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Erschienen in: Mathematics and Financial Economics 3/2020

12.03.2020

A generalized stochastic differential utility driven by G-Brownian motion

verfasst von: Qian Lin, Dejian Tian, Weidong Tian

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2020

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Abstract

This paper introduces a class of generalized stochastic differential utility (GSDU) models in a continuous-time framework to capture ambiguity aversion on the financial market. This class of GSDU models encompasses several classical approaches to ambiguity aversion and includes new models about ambiguity aversion. For a general GSDU model, we demonstrate its continuity, monotonicity, time consistency, concavity, and homotheticity. We investigate its comparative ambiguity aversion and direction aversion under sufficient conditions. We further solve an optimal portfolio choice problem in one GSDU model as an application.

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Fußnoten
1
There are other classes of ambiguity models in a dynamic setting in literature. See, for instance, Klibanoff et al. [22] and Ju and Miao [21] for a smooth ambiguity model and its applications.
 
2
As shown in El Karouri et al. [11], a SDU model is closely related to a backward stochastic differential equation (BSDE).
 
3
The equilibrium analysis of ambiguous volatility is recently developed in Beissner [2], and Beissner and Riedel [3].
 
4
Rosazza Gianin [32] and Jiang [20] also consider the relationships between risk measure theory and generator f in the sense of g-expectations.
 
5
See Easley and O’Hara [9], Easley and O’Hara [10], Buraschi et al. [4] for the effect of the covariance ambiguity or uncertainty on optimal portfolio choice and equilibrium asset price.
 
6
An introduction of G-expectation theory is presented in Appendix A.
 
7
For example, if f is Lipschitz continuous in its second argument, then for each \(c\in {\mathscr {C}}\), \(f(\cdot , 0, 0, 0)\in M^\beta _G (0, T)\).
 
8
If the aggregators f and \(g_{ij}\) do not satisfy the uniformly Lipschitz conditions (H2) in both y and z, the existence and the uniqueness solution to a corresponding GBSDE can be still proved under certain circumstances. For instance, assume \(f = 0\) and \(g(s,c,y,z)=u(c)-\frac{\gamma }{2} |z|^{2},\) and \(\xi \) and \(u(\cdot )\) satisfy some milder conditions. Then, the aggregator \(g(\cdot )\) is uniformly locally (but not uniformly) Lipschitz in z. By Hu et al. [19] (Theorem 5.3, Propositions 3.5 and 3.7), there exists a unique solution of the corresponding GBSDE (7). Therefore, the following stochastic differential utility model for \((\xi , c)\) is well defined:
$$\begin{aligned} Y_t^{c,\xi } =-\hat{{\mathbb {E}}}_{t}\left[ -\xi -\int _t^T \big (u(c_{s})-\frac{\gamma }{2} |Z^{c,\xi }_{s}|^{2} \big )d\langle B\rangle _{s}\right] . \end{aligned}$$
In this paper, we study the properties of a general GSDU model in which the aggregators satisfy (H1) and (H2), and the discussions of other stochastic differential utility models are beyond the scope of this paper.
 
9
The operator \(\tilde{{\mathbb {E}}}_t[\cdot ]\) can be represented in terms of the sublinear expectation \(\hat{{\mathbb {E}}}_{t}\) introduced in Epstein and Ji [13] and Hu et al. [18]. Precisely, \(\tilde{{\mathbb {E}}}_t[\xi ]=\hat{{\mathbb {E}}}_{t}^{{\tilde{G}}}\left[ \xi \exp \Big ( \int _{t}^{T}d_{s}dB_{s}-\frac{1}{2}\int _{t}^{T}|d_{s}|^{2}d\langle B\rangle _{s}-\int _{t}^{T}b_{s}d_{s}ds+\int _{t}^{T}|b_{s}|^{2}d\langle {\tilde{B}}\rangle _{s}\Big )\right] ,\) where \((B,{\tilde{B}})\) is an auxiliary extended \({\tilde{G}}\)-Brownian motion and
$$\begin{aligned} {\tilde{G}}(A)=\frac{1}{2}\sup _{{\underline{\sigma }}^{2}\le \nu \le {\bar{\sigma }}^{2}} tr\left[ A\left[ \begin{matrix} \nu &{} 1\\ 1&{} \nu ^{-1} \end{matrix}\right] \right] , \ A\in {\mathbb {S}}^{2}, \end{aligned}$$
where \({\mathbb {S}}^{2}\) denotes the set of \(2\times 2\) symmetric matrices.
 
10
Specifically, Epstein and Ji [13] formulate the recursive utility model under the G-expectation as
$$\begin{aligned} Y_t^{c,\xi } =-\hat{{\mathbb {E}}}_t\left[ -u(\xi )-\int _t^T f(s, c_s, Y_s^{c,\xi })ds\right] . \end{aligned}$$
where \(u(\cdot )\) is a standard utility function.
 
11
This is the \(\kappa \)-ignorance ambiguity model introduced by Chen and Epstein [5] with broad applications to economics and finance. See Bernard et al. [1], Nishimura and Ozaki [28], Schroder and Skiadas [33] and Tian and Tian [35]. Specifically,
$$\begin{aligned}\Theta =\left\{ Q_{\theta }: ~dQ_{\theta }/dP=\exp \left\{ -\int _{0}^{T}\theta _{s} dW_{s}-\frac{1}{2}\int _{0}^{T}\theta _{s}^{2}ds\right\} ,~ |\theta _{s}|\le \kappa \right\} .\end{aligned}$$
 
12
There exists a unique solution to this equation, see Chapter 5 in Peng [31]. Precisely, \(X_{t}=\Gamma _{t}\left( x-\int _{0}^{t}c_{s}\Gamma _{s}ds\right) \), where \(\Gamma _{t}=\exp \left( \int _{0}^{t}(r+\pi _{s}^{T}(\mu -r\mathbf{{I}}))ds +\frac{1}{2}\int _{0}^{t}\langle \pi _{s}\pi _{s}^{T}, d\langle B\rangle _{s} \rangle -\int _{0}^{t}\pi _{s}^{T}dB_{s}\right) , t\in [0,T].\)
 
13
By a polynomial growth condition, we mean that there exists a constant \(k_{0}>0\) and a positive integer \(k_{1}\) such that \(|\varphi (t, x)|\le k_{0}(1+|x|^{k_{1}}) \) for all \((t, x)\in Q\).
 
14
The intuition is similar to the expected return ambiguity situation in which the smallest expected return is associated with the worst-case scenario. See Chen and Epstein [5].
 
15
In a single period with both expected return and volatility ambiguity, Easley and O’Hara [9], Sect. 3, demonstrate that the agent’s investment decision depends on the joint effect of the expected return and the volatility ambiguity even though two risky assets are independent. See also Easley and O’Hara [10], Sect. 2. Moreover, Fouque et al. [15] discuss the uncertainty on \(\rho \) in the absence of uncertainty on \(\sigma _{1}^{2}\) and \(\sigma _{2}^{2}\).
 
16
We thank one anonymous referee for pointing out the following example.
 
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Metadaten
Titel
A generalized stochastic differential utility driven by G-Brownian motion
verfasst von
Qian Lin
Dejian Tian
Weidong Tian
Publikationsdatum
12.03.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 3/2020
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-020-00264-z

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