Backward nonlinear expectation equations

Abstract

Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form

$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$

We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.

Appendices

Auxiliary results

This appendix gathers some auxiliary results required in the main body of this article.

Lemma A.1

The space $({\mathrm {P}},\Vert {\cdot }\Vert _{{\mathrm {P}}})$ is a Banach space.

Proof

Since ${\mathrm {L}}_T$ is a Banach space, so is the space ${\mathbb {L}}^1(\mu ;{\mathrm {L}}_T)$ of ${\mathrm {L}}_T$-valued $\mu$-integrable functions by Theorem III.6.6 in Dunford and Schwartz [13]. Hence if $\{X^n\}_{n\in {\mathbb {N}}}$ is a Cauchy sequence in ${\mathcal {P}}\subset {\mathbb {L}}^1(\mu ;{\mathrm {L}}_T)$, there is $Y\in {\mathbb {L}}^1(\mu ;{\mathrm {L}}_T)$ such that $\Vert X^n-Y\Vert _{{\mathrm {P}}} = \int _{[0,T]}\Vert X^n_t-Y_t\Vert _{{\mathrm {L}}}\,\mu ({\mathrm {d}}t)\rightarrow 0$. Hence there are a subsequence $\{n_k\}_{k\in {\mathbb {N}}}$ and a $\mu$-null set N such that $X_t^{n_k}\rightarrow Y_t$ in ${\mathrm {L}}_T$ as $k\rightarrow \infty$ for all $t\in [0,T]{\setminus } N$. It follows that $X\triangleq 1_{[0,T]{\setminus } N}Y$ satisfies $\Vert X^n-X\Vert _{{\mathrm {P}}}\rightarrow 0$, and since $X^{n_k}_t\in {\mathrm {L}}_t$ for all $k\in {\mathbb {N}}$ and ${\mathrm {L}}_t$ is itself a Banach space, we have $X_t\in {\mathrm {L}}_t$ for all $t\in [0,T]$. $\square$

Lemma A.2

The integral

$$\begin{aligned} I : {\mathrm {P}}\rightarrow {\mathrm {D}}, \quad X \mapsto IX, \quad \text {where } (IX)_t \triangleq {\textstyle \int _0^t} X_s \, \mu ({\mathrm {d}}s),\ t \in [0,T], \end{aligned}$$

is a continuous linear operator.

Proof

By (2.6), IX is an adapted ${\mathscr {L}}$-process and $\sup _{t\in [0,T]}\Vert IX_t\Vert _{{\mathrm {L}}}\le \Vert X\Vert _{{\mathrm {P}}}$. Moreover, $\Vert (IX)_s - (IX)_t\Vert _{{\mathrm {L}}} \le \int _t^s \Vert X_u\Vert _{{\mathrm {L}}}\, \mu ({\mathrm {d}}u)$ implies that IX is càdlàg. $\square$

Lemma A.3

For each $\xi \in {\mathrm {L}}_T$ let $X^\xi : [0,T] \rightarrow {\mathrm {L}}_T, \ t \mapsto {\mathcal {E}}_t [\xi ]$. Then $\{{\mathcal {E}}_t\}_{t\in [0,T]}$ is regular (measurable) if and only if there exists a dense subset $M\subset {\mathrm {L}}_T$ such that $X^\xi$ is càdlàg (measurable) for each $\xi \in M$.

Proof

The operator satisfies

$$\begin{aligned} {\textstyle \sup _{t \in [0,T]}} \Vert X^\xi (t) - X^\eta (t)\Vert _{{\mathrm {L}}} \le \Vert \xi - \eta \Vert _{{\mathrm {L}}} \quad \text {for all } \xi , \eta \in {\mathrm {L}}_T \end{aligned}$$

by the projection property, so the claim follows from a density argument. $\square$

Lemma A.4

For each $n \in {\mathbb {N}}$, let $g^n:[0,T]\times {\mathrm {S}}\rightarrow {\mathrm {L}}_T$ be defined as in (3.10) and let $\xi ^n\in {\mathrm {L}}_T$. Then the pair $(g^n,\xi ^n)$ is a BNEE parameter, and, with the constant $L>0$ from (3.7), we have for all $n \in {\mathbb {N}}$

$$\begin{aligned} \Vert g^n(t,U) - g^n(t,V)\Vert _{{\mathrm {L}}} \le L \Vert U^t - V^t\Vert _{{\mathrm {S}}} \quad \text { for all }U,V\in {\mathrm {S}},\ t \in [0,T]. \end{aligned}$$

Proof

From (3.10) it is immediate that $g^n({\cdot },U)$ is a measurable and adapted ${\mathscr {L}}$-step process, so $g^n(\cdot , U) \in {\mathrm {D}}$ for all $U \in {\mathrm {S}}$. In particular $g^n(\cdot ,0)\in {\mathrm {P}}$ and $g^n(\cdot ,U)\in \mathcal X$ for all $U \in {\mathrm {S}}$. Moreover, for an arbitrary $t \in [t_{k}^n,t_{k+1}^n)$, the definition of $g^n$, (3.7) and the projection property (2.3) imply

$$\begin{aligned} \Vert g^n(t, U) - g^n(t,V)\Vert _{{\mathrm {L}}} \le L \Vert U_{t_{k+1}^n}- V_{t_{k+1}^n} \Vert _{{\mathrm {L}}} \le L \Vert U^t - V^t \Vert _{{\mathrm {S}}}, \end{aligned}$$

thus finishing the proof. $\square$

Lemma A.5

For each $n\in {\mathbb {N}}$ the ${\mathscr {L}}$-process $X^n$ given by (3.9) is a member of ${\mathrm {D}}$ and satisfies the BNEE

$$\begin{aligned} X_t^n = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g^n(s, X^n) \,\mu ({\mathrm {d}}s) + \xi ^n \Bigr ], \quad t \in [0,T], \end{aligned}$$

where $g^n:[0,T]\times {\mathrm {S}}\rightarrow {\mathrm {L}}_T$ is the generator defined in (3.10).

Proof

Let $n\in {\mathbb {N}}$ and $k\in \{0, \ldots ,N_n-1\}$. By (3.10) we have

$$\begin{aligned} f^n(t_k^n, {\mathcal {E}}_{t_k^n}[X^n_{t_{k+1}^n}]) =g^n(t, X^n) \quad \text {for all }t \in [t_k^n,t_{k+1}^n), \end{aligned}$$

and hence (3.9) yields

$$\begin{aligned} X_t^n = {\mathcal {E}}_{t} \Bigl [ \textstyle {\int _t^{t_{k+1}^n}} g^n(s, X^n) \,\mu ({\mathrm {d}}s) + X^n_{t^n_{k+1}} \Bigr ],\quad t \in [t_k^n, t_{k+1}^n),\ k= 0, \ldots N_n-1. \end{aligned}$$

Iterating this and using (TC) we obtain

$$\begin{aligned} X_t^n = {\mathcal {E}}_{t} \Bigl [ \textstyle {\int _t^{T}} g^n(s, X^n) \,\mu ({\mathrm {d}}s) + \xi ^n \Bigr ],\quad t\in [0,T]. \end{aligned}$$

Now Lemma 3.5 shows that $X^n \in {\mathrm {D}}$ since $\{{\mathcal {E}}_t\}_{t \in [0,T]}$ is regular and $(g^n,\xi )$ is a BNEE parameter by Lemma A.4. $\square$

Lemma A.6

The family ${\mathcal {Q}}$ defined in (4.1) is uniformly tight.

Proof

Since $\mu$ and $\sigma$ are bounded, this is a consequence of the moment criterion for tightness on Wiener space; see Corollary 16.9 in Kallenberg [23]. $\square$

Lemma A.7

With $f^n$ defined in (5.2), $\{(\varDelta ^n, f^n, \xi )\}_{n \in {\mathbb {N}}}$ is $(f_c,\xi )$-exhausting.

Proof

By (A2) it follows immediately that the Lipschitz condition (3.7) holds true, and thus it remains to prove (3.8). For any Y in ${\mathrm {S}}$, (A3) implies $f^n(t,Y_t) \rightarrow f_c(t,Y_t)$ for ${\mathrm {d}}t$-a.e. $t \in [0,T]$ since $c_{t}^n \rightarrow c_t$ for ${\mathrm {d}}t$-a.e. $t \in [0,T]$. Note that (A1) and (A2) imply that for all $\varDelta \ge 0$

$$\begin{aligned} \Vert f^\varDelta ( c, u)\Vert _{{\mathrm {L}}} \le C_0(1+\Vert c\Vert _{{\mathrm {L}}}+\Vert u\Vert _{{\mathrm {L}}})\quad \text {for all }c,u\in {\mathrm {L}}_T \end{aligned}$$

(A.1)

where $C_0\triangleq h(T) + L + \Vert f(0,0)\Vert _{{\mathrm {L}}}$. In particular, we have

$$\begin{aligned} \Vert f^n(t, Y_t) - f_c(t, Y_t)\Vert _{\mathrm {L}}\le C_0 (2+ \Vert c_t\Vert _{{\mathrm {L}}} + \Vert c^n_t\Vert _{{\mathrm {L}}} +2 \Vert Y\Vert _{{\mathrm {S}}}). \end{aligned}$$

Hence Vitali’s theorem yields

$$\begin{aligned} {\textstyle \sum _{k=0}^{N_n -1}} {\textstyle \int _{t_k^n}^{t_{k+1}^n}} \Vert f_c(s, Y_s) - f^n(t_k^n, Y_s) \Vert _{{\mathrm {L}}}\, {\mathrm {d}}s \rightarrow 0 \end{aligned}$$

for all $Y \in {\mathrm {S}}$, establishing (3.8). $\square$

Lemma A.8

(A priori estimate) For each $n\in {\mathbb {N}}$, let $U^n$ be the discrete-time recursive utility process defined in (5.1). There is a constant $C_1>0$ such that for all but finitely many $n\in {\mathbb {N}}$, we have

$$\begin{aligned} \max _{k=0,\ldots ,N_n}\Vert U^n_k\Vert _{{\mathrm {L}}} \le C_1 \bigl (1 + \Vert \xi \Vert _{{\mathrm {L}}} + \Vert c\Vert _{{\mathrm {P}}} \bigr ). \end{aligned}$$

Proof

For every $k = 0,\ldots ,N_n-1$ we have

$$\begin{aligned} U^n_k = {\mathcal {E}}_{t_k^n} \left[ \varDelta _k^n f^{\varDelta _k^n} (c_{t_k^n }^n, {\mathcal {E}}_{t_k^n} [U^n_{k+1}]) + U_{k+1}^n \right] \end{aligned}$$

(A.2)

by (SI) and therefore

$$\begin{aligned} \Vert U_k^n\Vert _{{\mathrm {L}}} \le (1+C_0 \varDelta ^n_{k})\Vert U_{k+1}^n\Vert _{{\mathrm {L}}} + C_0 \varDelta ^n_{k}\bigl ( 1+ \Vert c_{t_k^n}^n\Vert _{{\mathrm {L}}} \bigr ) \end{aligned}$$

by the projection property (2.3), the triangle inequality and (A.1). Iterating this and using $1+x\le e^x$ we obtain

$$\begin{aligned} \Vert U_k^n\Vert _{{\mathrm {L}}}&\le e^{C_0T} \Vert \xi \Vert _{{\mathrm {L}}} + C_0 e^{C_0T}{\textstyle \sum _{l=0}^{N_n-1}} \varDelta ^n_{l} (1 + \Vert c_{t_l^n}^n\Vert _{{\mathrm {L}}}). \end{aligned}$$

Since $\sum _{l=0}^{N_n-1} \varDelta ^n_{l} \Vert c_{t_l^n}^n\Vert _{{\mathrm {L}}} \rightarrow \Vert c\Vert _{{\mathrm {P}}}$ the proof is complete. $\square$

Regularity for classical G-expectations

Classical, non-random G-expectations as developed in Peng [35, 36] and Denis et al. [11] are subsumed in the random G-expectation framework of Sect. 4 as the special case where $\mu =0$ and $\sigma (r,X,\nu _r)=\nu _r$. Hence classical G-expectations are regular by Theorem 4.4. This appendix provides an alternative, simpler approach to establishing regularity of G-expectations.

We denote by $B_s(\omega ) = \omega (s)$ the canonical process on $\varOmega _t$. We write $C_\text {lLip}({\mathbb {R}}^{d\times n})$ for the space of all functions $\varphi : {\mathbb {R}}^{d\times n} \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} |\varphi (x) - \varphi (y)| \le C (1+ |x|^k+ | y|^k) |x - y|\quad \text {for all } x,y \in {\mathbb {R}}^{d\times n} \end{aligned}$$

for some $k,C>0$. In the following, we outline the construction of G-expectations and the associated appropriate domain ${\mathscr {L}}= \{ ({\mathrm {L}}_t, \Vert {\cdot }\Vert _{{\mathrm {L}}})\}_{t \in [0,T]}$. First, we define the G-expectation for members of the increasing family $\{{\mathcal {H}}_t\}_{t \in [0,T]}$ of cylindrical random variables

$$\begin{aligned} {\mathcal {H}}_t \triangleq \bigl \{ \varphi (B_{t_1}, \ldots , B_{t_n}) \, :\, n \in {\mathbb {N}},\ t_1, \ldots t_n \in [0,t],\ \varphi \in C_\text {lLip}({\mathbb {R}}^{d\times n}) \bigr \}. \end{aligned}$$

For this, fix a function $G: {\mathbb {S}}_d \rightarrow {\mathbb {R}}$ of the form

$$\begin{aligned} G(A) = \tfrac{1}{2}\sup _{\gamma \in \varGamma } {{\mathrm{{\mathrm {tr}}}}}[\gamma \gamma ^\top A],\quad A\in {\mathbb {S}}^d, \end{aligned}$$

where $\varGamma \subset {\mathbb {R}}^{d \times d}$ is bounded, non-empty and closed, and ${\mathbb {S}}_d$ denotes the set of symmetric $d\times d$ matrices. The sublinear expectation $\{{\mathcal {E}}^G_t\}_{t\in [0,T]}$ (the G-expectation) is defined on $\{{\mathcal {H}}_t\}_{t\in [0,T]}$ by the condition that, for each $\varphi \in C_\text {lLip}({\mathbb {R}}^{d\times n})$ and all $0 \le t_1 \le \cdots \le t_n \le T$, we have

$$\begin{aligned} {\mathcal {E}}^G_{t_{n-1}} \bigl [\varphi (B_{t_1}, B_{t_2}, \ldots , B_{t_n}- B_{t_{n-1}}) \bigr ] = \psi (B_{t_1}, \ldots , B_{t_{n-1}}), \end{aligned}$$

(B.1)

where $\psi (x_1, \ldots , x_{n-1}) \triangleq {\mathcal {E}}^G[ \varphi (x_1, x_2, \ldots , \sqrt{t_n -t_{n-1}} B_1)]$. Here $B_1$ is G-normal, i.e. for each $\varphi \in C_\text {lLip}({\mathbb {R}}^{d})$ one defines ${\mathcal {E}}^G[\varphi (x+ \sqrt{t} B_1)]\triangleq u(t,x)$ where u is the unique continuous viscosity solution of the G-heat equation

$$\begin{aligned} u_t - G(u'') = 0, \quad u(0,{\cdot }) = \varphi . \end{aligned}$$

Next, for each $t \in [0,T]$ let ${\mathrm {L}}_t$ denote the closure of ${\mathcal {H}}_t$ with respect to the norm $\Vert {\cdot }\Vert _{{\mathrm {L}}} = {\mathcal {E}}^G[|{\cdot }|]$. Denis et al. [11] show that⁸ there exists a weakly compact family ${\mathcal {Q}}$ of $\sigma$-additive probability measures on $(\varOmega ,{\mathcal {F}})$ such that

$$\begin{aligned} {\mathcal {E}}^G[X] = \sup _{{\mathbb {Q}}\in {\mathcal {Q}}} {{\mathrm{{\mathrm {E}}}}}^{\mathbb {Q}}[X] \quad \text {for all } X \in {\mathcal {H}}_T \end{aligned}$$

and identify the Banach space ${\mathrm {L}}_T$ as a subspace of the space of measurable functions $\varOmega \rightarrow {\mathbb {R}}$, modulo equality ${\mathcal {Q}}$-quasi surely. Thus ${\mathscr {L}}= \{ ({\mathrm {L}}_t, \Vert {\cdot }\Vert _{{\mathrm {L}}})\ : \, t \in [0,T]\}$ is an appropriate domain for $\{{\mathcal {E}}^G_t\}_{t \in [0,T]}$.

Given the above construction of the appropriate domain ${\mathscr {L}}$, we can now show that G-expectations are regular in the sense of Definition 3.2.

Lemma B.1

Let $\xi = \varphi (B_{t_1}, \ldots B_{t_n}) \in {\mathcal {H}}_T$ where $\varphi$ is bounded and Lipschitz continuous. Then there exists a constant $K>0$ such that

$$\begin{aligned} \bigl \Vert {\mathcal {E}}^G_s[\xi ] - {\mathcal {E}}^G_t[\xi ]\bigr \Vert _{{\mathrm {L}}} \le K |s-t|^{\frac{1}{2}} \quad \text {for all }s,t\in [0,T]. \end{aligned}$$

Proof

Let $L>0$ denote the Lipschitz constant of $\varphi$ and let $0\le t < s \le T$. Without loss of generality⁹ we may assume that $t_i = s < t =t_j$ for $1 \le i < j \le n$. Iterating (B.1) we see that ${\mathcal {E}}^G_{t_k} [\xi ] = \psi ^k (B_{t_1}, \ldots , B_{t_k})$, where $\psi ^{n} \triangleq \varphi$,

$$\begin{aligned} \psi ^k(x_1, \ldots , x_k) \triangleq {\mathcal {E}}^G\bigl [\psi ^{k+1} (x_1, \ldots , x_k, \sqrt{t_{k+1}- t_k} B_1 + x_k)\bigr ], \quad k= n-1, \ldots , 1, \end{aligned}$$

and each $\psi ^k$ has the same Lipschitz constant L as $\varphi$. Thus

$$\begin{aligned} \bigl | {\mathcal {E}}^G_{t_{k}}[\xi ] - {\mathcal {E}}^G_{t_{k-1}}[\xi ] \bigr | \le | B_{t_{k}} - B_{t_{k-1}} | + \sqrt{t_k - t_{k-1}} {\mathcal {E}}^G[|B_1|]. \end{aligned}$$

Hence writing $j = i+k$ we have

$$\begin{aligned} \bigl |{\mathcal {E}}^G_s[\xi ] - {\mathcal {E}}^G_t[\xi ]\bigr | \le L{\textstyle \sum _{\ell =1}^{k}} \bigl ( | B_{t_{i+ \ell }} - B_{t_{i+\ell -1}}| + \sqrt{t_{i+\ell } - t_{i+\ell -1}} {\mathcal {E}}^G[|B_1|] \bigr ). \end{aligned}$$

Note that $t_{i+\ell } - t_{i+ \ell -1} \le |s-t|$ so that

$$\begin{aligned} \bigl |{\mathcal {E}}^G_s[\xi ] - {\mathcal {E}}^G_t[\xi ]\bigr | \le L {\textstyle \sum _{\ell =1}^{k}} | B_{t_{i+ \ell }} - B_{t_{i+\ell -1}}| + k L \sqrt{|s-t|} {\mathcal {E}}^G[|B_1|]. \end{aligned}$$

By (B.1) we have ${\mathcal {E}}^G[|B_{u+ \varDelta } - B_u|] = \varDelta ^\frac{1}{2} {\mathcal {E}}^G[|B_1|]$. Since $|B_1| \in {\mathcal {H}}_T$ we have $c \triangleq {\mathcal {E}}^G[|B_1|]<\infty$ and therefore

$$\begin{aligned} {\mathcal {E}}^G\bigl [|{\mathcal {E}}^G_s[\xi ] - {\mathcal {E}}^G_t[\xi ]| \bigr ] \le L(k+1) |s-t|^\frac{1}{2} c, \end{aligned}$$

and the proof is complete. $\square$

Corollary B.2

(Regularity of classical $\varvec{G}$-expectations) The classical G-expectation is a regular nonlinear expectation.

Proof

The collection $M \subset {\mathrm {L}}_T$ of all $\xi = \varphi (B_{t_1}, \ldots B_{t_n}) \in {\mathcal {H}}_T$ where $\varphi$ is bounded and Lipschitz continuous is dense in ${\mathrm {L}}_T$ by weak compactness of ${\mathcal {Q}}$. For every $\xi \in M$ the map $t \mapsto {\mathcal {E}}^G_t[\xi ]$ is uniformly continuous by Lemma B.1. Hence Lemma A.3 applies. $\square$

Remark B.3

If we are given a continuous and positively homogeneous function $\tilde{G}:{\mathbb {S}}_d\rightarrow {\mathbb {R}}$ that is dominated by G in the sense that

$$\begin{aligned} \tilde{G}(A) - \tilde{G}(B) \le G(A-B),\qquad A,B\in {\mathbb {S}}_d, \end{aligned}$$

a similar construction as above leads to a fully nonlinear $\tilde{G}$-expectation $\{{\mathcal {E}}^{\tilde{G}}_t\}_{t\in [0,T]}$ on ${\mathscr {L}}$ defined in terms of $\tilde{G}$. By construction, $\{{\mathcal {E}}^{\tilde{G}}_t\}_{t\in [0,T]}$ is dominated by $\{{\mathcal {E}}^{G}_t\}_{t\in [0,T]}$ in the sense of Definition 2.4. Nonlinear expectations of this form have been introduced by Peng [37] and Guo et al. [19]. One can verify as in Lemma B.1 and Corollary B.2 that $\{{\mathcal {E}}^{\tilde{G}}_t\}_{t\in [0,T]}$ is regular, i.e. our results also apply to $\tilde{G}$-expectations.