mathcal{F})\) that are measurable with respect to the sigma-algebras \(\mathcal{F}_{t}\) and \(\mathcal{G}\subset\mathcal{F}\), respectively, and by \(\mathcal{L} ^{\infty}\), \(\mathcal{L}^{\infty}_{t}\), \(\mathcal{L}^{\infty}( \mathcal{G})\) the collections of bounded elements in \(\mathcal{L}^{p}\), \(\mathcal{L}^{p}_{t}\) and \(\mathcal{L}^{p}(\mathcal{G})\). Let \(\mathcal{S}^{2}(\mathcal{I})\) denote the space of \(\mathbf{F}\)-adapted semimartingales \(Y = (Y_{t})_{t\in\mathcal{I}}\) that are square-integrable in the sense that \(\|Y\|^{2}_{\mathcal{S} ^{2}(\mathcal{I})}<\infty\), where For a given measure \(\mu\) on a measurable space \((\mathbb{U}, \mathcal{U})\), we denote by \(\mathcal{L}^{p}(\mu)\), \(p>0\), the set of Borel functions \(v: \mathbb{U}\to\mathbb{R}\) with \(|v|_{p,\mu}< \infty\), where and by \(\mathcal{L}_{+}^{p}(\mu)\) the set of nonnegative elements in \(\mathcal{L}^{p}(\mu)\).

Dynamic coherent risk measures and (strong) time-consistency are defined in an \(\mathcal{L}^{2}\)-setting as follows.

More generally, a map \(\rho:\mathcal{L}^{2}\to\mathcal{S}^{2}( \mathcal{I})\) is called a time-consistent dynamic risk measure if \(\rho\) satisfies conditions (i) and (v). For a proof of the equivalence of (v) and (vi), we refer to Föllmer and Schied [26, Lemma 11.11]; for a discussion of (the unconditional version of) the properties (i)‐(iv), see [2, 3]. One way to construct a time-consistent dynamic risk measure is as solution to an associated backward stochastic differential equation (BSDE) or backward stochastic difference equation (BS\(\Delta \)E). To ensure that such dynamic risk measures satisfy (iii) and (iv), the corresponding driver functions must be positively homogeneous, subadditive and should not depend on the value of the risk measure (see [42, Proposition 11] and [15, Lemma 2.1]). Furthermore, a necessary condition to ensure that a comparison principle (and hence the monotonicity in (ii)) holds is that the driver function in addition satisfies a gradient condition (see [17, Theorem 3.2.2]). For background on the notion of strong time-consistency and its relation to \(g\)-expectations, we refer to [6, 7, 39, 42]. Specifically, in our setting, such driver functions are defined as follows.

We describe next the dynamic (coherent) risk measures defined via BSDEs (if \(\mathcal{I}=[0,T]\)) and BS\(\Delta\)Es (if ℐ is a finite partition of \([0,T]\)).

We turn first to the discrete-time lattice setting, fixing a uniform partition \(\pi=\pi_{\Delta}\) of \([0,T]\) with \(\Delta= T/N\) for some \(N\in\mathbb{N}\). Let \(L^{(\pi)}=(L^{(\pi)}_{t})_{t\in\pi}\) denote a square-integrable zero-mean random walk starting at zero and taking values in \((\sqrt{\Delta}\mathbb{Z})^{k}\), and let \(\mathbf{F}^{( \pi)} = (\mathcal{F}^{(\pi)}_{t})_{t\in\pi}\) denote the filtration generated by \(L^{(\pi)}\). Furthermore, we let \(g^{(\pi)}\) be a coherent driver function as in Definition 2.3 with \(\mathcal{I}=\pi\) and \(\mu(\mathrm{d}x)\) equal to the scaled law \(\nu^{(\pi)}(\mathrm{d}x)\) of \(\Delta L^{(\pi)}_{t} = L^{(\pi)} _{t + \Delta} - L^{(\pi)}_{t}\), \(t\in\pi\backslash\{T\}\), given by In view of [37, Proposition 3.2], the BS\(\Delta\)E for \((Y^{(\pi)}, Z^{(\pi)})\) corresponding to the final value \(- X^{(\pi)}\in\mathcal{L}^{2}(\mathcal{F}_{T}^{(\pi)})\) and the driver function \(g^{(\pi)}\) takes the form, analogous to the one in the continuous-time case given in (2.6) below, for \(t\in\pi\backslash\{T\}\) and with \(Y_{T}^{(\pi)} = - X^{( \pi)}\), where \(I_{A}\) denotes the indicator of a set \(A\). In difference notation, the BS\(\Delta\)E (2.2) is for \(t\in\pi\backslash \{T\}\) given by with \(Y_{T}^{(\pi)} = - X^{(\pi)}\). A pair \((Y^{(\pi)}, Z^{(\pi)})\) is a solution of the BS\(\Delta\)E if for any \(t\in\pi\), it satisfies (2.2) with If the Lipschitz constant \(K=K^{(\pi)}\) of the driver function \(g^{(\pi)}\) is strictly smaller than the reciprocal \(1/\Delta\) of the mesh size, we can use [37, Propositions 3.1 and 3.2] and the fact that in the notation of [37] \(F^{(\pi)}\) is independent of \(W^{(\pi)}\), to conclude that there exists a unique solution \((Y^{(\pi)}, Z^{(\pi)})\) to the BS\(\Delta\)E which satisfies for \(t\in\pi\) the relations for \(x\in(\sqrt{\Delta}\mathbb{Z})^{k}\), where \(\mathcal{F}_{t} ^{(\pi)}\vee\{\Delta L^{(\pi)}_{t} = x\} := \mathcal{F}_{t}^{( \pi)}\vee\sigma(\{\Delta L^{(\pi)}_{t} = x\})\) denotes the smallest sigma-algebra containing \(\mathcal{F}_{t}^{(\pi)}\) as well as the sigma-algebra \(\sigma(\{\Delta L^{(\pi)}_{t} = x\})\) generated by the set \(\{\Delta L^{(\pi)}_{t} = x\}\). In analogy with the continuous-time case (reviewed below), the dynamic risk measure associated to the solution to the BS\(\Delta\)E is defined as follows.

In the continuous-time case \(\mathcal{I}=[0,T]\), we consider risky positions described by random variables \(X\) that are measurable with respect to \(\mathcal{F}_{T}\), where \(\mathbf{F}=(\mathcal{F}_{t})_{t \in[0,T]}\) denotes the right-continuous and completed filtration generated by a Poisson random measure \(N\) on \([0,T]\times\mathbb{R} ^{k}\backslash\{0\}\) for some \(k\in\mathbb{N}\). We suppose throughout that the associated Lévy measure \(\nu\) satisfies the following condition:

We denote by \(\tilde{N}(\mathrm{d}t\times\mathrm{d}x)= N(\mathrm{d}t \times\mathrm{d}x) - \nu(\mathrm{d}x)\,\mathrm{d}t\) the compensated Poisson random measure and by \(L=(L_{t})_{t\in[0,T]}\) the (column-vector) Lévy process given by Under Assumption 2.5, we have \({\mathbb{E}}[|L_{t}|^{2+ \varepsilon_{0}}]<\infty\) for any \(t\in[0,T]\) (see [44, Theorem 25.3]).

Let \(\tilde{\mathcal{H}}^{2}\) denote the set of \(\tilde{\mathcal{P}}\)-measurable square-integrable processes where, with \(\mathcal{P}\) denoting the predictable sigma-algebra, \(\tilde{\mathcal{P}}=\mathcal{P}\otimes\mathcal{B}(\mathbb{R}^{k} \backslash\{0\})\), and let \(\mathcal{U}\) denote the Borel sigma-algebra induced by the \(L^{2}(\nu(\mathrm{d}x))\)-norm. In particular, \(U\in\tilde{\mathcal{H}}^{2}\) is such that \(\|U\|_{ \tilde{\mathcal{H}}^{2}} <\infty\), where Moreover, let \(\mathcal{M}_{2}\) denote the set of probability measures \(\mathbb{Q} = \mathbb{Q}^{\xi}\) on \((\varOmega,\mathcal{F}_{T})\) that are absolutely continuous with respect to ℙ with square-integrable Radon–Nikodým derivative \(\xi\in\mathcal{L} ^{2}_{+}(\mathcal{F}_{T})\), and write \(\mathcal{S}^{2}:= \mathcal{S} ^{2}[0,T]\).

Let us next consider a coherent driver function \(g\) as in Definition 2.3 with \(\mu=\nu\) and \(\mathcal{I}=[0,T]\) and fix a final condition \(X\in\mathcal{L}^{2}\). The associated BSDE for the pair \((Y,Z)\in\mathcal{S}^{2}\times\tilde{\mathcal{H}}^{2}\) is given by for \(t\in[0,T]\). This BSDE, as we recall from [4], 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.

We describe next a representation result for a dynamic risk measure \(\rho^{g}\) in terms of the representing processes \((H^{\xi})\) of the stochastic logarithms of the Radon–Nikodým derivatives \(\xi\in L^{2}_{+}(\mathcal{F}_{T})\) of the measures \(\mathbb{Q}^{ \xi}\in\mathcal{M}_{2}\), which are given by where \(\mathcal{E}(\cdot)\) denotes the Doléans-Dade stochastic exponential. A \(\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{U}\)-measurable set family \(C=(C_{t})_{t\in[0,T]}\) is called convex or closed if for any \(t\in[0,T]\), the set \(C_{t}\) is convex or closed.

The proof follows by an adaptation of the arguments given in [36, Theorem A.25] and is omitted.

We next turn to the question of convergence of a sequence \(( \rho^{g^{(\pi)}, (\pi)})_{\pi}\) of dynamic coherent risk measures as in Definition 2.4 when the mesh size \(\Delta=\Delta_{ \pi}\) tends to zero. Let us suppose that \((\rho^{g^{(\pi)}, (\pi)})_{ \pi}\) are driven by the random walks \((L^{(\pi)})_{\pi}\) that are defined as for some probability distribution \((p_{j}^{\Delta}, j\in\mathbb{Z} ^{k})\) on \(\mathbb{Z}^{k}\) that is given, in terms of a constant \(c\ge1\) (to be specified shortly) and a partition \((B^{ \Delta}_{j}, j = (j_{1},\ldots, j_{k}) \in\mathbb{Z}^{k})\) of \((\sqrt{\Delta}\mathbb{Z})^{k}\) into block sets of the form where \(A_{k}^{\Delta}= [k\sqrt{\Delta}, (k+1)\sqrt{\Delta})\) if \(k>0\), \(A_{k}^{\Delta}= ((k-1)\sqrt{\Delta}, k\sqrt{\Delta}]\) if \(k<0\) and \(A_{0}^{\Delta}= (-\sqrt{\Delta}, \sqrt{\Delta})\), by where where, as before, \(\nu_{2} = \int_{\mathbb{R}^{k}\backslash\{0\}}|x|^{2} \nu(\mathrm{d}x)\). When \(\Delta\searrow0\), we have by the dominated convergence theorem that where \(L_{T}^{(\pi),m}\) and \(x_{m}\), \(m\in\{1,\ldots,k\}\), denote the \(m\)th coordinates of \(L_{T}^{(\pi)}\) and \(x\in{\mathbb{R}}^{k}\).

Moreover, we have by functional weak convergence theory (see e.g. [29, Theorem VII.3.7]) where \(\stackrel{\mathrm{d}}{\longrightarrow}\) denotes convergence in law in the Skorokhod \(J_{1}\)-topology on the space \(\mathbb{D}([0,T], \mathbb{R}^{k})\) of \(\mathbb{R}^{k}\)-valued RCLL functions.

On a suitably chosen probability space, \(L_{T}^{(\pi)}\) converges to \(L_{T}\) almost surely as \(\Delta\searrow0\). The latter convergence also holds in a stronger sense thanks to moment conditions satisfied by \(L^{(\pi)}_{T}\) that we show next. We define the value of \(c\) in terms of \(\varepsilon_{0}>0\) given in Assumption 2.5 by

The moment conditions in Lemma 2.10 carry over to those of path-functionals as follows.

To guarantee that the convergence of the random walks \((L^{(\pi)})_{ \pi}\) carries over to the convergence of the corresponding BS\(\Delta \)Es, we impose the following condition on the sequence of coherent driver functions \((g^{(\pi)})_{\pi}\) and their piecewise constant RCLL interpolations \((\tilde{g}^{(\pi)})_{\pi}\).

The convergence result for BS\(\Delta\)Es [37, Theorem 4.1] is phrased as follows in the current setting.

We describe next the Choquet-type integrals that feature in the definition of dynamic spectral risk measures given in the next section. We refer to [18] for a treatment of the theory of nonlinear integration. The Choquet-type integrals we consider are given in terms of measure distortions that we define next.

On a given measure space \((\mathbb{U},\mathcal{U},\mu)\), a set \(A\in\mathcal{U}\) with \(\mu(A)>0\) is called an atom if \(C\subseteq A\) implies \(\mu(C)\in\{0,\mu(A)\}\). We assume throughout that the measure distortions and associated measure spaces are of the following type:

The Choquet-type integrals that we consider are defined as follows.

We record next a robust representation result for Choquet-type integrals that plays an important role in the sequel. Let \({\mathcal{M}}_{p, \mu}\), \(p\ge1\), denote the set of measures \(m\) on \((\mathbb{U}, \mathcal{U})\) that are absolutely continuous with respect to a given measure \(\mu\) on this space with Radon–Nikodým derivatives such that \(\frac{\mathrm{d}m}{\mathrm{d}\mu}\in\mathcal{L}^{p}_{+}( \mu)\).

Analogously, we define \(\mathcal{F}_{t}\)-conditional Choquet-type integrals as follows.

One way to define a sequence of conditional spectral risk measures adapted to the filtration \(\mathbf{F}^{(\pi)}=(\mathcal{F}_{t}^{( \pi)})_{t\in\pi}\) is recursive in terms of conditional Choquet-integrals, as follows.

The class of iterated spectral risk measures defined above contains in particular the iterated tail conditional expectation proposed in [28] and is closely related to the dynamic weighted V@R defined in [12] 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 [8] for bounded random variables; see also [26, Theorems 4.79 and 4.94], and see [11] for the extension to the set of measurable random variables (we refer to [22] for families of dynamic risk measures defined via stochastic distortion probabilities in a binomial tree setting; see [13] 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 we identify the driver function of the associated BS\(\Delta\)E.

We consider next the stochastic optimisation problem given in terms of DSR by the criterion to minimise, for \(t\in[0,T]\), the quantity The investor’s problem is to identify a stochastic process \(\tilde{\mathcal{J}}^{*}=(\tilde{\mathcal{J}}^{*}_{t})_{t\in[0,T]}\) and an allocation strategy \(\theta^{*}\in{\mathcal{A}}\) such that While the problem in (6.2) may be solved via a BSDE approach (as used for instance in [5, 36] to analyse utility optimisation and robust portfolio choice problems), due to its Markovian nature, it may also be approached via classical methods based on an associated Hamilton–Jacobi–Bellman equation; this is the method we present here. One class of allocation strategies are those of feedback type, defined as follows.

Associated to a given allocation strategy \(\bar{\theta}\in\bar{ \varTheta}\) of feedback type, there exists a value function \(V^{\bar{ \theta}}\) satisfying \(J^{*}_{t} = V^{\bar{\theta}}(t,X^{\bar{\theta }}_{t})\) for \(t\in[0,T]\) (as a consequence of the Markov property). If sufficiently regular, the function \(V^{\bar{\theta}}\) satisfies a semilinear PIDE that is given in terms of certain operators \(\mathcal{D}^{\theta}\) and \({\mathcal{G}}^{\theta}\) indexed by \(\theta\in\mathcal{B}\). For any function \(f\in C^{1,1}([0,T]\times {\mathbb{R}})\), these operators yield the functions \(\mathcal{D}^{ \theta}_{t,x}f:{\mathbb{R}}^{k}\to{\mathbb{R}}\) and \(\mathcal{G} ^{\theta}f:[0,T]\times{\mathbb{R}}_{+}\backslash\{0\}\to{\mathbb{R}}\) given in terms of by (denoting \(f' = \frac{\partial f}{\partial x}\)) The nonlinear Feynman–Kac formula (see Remark 2.7) implies that if the following semilinear PIDE has a sufficiently regular solution, it is equal to \(V^{\bar{\theta}}\): Standard arguments suggest that if the optimal allocation strategy \(\theta^{*}\) is of feedback type and the corresponding value function \(V\) is sufficiently regular, then \(V\) satisfies the Hamilton–Jacobi–Bellman (HJB) equation Next we verify that a sufficiently smooth solution of the HJB equation gives rise to a solution of the optimisation problem in (6.2). Let \(C_{b}^{1,1}([0,T]\times{\mathbb{R}})\) denote the set of \(C^{1,1}\)-functions \(f:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}\) with bounded first order derivatives.