Stochastic Decision Problems with Multiple Risk-Averse Agents

Abstract

We consider a stochastic decision problem, with dynamic risk measures, in which multiple risk-averse agents make their decisions to minimize their individual accumulated risk-costs over a finite-time horizon. Specifically, we introduce multi-structure dynamic risk measures induced from conditional g-expectations, where the latter are associated with the generator functionals of certain BSDEs that implicitly take into account the risk-cost functionals of the risk-averse agents. Here, we also assume that the solutions for such BSDEs almost surely satisfy a stochastic viability property w.r.t. a certain given closed convex set. Using a result similar to that of the Arrow–Barankin–Blackwell theorem, we establish the existence of consistent optimal decisions for the risk-averse agents, when the set of all Pareto optimal solutions, in the sense of viscosity solutions, for the associated dynamic programming equations is dense in the given closed convex set. Finally, we comment on the characteristics of acceptable risks w.r.t. some uncertain future outcomes or costs, where results from the dynamic risk analysis are part of the information used in the risk-averse decision criteria.

A preliminary version of this paper was presented at the Modeling and Optimization: Theory and Applications Conference, August 17–19, 2016, Bethlehem, PA, USA.

Appendix: Proofs

In this section, we give the proofs for the main results.

Proof of Proposition 1

Notice that m and σ are bounded and Lipschitz continuous w.r.t. $(t,x) \in [0,T] \times \mathbb{R}^{d}$ and uniformly for $u \in \prod \nolimits _{i=1}^{n}U^{i}$. Then, for any $(t,x) \in [0,T] \times \mathbb{R}^{d}$ and u _⋅ ^¬j, for j = 1, 2, …, n, are progressively measurable processes, there always exists a unique path-wise solution $X_{\cdot }^{t,x;u_{\cdot }^{\neg j} } \in \mathcal{S}^{2}\big(t,T; \mathbb{R}^{d}\big)$ for the forward SDE in (9). On the other hand, consider the following BSDEs,

$$\displaystyle\begin{array}{rcl} -d\hat{Y }_{s}^{j,t,x;u_{\cdot }^{\neg j} } = g_{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} },\hat{Y }_{s}^{j,t,x;u_{\cdot }^{\neg j} },Z_{s}^{j,t,x;u_{\cdot }^{\neg j} }\big)ds - Z_{s}^{j,t,x;u_{\cdot }^{\neg j} }dB_{s},& & \\ j = 1,2,\ldots,n,& &{}\end{array}$$

(43)

where

$$\displaystyle{ \hat{Y }_{T}^{j;t,x;u_{\cdot }^{\neg j} } =\int _{ t}^{T}c_{ j}\big(\tau,X_{\tau }^{t,x;u_{\cdot }^{\neg j} },u_{\tau }^{j}\big)d\tau +\varPsi _{ j}(X_{T}^{t,x;u_{\cdot }^{\neg j} }). }$$

From Lemma 2, Eq. (43) admits unique solutions $\big(\bar{Y }_{\cdot }^{j,t,x;u_{\cdot }^{\neg j} },Z_{\cdot }^{j,t,x;u_{\cdot }^{\neg j} }\big)$, for j = 1, 2, …, n, in $\mathcal{S}^{2}\big(t,T; \mathbb{R}\big) \times \mathcal{H}^{2}\big(t,T; \mathbb{R}^{d}\big)$. Furthermore, if we introduce the following

$$\displaystyle{ Y _{s}^{j,t,x;u_{\cdot }^{\neg j} } =\hat{ Y }_{s}^{j,t,x;u_{\cdot }^{\neg j} } -\int _{t}^{s}c_{ j}\big(\tau,X_{\tau }^{t,x;u_{\cdot }^{\neg j} },u_{\tau }^{j}\big)d\tau,\quad s \in [t,T]. }$$

Then, the family of forward of the BSDEs in (14) holds with $\big(Y _{\cdot }^{j,t,x;u_{\cdot }^{\neg j} },Z_{\cdot }^{j,t,x;u_{\cdot }^{\neg j} }\big)$, for j = 1, 2, …, n. Moreover, we also observe that $Y _{t}^{j,t,x;u_{\cdot }^{\neg j} }$, for j = 1, 2, …, n, are deterministic. This completes the proof of Proposition 1. □

Proof of Proposition 2

For any r ∈ [t, T], with t ∈ [0, T], we consider the following probability space $\big(\varOmega,\mathcal{F}, \mathbb{P}\big(\cdot \vert \mathcal{F}_{r}^{t}\big),\{\mathcal{F}^{t}\}\big)$ and notice that η is deterministic under this probability space. Then, for any s ≥ r, there exist progressively measurable process ψ such that

$$\displaystyle\begin{array}{rcl} u_{s}^{j}(\varOmega )& =& \psi (\varOmega,B_{ \cdot \wedge s}(\varOmega )), \\ & =& \psi (s,\bar{B}_{\cdot \wedge s}(\varOmega ) + B_{r}(\varOmega )),{}\end{array}$$

(44)

where $\bar{B}_{s} = B_{s} - B_{r}$ is a standard d-dimensional Brownian motion. Note that n tuple u _⋅ ^¬j, for j = 1, 2, …, n, are $\mathcal{F}_{r}^{t}$-adapted processes, then we have the following restriction w.r.t. Σ _[t, T]

$$\displaystyle{ \big(\varOmega,\mathcal{F},\{\mathcal{F}^{t}\}, \mathbb{P}\big(\cdot \vert \mathcal{F}_{ r}^{t}\big)(\omega '),B_{ \cdot },u_{\cdot }^{\neg j}\big) \in \varSigma _{ [t,T]},\quad j = 1,2,\ldots,n, }$$

(45)

where ω′ ∈ Ω′ such that $\varOmega ' \in \mathcal{F}$, with $\mathbb{P}(\varOmega ') = 1$. Furthermore, noting Lemma 2, if we work under the probability space $\big(\varOmega ',\mathcal{F}, \mathbb{P}\big(\cdot \vert \mathcal{F}_{r}^{t}\big)\big)$, then the statement in (34) holds $\mathbb{P}$- almost surely. This completes the proof of Proposition 2. □

Proof of Proposition 4

Suppose there exists an n-tuple of optimal risk-averse decisions $(\hat{u}_{\cdot }^{1},\hat{u}_{\cdot }^{2},\ldots,\hat{u}_{\cdot }^{n}) \in \prod \nolimits _{i=1}^{n}\mathcal{U}_{[0,T]}^{i}$ satisfying the statements in Definition 3. Assume that $(t,x) \in [0,T] \times \mathbb{R}^{d}$ is fixed. For any $u_{\cdot }^{j} \in \mathcal{U}_{[t,T]}^{j}$, restricted to Σ _[t, T], for j ∈ {1, 2, …, n}, we consider an $\mathbb{R}^{n}$-valued process $\varphi \big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big)$, with

$$\displaystyle{ u_{\cdot }^{\neg j} = (\begin{array}{ccccccc} \hat{u}_{ \cdot }^{1},&\ldots,&\hat{u}_{ \cdot }^{j-1},&u_{ \cdot }^{j},&\hat{u}_{ \cdot }^{j+1},&\cdots \,,&\hat{u}_{ \cdot }^{n} \end{array} ) \in \prod \nolimits _{i=1}^{n}\mathcal{U}_{ [t,T]}^{i}, }$$

which is restricted to Σ _[t, T]. Then, using Itô integral formula, we can evaluate the difference between $\varphi _{j}\big(T,X_{T}^{t,x;u_{\cdot }^{\neg j} }\big)$ and $\varphi _{j}\big(t,x\big)$, for j = 1, 2, …, n, as follows:⁴

$$\displaystyle\begin{array}{rcl} & & \varphi _{j}\big(T,X_{T}^{t,x;u_{\cdot }^{\neg j} }\big) -\varphi _{j}\big(t,x\big) =\int _{ t}^{T}\Big[ \dfrac{\partial } {\partial t}\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) + \mathcal{L}_{t}^{u^{\neg j} }\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big)\Big]ds \\ & & \phantom{\varphi _{j}\big(T,X_{T}^{t,x;u_{\cdot }^{\neg j} }\big) -}\quad +\int _{ t}^{T}D_{ x}\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) \cdot \sigma (s,X_{s}^{t,x;u_{\cdot }^{\neg j} },u_{s}^{\neg j})dB_{ s}. {}\end{array}$$

(46)

Using (20), we further obtain the following

$$\displaystyle\begin{array}{rcl} & & \dfrac{\partial } {\partial t}\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) + \mathcal{L}_{t}^{u^{\neg j} }\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) \\ & & \quad + g_{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} },\varphi \big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big),D_{x}\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) \cdot \sigma (s,X_{s}^{t,x;u_{\cdot }^{\neg j} },u_{s}^{\neg j})\big) \geq 0, \\ & & \quad \quad \quad \quad \quad j = 1,2,\ldots,n. {}\end{array}$$

(47)

Furthermore, if we combine (46) and (47), then we obtain

$$\displaystyle\begin{array}{rcl} & & \varphi _{j}\big(t,x\big) \leq \varPsi _{j}\big(X_{T}^{t,x;u_{\cdot }^{\neg j} }\big) \\ & & \quad \quad +\int _{ t}^{T}g_{ j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} },\varphi (s,X_{s}^{t,x;u_{\cdot }^{\neg j} }),D_{x}\varphi _{j}(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }) \cdot \sigma (s,X_{s}^{t,x;u_{\cdot }^{\neg j} },u_{s}^{\neg j})\big)ds \\ & & \quad \quad \quad -\int _{t}^{T}D_{ x}\varphi _{j}(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }) \cdot \sigma (s,X_{s}^{t,x;u_{\cdot }^{\neg j} },u_{s}^{\neg j})dB_{ s}. {}\end{array}$$

(48)

Define $Z_{s}^{j,t,x;u_{\cdot }^{\neg j} } = D_{x}\varphi _{j}(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }) \cdot \sigma (s,X_{s}^{t,x;u_{\cdot }^{\neg j} },(\hat{u}_{s},v_{s}))$, for s ∈ [t, T] and for j = 1, 2, …, n, then $\varphi _{j}\big(t,x\big) \leq Y _{t}^{j,t,x;u_{\cdot }^{\neg j} }$ follows, where $(Y _{\cdot }^{j,t,x;u_{\cdot }^{\neg j} },Z_{\cdot }^{j,t,x;u_{\cdot }^{\neg j} })$ is a solution to BSDE in (14) (cf. Eq. (13)). As a result of this, we have

$$\displaystyle{ \varphi _{j}\big(t,x\big) \leq V _{j}^{u^{j} }\big(t,x\big),\quad j = 1,2,\ldots,n. }$$

Moreover, if there exists at least one $\hat{u}^{j}$ satisfying (40), i.e., if $\hat{u}^{j}$ is a measurable selector of

$$\displaystyle\begin{array}{rcl} & & \mathop{\mathrm{arg\,max}}\limits \Big\{\mathcal{L}_{s}^{u^{\neg j} }\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) {}\\ & & \quad \quad + g_{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} },\varphi \big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big),D_{x}\varphi _{j}\big(s,X_{s}^{t,x;u_{\cdot }^{\neg j} }\big) \cdot \sigma \big (s,X_{s}^{t,x;w},u_{ s}^{\neg j}\big)\big)\Big\}, {}\\ & & \quad \quad \quad \quad \quad \quad j \in \{ 1,2,\ldots,n\}. {}\\ \end{array}$$

Then, for $u_{\cdot }^{j} =\hat{ u}_{\cdot }^{j}$, for j ∈ {1, 2, …, n}, the inequality in (48) becomes an equality, i.e.,

$$\displaystyle\begin{array}{rcl} \varphi _{j}(t,x)& =& V _{j}^{\hat{u}^{j} }\big(t,x\big) {}\\ & & \triangleq Y _{T}^{j,t,x;\hat{u}_{\cdot }^{\neg j} },\,\,j \in \{ 1,2,\ldots,n\},\,\,\text{(cf. Eqs. (11) and (13))}. {}\\ \end{array}$$

Note that the corresponding path-wise solution $X_{s}^{t,x;\hat{u}_{\cdot }}$ is progressively measurable, since $\hat{u}_{\cdot }\in \prod \nolimits _{i=1}^{n}\mathcal{U}_{[0,T]}^{i}$ is also restricted to Σ _[t, T].

On the other hand, noting the relations in (11) and (13), for any $(t,x) \in [0,T] \times \mathbb{R}^{n}$, with restriction to Σ _[t, T], define the following utility function over the closed convex set K

$$\displaystyle\begin{array}{rcl} K \ni \big (\begin{array}{cccc} \rho _{t,T}^{g_{1}}\big[\xi _{ t,T}^{1}\big(u^{\neg 1}\big)\big],&\rho _{ t,T}^{g_{2}}\big[\xi _{ t,T}^{2}\big(u^{\neg 2}\big)\big],&\cdots \,,&\rho _{ t,T}^{g_{n}}\big[\xi _{ t,T}^{n}\big(u^{\neg n}\big)\big] \end{array} \big)& & {}\\ \quad \quad \rightarrow \mathcal{J} (u) =\sum \nolimits _{ i=1}^{n}\pi ^{i}\rho _{ t,T}^{g_{i} }\big[\xi _{t,T}^{i}\big(u^{\neg i}\big)\big],& & {}\\ \end{array}$$

where π ⁱ > 0, for i = 1, 2, …, n.

Note that the utility function $\mathcal{J} (u) =\sum \nolimits _{ i}^{n}\pi ^{i}\rho _{t,T}^{g_{i}}\big[\xi _{t,T}^{i}\big(u^{\neg i}\big)\big]$ satisfies the following property

$$\displaystyle{ \sum \nolimits _{i=1}^{n}\pi ^{i}\rho _{ t,T}^{g_{i} }\big[\xi _{t,T}^{i}\big(\tilde{u}^{\neg i}\big)\big] <\sum \nolimits _{ i=1}^{n}\pi ^{i}\rho _{ t,T}^{g_{i} }\big[\xi _{t,T}^{i}\big(u^{\neg i}\big)\big], }$$

i.e., $\mathcal{J} (\tilde{u}) <\mathcal{J} (u)$, whenever

$$\displaystyle\begin{array}{rcl} & & \big(\begin{array}{cccc} \rho _{t,T}^{g_{1}}\big[\xi _{ t,T}^{1}\big(\tilde{u}^{\neg 1}\big)\big],&\rho _{ t,T}^{g_{2}}\big[\xi _{ t,T}^{2}\big(\tilde{u}^{\neg 2}\big)\big],&\cdots \,,&\rho _{ t,T}^{g_{n}}\big[\xi _{ t,T}^{n}\big(\tilde{u}^{\neg n}\big)\big] \end{array} \big) {}\\ & & \quad \quad \quad \prec \big (\begin{array}{cccc} \rho _{t,T}^{g_{1}}\big[\xi _{ t,T}^{1}\big(u^{\neg 1}\big)\big],&\rho _{ t,T}^{g_{2}}\big[\xi _{ t,T}^{2}\big(u^{\neg 2}\big)\big],&\cdots \,,&\rho _{ t,T}^{g_{n}}\big[\xi _{ t,T}^{n}\big(u^{\neg n}\big)\big] \end{array} \big),{}\\ \end{array}$$

w.r.t. the class of admissible control processes $\prod \nolimits _{i=1}^{n}\mathcal{U}_{[t,T]}^{i}$ (cf. Eqs. (36)– (39)). Then, from the Arrow–Barankin–Blackwell theorem (e.g., see [3]), for all t ∈ [0, T], one can see that the set in

$$\displaystyle\begin{array}{rcl} & & \Bigg\{\big(\begin{array}{cccc} \rho _{t,T}^{g_{1}}\big[\xi _{ t,T}^{1}\big(u^{\neg 1}\big)\big],&\rho _{ t,T}^{g_{2}}\big[\xi _{ t,T}^{2}\big(u^{\neg 2}\big)\big],&\cdots \,,&\rho _{ t,T}^{g_{n}}\big[\xi _{ t,T}^{n}\big(u^{\neg n}\big)\big] \end{array} \big) \in K\,\Big\vert \, {}\\ & & \,\,\exists \,\pi ^{i}> 0,\,\,i = 1,2,\ldots,n,\,\,\min \sum \nolimits _{ i=1}^{n}\pi ^{i}\rho _{ t,T}^{g_{i} }\big[\xi _{t,T}^{i}\big(u^{\neg i}\big)\big] =\sum \nolimits _{ i=1}^{n}\pi ^{i}\rho _{ t,T}^{g_{i} }\big[\xi _{t,T}^{i}\big(\hat{u}^{\neg i}\big)\big]\Bigg\} {}\\ \end{array}$$

is dense in the set of all Pareto equilibria. This further implies that, for any choice of π ⁱ > 0, i = 1, 2, …, n, the minimizer $\mathcal{J} (\hat{u}) =\sum \nolimits _{ i}^{n}\pi ^{i}\rho _{0,T}^{g_{i}}\big[\xi _{0,T}^{i}\big(\hat{u}^{\neg i}\big)\big]$ over K satisfies the Pareto equilibrium condition w.r.t. some n-tuple of optimal risk-averse decisions $(\hat{u}_{\cdot }^{1},\hat{u}_{\cdot }^{2},\ldots,\hat{u}_{\cdot }^{n}) \in \prod \nolimits _{i=1}^{n}\mathcal{U}_{[0,T]}^{i}$. This completes the proof of Proposition 4. □