Time-consistent mean-variance portfolio selection in discrete and continuous time

Abstract

It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellman’s optimality principle and therefore the usual dynamic programming approach fails. We develop a time-consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This complements and generalises the formulation by Basak and Chabakauri (2010) and the corresponding example in Björk and Murgoci (2010), where the treatment and the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the Föllmer–Schweizer decomposition of the mean-variance trade-off. As a by-product, this also gives new convergence results for the Föllmer–Schweizer decomposition, i.e., for locally risk-minimising strategies.

Appendix: Representative square-integrable portfolios

In this appendix, we show the existence of representative square-integrable portfolios as announced in Sect. 2. As stated in Lemma A.1 below, these are square-integrable strategies φ ⁱ∈Θ _S for i=1,…,d that are representative in the sense that the financial market $(\widetilde{S}, \varTheta_{\widetilde{S}})$ with for i=1,…,d generates the same family of wealth processes as the financial market (S,Θ _S ), i.e., . For this we use the notion of σ-square-integrability. A semimartingale X is σ-square-integrable, which we denote by $X\in\mathcal{H}^{2}_{\sigma }(P)$, if there exists an increasing sequence (D _n ) of predictable sets such that D _n ↑Ω×[0,T] and for each n; see [30] for the concept of σ-localisation. If there exists a sequence of stopping times (σ _n ) such that we can choose $D_{n}=[\mskip-2mu[0,\sigma_{n}]\mskip-2mu]$ for each n∈ℕ, the concept of σ-square-integrability coincides with the classical notion of local square-integrability. The latter is for example always the case if S is continuous. The basic idea for the proof is then the following. Even though square-integrability is a global property of the strategy ϑ, it implies that ϑ is σ-square-integrable, i.e., , which can be characterised (ω,t)-pointwise. Since there exists a one-to-one correspondence between σ-square-integrable and square-integrable integrands by Proposition 2 in [23] (see below), the (ω,t)-pointwise characterisation of σ-square-integrability is sufficient to find the representative square-integrable portfolios. To derive this characterisation, we need to work with the notion of predictable characteristics which we introduce next.

As in Theorem II.2.34 in [27], each semimartingale S has the canonical representation

with the jump measure μ of S and its predictable compensator ν. Then the quadruple (b,c,F,B) of predictable characteristics of S consists of a predictable ℝ^d-valued process b, a predictable nonnegative-definite symmetric matrix-valued process c, a predictable process F with values in the set of Lévy measures on ℝ^d, and a predictable nondecreasing process B null at zero such that

Using this local description of the semimartingale S, we can prove the existence of representative square-integrable portfolios.

Lemma A.1

There exist square-integrable strategies φ ⁱ∈Θ _S for i=1,…,d such that the financial markets (S,Θ _S ) and $(\widetilde{S},\varTheta_{\widetilde{S}})$ with for i=1,…,d admit the same families of wealth processes, i.e., .

Proof

By Proposition 2 in [23] (and the paragraph preceding that), σ-square-integrability of a semimartingale X is equivalent to the existence of a strictly positive, bounded predictable process ψ such that . As ψ is bounded and strictly positive, we can therefore always switch back and forth between σ-square- integrable X and square-integrable semimartingales Y by using the associativity of the stochastic integral, i.e., and . Moreover, this also allows to reduce our problem to σ-square-integrability, which we consider first. Like any semimartingale, a stochastic integral of an S-integrable process ϑ is σ-square-integrable if and only if the sum $Z:=\sum_{0<s\leq\cdot}(\vartheta_{s}^{\top}\Delta S_{s})^{2}$ of its squared jumps is σ-integrable, i.e., there exists an increasing sequence (D _n ) of predictable sets such that D _n ↑Ω×[0,T] and has integrable total variation $\int_{0}^{T}|dZ^{n}_{s}|$ for each n. By Theorem II.1.8 in [27], the latter condition is equivalent to the process $\int_{0}^{\cdot}\int_{\mathbb{R}^{d}}(\vartheta_{s}^{\top}{x})^{2}F_{s}(dx)\,dB_{s}$ being σ-integrable, which holds if and only if $\int_{\mathbb{R}^{d}}(\vartheta_{s}^{\top}{x})^{2}F_{s}(dx)<+\infty$ P _B -a.e. If S is one-dimensional, i.e., d=1, we can write $\vartheta_{s}^{2}\int_{\mathbb{R}^{d}}x^{2}F_{s}(dx)=\int_{\mathbb {R}^{d}}(\vartheta_{s}^{\top}{x})^{2}F_{s}(dx)<+\infty$ P _B -a.e., which basically tells us that we must have ϑ=0 P _B -a.e. on the set $D^{c}:=\{\int_{\mathbb{R}^{d}}x^{2}F(dx)=+\infty\}\in\mathcal{P}$. Therefore setting , where ψ is the integrand from Proposition 2 in [23] for the σ-square-integrable semimartingale , gives the desired strategy.

In the multidimensional case, the situation is more involved due to the linear dependence between the different components of S. To deal with this issue, we use similar techniques as in [14], where we also send the reader for more explanations on problems arising from this. For the rest of the proof, we consider integrands ϑ∈L(S) as elements of $L^{0}(\varOmega\times[0,T],\mathcal{P},P_{B};\mathbb {R}^{d})$ and define the linear subspace V by

$$V= \biggl\{\vartheta\in L^0\bigl(\varOmega\times[0,T],\mathcal {P},P_B;\mathbb{R}^d\bigr)\biggm{|}\int_{\mathbb{R}^d} \bigl(\vartheta^\top {x}\bigr)^2F(dx)<+\infty\ P_B\text{-a.e.} \biggr\}. $$

By definition, V satisfies the stability property that for all ϑ ¹,ϑ ²∈V and $D\in\mathcal{P}$, and it is closed with respect to convergence in P _B -measure by Fatou’s lemma. So there exist by Lemma 6.2.1 in [17] (see also Lemma 5.2 in [14]) v ⁱ∈V for i=1,…,d such that

• (1) {v ⁱ⁺¹≠0}⊆{v ⁱ≠0} for i=1,…,d−1,

• (2) |v ⁱ(ω,t)|=1 or |v ⁱ(ω,t)|=0,

• (3) (v ⁱ)^⊤ v ^k=0 for i≠k,

• (4) ϑ∈V if and only if $\vartheta=\sum_{i=1}^{d} (\vartheta^{\top}{v}^{i}) v^{i}$ P _B -a.e.

Since v ⁱ is in V and bounded according to (2), we have that v ⁱ∈L(S) and is σ-square-integrable for i=1,…,d. By Proposition 2 in [23], there exist strictly positive, bounded predictable processes ψ ⁱ such that for i=1,…,d, and we set φ ⁱ=ψ ⁱ v ⁱ and . Since we can write each ϑ∈Θ _S ⊆V as $\vartheta=\sum_{i=1}^{d} (\vartheta^{\top}{v}^{i}) v^{i}=\sum_{i=1}^{d} \frac{(\vartheta^{\top}{v}^{i})}{\psi^{i}} \varphi^{i}$ P _B -a.e. by (4), this gives $\widetilde{\vartheta}=(\frac{(\vartheta^{\top}{v}^{1})}{\psi^{1}},\ldots,\frac {(\vartheta^{\top}{v}^{d})}{\psi^{d}})=:\varPsi\vartheta\in\varTheta_{\widetilde{S}}$, where $\varPsi:= (\frac{v^{1}}{\psi^{1}},\ldots,\frac{v^{d}}{\psi^{d}} )^{\top}$ is a predictable ℝ^d×d-valued process, and that by the associativity of the stochastic integral. Conversely, we have for each $\widetilde{\vartheta}\in\varTheta_{\widetilde{S}}$ that $\vartheta=\sum_{i=1}^{d} \widetilde{\vartheta}^{i} \varphi^{i}=\varPhi\widetilde{\vartheta}\in\varTheta_{S}$ with , where Φ:=(φ ¹,…,φ ^d) is an ℝ^d×d-valued predictable process, which allows us to conclude that and this completes the proof. □

Remark A.2

As an alternative to the proof above one can introduce a predictable correspondence C by

$$C(\omega,t):= \biggl\{y\in\mathbb{R}^d\biggm{|}\int_{\mathbb {R}^d} \bigl(y^\top {x}\bigr)^2F(dx)<+\infty \biggr\} $$

for all (ω,t)∈Ω×[0,T]. Then the condition ϑ∈V can be formulated as the pointwise constraint that ϑ(ω,t)∈C(ω,t) P _B -a.e. As the values of C are linear subspaces, one can deduce the existence of representative σ-square-integrable portfolios by using (the arguments in the proof of) Theorem B.3 in Nutz [39]. The correspondence of the transformed constraints $\widetilde{C}$ is then of course equal to ℝ^d for all (ω,t)∈Ω×[0,T] and the representative σ-square-integrable portfolios are the representative portfolios.