Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

Abstract

For an $\mathcal{F}_T$-measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an $\mathcal{F}_T$-measurable stochastic process (called a position process, not necessarily $\mathbb {F}$-adapted), mimicking BSDE’s approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in Z(t, s)). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed.

Appendix

In this appendix, we will sketch an argument supporting the BSVIE model for the equilibrium recursive utility process/equilibrium dynamic risk measure of a position process $\psi (\cdot )$. The idea is adopted from [49]. Let $\psi (\cdot )$ be a continuous $\mathcal{F}_T$-measurable process. Let $\Pi =\{t_k~|~0\leqslant k\leqslant N\}$ be a partition of [0, T] with $0=t_0<t_1<\cdots<t_{N-1}<t_N=T$. The mesh size of $\Pi$ is denoted by $\Vert \Pi \Vert \triangleq \displaystyle \max _{0\leqslant i\leqslant N-1}|t_{i+1}-t_i|$. Let

$$\begin{aligned}\psi ^\Pi (t)=\sum _{k=1}^N\psi _k\mathbf{1}_{(t_{k-1},t_k]}(t),\end{aligned}$$

with

$$\begin{aligned}\psi _k=\psi (t_k)\in L^2_{\mathcal{F}_T}(\Omega ;\mathbb {R}),\qquad k=1,2,\ldots ,N.\end{aligned}$$

We assume that

$$\begin{aligned}\lim _{\Vert \Pi \Vert \rightarrow 0}\sup _{t\in [0,T]}\mathbb {E}|\psi ^\Pi (t)-\psi (t)|^2=0.\end{aligned}$$

We first try to specify the time-consistent recursive utility process for $\psi ^\Pi (\cdot )$, making use of BSDEs. Then let $\Vert \Pi \Vert \rightarrow 0$ to get our BSVIE time-consistent recursive utility process model for $\psi (\cdot )$.

For $\{\psi ^\Pi (t)~|~t\in (t_{N-1},t_N]\}=\{\psi _N\}$, its recursive utility at $t\in [t_{N-1},t_N]$ is given by $Y^N(t)$, where $(Y^N(\cdot ),Z^N(\cdot ))$ is the adapted solution to the following BSDE:

$$\begin{aligned} Y^N(t)= & {} \psi _N+\int _t^Tg(s,Y^N(s),Z^N(s))ds \nonumber \\&- \int _t^TZ^N(s)dW(s),\qquad t\in [t_{N-1},t_N]. \end{aligned}$$

(8.1)

Here, $g:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is an aggregator. Next, for $\{\psi ^\Pi (t)~|~t\in (t_{N-2},t_N]\}$, the recursive utility at $t\in (t_{N-2},t_{N-1}]$ is denoted by $Y^{N-1}(t)$ and we should have

$$\begin{aligned}&\displaystyle Y^{N-1}(t)\nonumber \\&\quad =\psi _{N-1}+\int _{t_{N-1}}^Tg(s,Y^N(s),Z^{N-1}(s))ds+\int _t^{t_{N-1}} g(s,Y^{N-1}(s),Z^{N-1}(s))ds\nonumber \\&\displaystyle \qquad -\int _t^TZ^{N-1}(s)dW(s),\qquad t\in (t_{N-2},t_{N-1}]. \end{aligned}$$

(8.2)

Note that due to the time-consistent requirement, we have to use the already determined $Y^N(\cdot )$ in the drift term over $[t_{N-1},T]$. On the other hand, since $\psi _{N-1}$ is still merely $\mathcal{F}_T$-measurable, (8.2) has to be solved in [t, T] although $t\in (t_{N-2},t_{N-1}]$. Hence, in the martingale term, $Z^{N-1}(\cdot )$ has to be free to choose over the entire $[t_{N-2},T]$ and the already determined $Z^N(\cdot )$ cannot be forced to use there (on $[t_{N-1},T]$). Whereas, in the drift term over $[t_{N-1},T]$, it seems to be fine to either use already determined $Z^N(\cdot )$ or to freely choose $Z^{N-1}(\cdot )$, since the time-inconsistent requirement is not required for Z part. However, we use $Z^{N-1}(\cdot )$ in the drift, which will enable us to avoid a technical difficulty for BSVIEs later.

Similarly, the recursive utility on $(t_{N-3},t_{N-2}]$ should be

$$\begin{aligned}&\displaystyle Y^{N-2}(t)\\&\quad =\psi _{N-2}+\int _{t_{N-1}}^Tg(s,Y^N(s),Z^{N-2}(s))ds+\int _{t_{N-2}}^{t_{N-1}} g(s,Y^{N-1}(s),Z^{N-2}(s))ds\\&\displaystyle \qquad +\int _t^{t_{N-2}}g(s,Y^{N-2}(s),Z^{N-2}(s))ds-\int _t^TZ^{N-2}(s)dW(s), \quad t\in (t_{N-3},t_{N-2}]. \end{aligned}$$

This procedure can be continued inductively. In general, we have

$$\begin{aligned} \displaystyle Y^k(t)&=\psi _k+\sum _{i=k+1}^N\int _{t_{i-1}}^{t_i}g(s,Y^i(s),Z^k(s))ds +\int _t^{t_k}g(s,Y^k(s),Z^k(s))ds\\&\qquad -\int _t^TZ^k(s)dW(s), \quad t\in (t_{k-1},t_k]. \end{aligned}$$

Let us denote

$$\begin{aligned} Y^\Pi (t)=\sum _{k=1}^NY^k(t)\mathbf{1}_{(t_{k-1},t_k]}(t),\qquad Z^\Pi (t,s)=\sum _{k=1}^NZ^k(s)\mathbf{1}_{(t_{k-1},t_k]}(t). \end{aligned}$$

Then

$$\begin{aligned} Y^\Pi (t)=\psi ^\Pi (t)+\int _t^Tg(s,Y^\Pi (s),Z^\Pi (t,s))ds-\int _t^TZ^\Pi (t,s)dW(s),\quad t\in [0,T]. \end{aligned}$$

Let $\Vert \Pi \Vert \rightarrow 0$, by the stability of adapted solutions to BSVIEs [48], we obtain

$$\begin{aligned} Y(t)=\psi (t)+\int _t^Tg(s,Y(s),Z(t,s))ds-\int _t^TZ(t,s)dW(s),\quad t\in [0,T],\qquad \end{aligned}$$

(8.3)

which is the BSVIE that we expected. Moreover, it is found that if $Y(\cdot )$ is a utility process for $\psi (\cdot )$, the current utility Y(t) depends on the (realistic) future utilities $Y(r);\,t\leqslant r\leqslant T$, which is the main character of recursive utility process. Finally, we note that if we restrict $Z^{N-1}(\cdot )$ on $[t_{N-1},T]$ in (8.2), etc., then we will end up with the following BSVIE:

$$\begin{aligned}Y(t)=\psi (t)+\int _t^Tg(s,Y(s),Z(s,s))ds-\int _t^TZ(t,s)dW(s),\quad t\in [0,T],\end{aligned}$$

which is technically difficult since in general, $s\mapsto Z(s,s)$ is not easy to define.

Finally, we would like to point out a fact about BSVIEs and BSDEs. Let us first look at the following general BSDE:

$$\begin{aligned} Y(t)=\xi +\int _t^Tg(s,Y(s),Z(s))ds-\int _t^TZ(s)dW(s),\qquad t\in [0,T].\end{aligned}$$

(8.4)

Under standard conditions, for any $\xi$ in a proper space, the above BSDE admits a unique solution $(Y(\cdot ),Z(\cdot ))\equiv (Y(\cdot \,;T,\xi ),Z(\cdot \,;T,\xi ))$. By the uniqueness of adapted solutions of BSDEs, we have

$$\begin{aligned} \begin{aligned} Y(t;T,\xi )=Y(t;\tau ,Y(\tau ;T,\xi )),\\ Z(t;T,\xi )=Z(t;\tau ,Y(\tau ;T,\xi )),\end{aligned}\qquad \forall 0 \leqslant t<\tau \leqslant T. \end{aligned}$$

This can be referred to as a (backward) semi-group property of BSDEs [34]. However, there is no way to talk about the (backward) semi-group property for BSVIEs. To illustrate this point, let us look at the following simple BSVIE:

$$\begin{aligned}Y(t)=tW(T)-\int _t^TZ(t,s)dW(s),\qquad t\in [0,T].\end{aligned}$$

We can directly check that the adapted solution is given by

$$\begin{aligned}Y(t)=tW(t),\quad Z(t,s)=t,\qquad (t,s)\in \Delta [0,T].\end{aligned}$$

We see that the above $Y(\cdot )$ really could not be related to any (backward) semi-group property. The point that we want to make is that time-consistency and semi-group property are irrelevant.