0 引言

财富优化与市场均衡是金融数学研究的核心问题之一，文献[1]揭示了如何通过投资组合的有效边界来选择最优组合和如何通过分散投资来降低风险，开创了现代投资分析和金融理论的先河.假设市场中有m个行为人，其效用函数U_k(k=1, …, m)为严格递增，其资源禀赋过程记为{ε_k(t); 0≤t≤T}, (k=1, …, m)，则总资源禀赋ε(t)=$\sum\limits_{k = 1}^m {} $ε_k(t), 0≤t≤T，定义生存消费为c_k=inf{c∈R; U_k(c)>-∞}, (k=1, …, m)，则总生存消费为c=$\sum\limits_{k = 1}^m {} $c_k.本文讨论$\mathop {\sup }\limits_{\left( {{c_k},{\pi _k}} \right) \in {A_k}} E\int_0^t {{{\rm{e}}^{ - \int_0^t {\beta \left( u \right){\rm{d}}u} }}} $U_k(c_k(t))dt的财富优化问题, 其中: A_k是可容许投资组合的集合; c_k(t)是消费过程; U_k(x)是效用函数; β(t)为贴现率.

许多学者对类似的优化问题进行了研究.文献[2]讨论了在完备市场的资产连续.文献[3]讨论了资产服从poisson跳扩散时的财富最大化问题.文献[4-8]讨论了市场连续或跳扩散时的风险最小化问题.文献[9-12]讨论了风险资产的方差满足Heston模型下的优化问题，并将跳过程进行了推广.本文研究在完备市场中，股票支付连续红利的条件下，跳过程为计数过程的跳扩散模型时的财富优化问题^[13]，讨论所建金融市场下的均衡问题.

1 金融市场

设市场m=(r(t), ρ_i(t), μ_i(t), σ_i(t), φ_i(t), S_i(t), i=1, 2) 是标准且完备的，无风险资产B(t)和风险资产S_i(t)满足微分方程(F_t, 0≤t≤T)，{W(t), 0≤t≤T}，且

$ {\rm{d}}{\mathit{S}_i}\left( t \right) = {S_i}\left( {{t^ - }} \right)\left( {\left( {{\mathit{\mu }_i}\left( t \right) - {\mathit{\rho }_i}\left( t \right)} \right){\rm{d}}\mathit{t + }{\mathit{\sigma }_i}\left( t \right){\rm{d}}\mathit{W}\left( t \right) + {\varphi _i}\left( t \right){\rm{d}}\mathit{M}\left( t \right),i = 1,2,} \right. $

(1)

其中：r(t)为无风险利率；ρ_i(t)为股票的红利率；μ_i(t)为预期收益率；σ_i(t)为股票的波动率；φ_i(t)为股票价格跳跃高度. {W(t), 0≤t≤T}是定义在完备的概率空间(Ω, F, P)上的标准Brown运动. M(t)=N(t)-$\int_0^t {} $λ(s)ds, 0≤t≤T ({N(t), 0≤t≤T}；是与W(t)相互独立的非爆炸性的计数过程，其强度为λ(t).设总资源禀赋满足ε(t)=ε(0)+$\int_0^t {} $μ_ε(u)du+$\int_0^t {} $σ_ε(u)dW(u)+$\int_0^t {} $φ_ε(u)dM(u), 0≤t≤T.

假设  函数λ(t)，r(t)，μ_i(t)，σ_i(t)，φ_i(t)，ρ_i(t)都是可测、有界的，且满足：

1) λ(t)>0，r(t)≥0，σ_i(t)>0，ρ_i(t)≥0，φ_i(t)>-1，φ_i(t)≠0 (i=1, 2)；

2) 存在c₁∈(0, +∞)，使得|σ₁(t)φ₂(t)-σ₂(t)φ₁(t)|≥c₁, t∈[0, T];

3) 存在c₂∈(0, +∞)，使得

$ \frac{{\left( {{\mathit{\mu }_2}\left( t \right) - {\mathit{\rho }_2}\left( t \right) - \lambda \left( t \right){\mathit{\varphi }_2}\left( t \right) - r\left( t \right)} \right){\mathit{\sigma }_1}\left( t \right) - \left( {{\mathit{\mu }_1}\left( t \right) - {\mathit{\rho }_1}\left( t \right) - \lambda \left( t \right){\mathit{\varphi }_1}\left( t \right) - r\left( t \right)} \right){\mathit{\sigma }_2}\left( t \right)}}{{\mathit{\lambda }\left( t \right)\left( {{\mathit{\sigma }_2}\left( t \right){\mathit{\varphi }_1}\left( t \right) - {\mathit{\sigma }_1}\left( t \right){\mathit{\varphi }_2}\left( t \right)} \right)}} \ge {c_2},t \in \left[ {0,T} \right]. $

由假设知，存在θ₁(t), θ₂(t)满足μ_i(t)-λ(t)φ_i(t)-r(t)-σ_i(t)θ₁(t)+λ(t)φ_i(t)θ₂(t)=0, i=1, 2.得到命题1和命题2.

命题1  设L(t)=exp{-$\int_0^t {} $θ₁(s)dW(s)-$\frac{1}{2}$$\int_0^t {} $θ₁²(s)ds}· exp[$\int_0^t {} $log θ₂(s)dN(s)+$\int_0^t {} $λ(s)(1-θ₂(s))ds]，若令$\frac{{{\rm{d}}{P^*}}}{{{\rm{d}}P}}$=L(T)，则W^*(t)=W(t)+$\int_0^t {} $θ₁(s)ds为P^*下的标准Brown运动，N(t)为P^*下强度为λ(t)θ₂(t)的计数过程，且M^*(t)=N(t)-$\int_0^t {} $λ(s)θ₂(s)ds(0≤t≤T)为P^*鞅.

命题2  令${{\tilde S}_i}$(t)=S_i(t)/B(t)(i=1, 2)，则(1) 式等价于d${{\tilde S}_i}$(t)=${{\tilde S}_i}$(t^-)(σ_i(t)dW^*(t)+φ_i(t)dM^*(t))(i=1, 2).且${{\tilde S}_i}$(t)(0≤t≤T, i=1, 2) 为P^*鞅，即P^*为风险中性鞅测度^[14].

假设投资者k消费过程为c_k(t)，投资者建立的策略是一个自融资策略^[15]，则其财富过程X_k^{c_k, π_k}(t)为X_k^{c_k, π_k}(t)=m_k0B(t)+m_k1S₁(t)+m_k2S₂(t)+$\int_0^t {} $(ε_k(u)-c_k(u))du.若令m_k1S₁(t)=π_k1(t), m_k2S₂(t)=π_k2(t)，则X_k^{c_k, π_k}(t)满足微分方程dX_k^{c_k, π_k}(t)=r(t)X_k^{c_k, π_k}(t)dt+$\sum\limits_{i = 1}^2 {} $[π_ki(t)((σ_i(t)dW^*(t)+φ_i(t)dM^*(t))]+ε_k(t)-c_k(t).从而X_k^{c_k, π_k}(t)的贴现价值过程满足

$ \frac{{X_k^{{c_k},{\pi _k}}\left( t \right)}}{{B\left( t \right)}} = \int_0^t {\frac{{{\varepsilon _k}\left( u \right) - {c_k}\left( u \right)}}{{B\left( t \right)}}} {\rm{d}}\mathit{u + }\int_0^t {\sum\limits_{i = 1}^2 {{{\mathit{\tilde \pi }}_{ki}}} } \left( u \right){\mathit{\sigma }_i}\left( u \right){\rm{d}}{\mathit{W}^*}\left( u \right) + \int_0^t {\sum\limits_{i = 1}^2 {{{\mathit{\tilde \pi }}_{ki}}\left( u \right){\mathit{\varphi }_i}\left( u \right)} } {\rm{d}}{\mathit{M}^*}\left( u \right). $

(2)

2 主要结论

定理1  存在(${{\hat c}_k},{{\hat \pi }_k}$)∈A_k，是优化问题$\mathop {\sup }\limits_{\left( {{c_k},{\pi _k}} \right) \in {A_k}} E\int_0^t {{{\rm{e}}^{ - \int_0^t {\beta \left( u \right){\rm{d}}u} }}} $U_k(c_k(t))dt的解.投资者k的最优财富过程为

$ X_k^{{{\hat c}_k},{{\hat \pi }_k}}\left( t \right) = \frac{{B\left( t \right)}}{{L\left( t \right)}}\mathit{E}\left[ {\int_t^T {\frac{{L\left( u \right)}}{{B\left( u \right)}}\left( {{{\hat c}_k}\left( u \right) - {\varepsilon _k}\left( u \right)} \right){\rm{d}}\mathit{u}\left| {{F_t}} \right.} } \right]. $

(3)

证明  根据效用函数的定义，U_k(·)的导函数U_k′(·)的反函数是存在的，用I_k(·)来表示，则

$ \begin{array}{l} E\int_0^t {{e^{ - \int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}} {U_k}\left( {{c_k}\left( t \right)} \right){\rm{d}}\mathit{t = E}\left[ {\int_0^t {{e^{ - \int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}} \left[ {{U_k}\left( {{c_k}\left( t \right)} \right) - y{{\rm{e}}^{\int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}{c_k}\left( t \right)} \right]{\rm{d}}t} \right] + \\ E\left[ {\int_0^t {y\frac{{L\left( t \right)}}{{B\left( t \right)}}{c_k}\left( t \right){\rm{d}}t} } \right] \le \\ \sup \left\{ {E\left[ {\int_0^t {{e^{ - \int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}} \left[ {{U_k}\left( {{c_k}\left( t \right)} \right) - y{{\rm{e}}^{\int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}{c_k}\left( t \right)} \right]{\rm{d}}\mathit{t}} \right]} \right\} + E\left[ {\int_0^t {y\frac{{L\left( t \right)}}{{B\left( t \right)}}{c_k}\left( t \right){\rm{d}}t} } \right] = \\ E\left[ {\int_0^t {{e^{ - \int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}} {U_k}\left( {{I_k}\left( {y{{\rm{e}}^{\int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right)} \right){\rm{d}}\mathit{t}} \right],y > 0. \end{array} $

故c_k(t)=I_k$\left( {y{{\rm{e}}^{\int_0^t {\beta \left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right)$.定义函数Y_k(y)=$E\int_0^t {\frac{{L\left( t \right)}}{{B\left( t \right)}}{I_k}} \left( {y{{\rm{e}}^{\int_0^t {\beta \left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right){\rm{d}}t$.

由(2) 式可知$\frac{{X_k^{{c_k},{\pi _k}}\left( t \right)}}{{B\left( t \right)}} - \int_0^t {\frac{{{\varepsilon _k}\left( u \right) - {c_k}\left( u \right)}}{{B\left( t \right)}}} $du是一个P^*上鞅，则有${E^*}\left[ {\int_0^t {\frac{{{c_k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}t} } \right] \le {E^*}\left[ {\int_0^t {\frac{{{\varepsilon _k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}t} } \right]$, 取y_k=Y_k^-1($E\int_0^t {\frac{{L\left( t \right)}}{{B\left( t \right)}}} $ε_k(t)dt)，则最优消费过程为

$ {{\hat c}_k}\left( t \right) = {I_k}\left( {{y_k}{{\rm{e}}^{\int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right). $

(4)

故

$ E\int_0^t {\frac{{L\left( t \right)}}{{B\left( t \right)}}} {{\hat c}_k}\left( t \right){\rm{d}}\mathit{t = E}\int_0^t {\frac{{L\left( t \right)}}{{B\left( t \right)}}} {\varepsilon _k}\left( t \right){\rm{d}}t. $

(5)

由Bayes′s法则，可以得到${E^*}\int_0^t {\frac{{{{\hat c}_k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}t} = {E^*}\int_0^t {\frac{{{\varepsilon _k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}t} $.令K(t)=E[L(T)$\int_0^t {\frac{{{{\hat c}_k}\left( t \right) - {\varepsilon _k}\left( t \right)}}{{B\left( t \right)}}} $dt|F_t].则K(t)是一个P鞅.根据鞅表示定理, 存在可测的过程θ₁^*(t), θ₂^*(t)，使得K(t)=$\int_0^t {} $θ₁^*(s)dW(s)+$\int_0^t {} $θ₂^*(s)dM(s).由Ito′s引理，${\rm{d}}\left[ {\frac{{K\left( t \right)}}{{L\left( t \right)}}} \right] = \frac{{\theta _1^* + K\left( t \right){\theta _1}}}{{L\left( t \right)}}{\rm{d}}{W^*} + \frac{{\theta _2^*{\theta _2} + K\left( t \right)\left( {1 - {\theta _2}} \right)}}{{L\left( t \right){\theta _2}}}{\rm{d}}{M^*}$, 令

$ \left\{ \begin{array}{l} {\mathit{\pi }_{k1}}{\mathit{\sigma }_1} + {\mathit{\pi }_{k2}}{\mathit{\sigma }_2} = \frac{{\theta _1^* + K\left( t \right){\theta _1}}}{{L\left( t \right)}}B\left( t \right),\\ {\mathit{\pi }_{k1}}{\mathit{\varphi }_1} + {\mathit{\pi }_{k2}}{\mathit{\varphi }_2} = \frac{{\theta _2^*{\theta _2} + K\left( t \right)\left( {1 - {\theta _2}} \right)}}{{L\left( t \right){\theta _2}}}B\left( t \right). \end{array} \right. $

(6)

由假设条件知方程组(6) 有唯一解π_k=(π_k1, π_k2), (c_k, π_k)∈A_k，使得

$ \begin{array}{l} \frac{{X_k^{{{\hat c}_k},{{\hat \pi }_k}}\left( t \right)}}{{B\left( t \right)}} - \int_0^t {\frac{{{\varepsilon _k}\left( u \right) - {{\hat c}_k}\left( u \right)}}{{B\left( t \right)}}} {\rm{d}}u = \frac{{K\left( t \right)}}{{L\left( t \right)}} = \frac{1}{{L\left( t \right)}}E\left[ {L\left( T \right)\int_0^t {\frac{{{{\hat c}_k}\left( t \right) - {\varepsilon _k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}\mathit{t}\left| {{F_t}} \right.} } \right] = \\ {E^*}\left[ {\int_0^t {\frac{{{{\hat c}_k}\left( t \right) - {\varepsilon _k}\left( t \right)}}{{B\left( t \right)}}{\rm{d}}\mathit{t}\left| {{F_t}} \right.} } \right] = {E^*}\left[ {\int_0^T {\frac{{{{\hat c}_k}\left( t \right) - {\varepsilon _k}\left( t \right)}}{{B\left( u \right)}}{\rm{d}}\mathit{u}\left| {{F_t}} \right.} } \right] - \int_0^t {\frac{{{\varepsilon _k}\left( u \right) - {{\hat c}_k}\left( u \right)}}{{B\left( t \right)}}} {\rm{d}}u. \end{array} $

由Bayes′s法则，有$\frac{{X_k^{{{\hat c}_k},{{\hat \pi }_k}}\left( t \right)L\left( t \right)}}{{B\left( t \right)}} = L\left( t \right){E^*}\left[ {\int_t^T {\frac{{{{\hat c}_k}\left( u \right) - {\varepsilon _k}\left( u \right)}}{{B\left( u \right)}}{\rm{d}}u\left| {{F_t}} \right.} } \right] = E\left[ {\int_t^T {\frac{{L\left( u \right)\left( {{{\hat c}_k}\left( u \right) - {\varepsilon _k}\left( u \right)} \right)}}{{B\left( u \right)}}{\rm{d}}u\left| {{F_t}} \right.} } \right]$.即(3) 式为最优财富过程.定义两个函数f$\left( {\vec \alpha } \right) \buildrel \Delta \over = \mathop {\max }\limits_{\left\{ {k\left| {{\delta _k} > 0} \right.} \right\}} $(δ_kU_k′(c_k)), ${\vec \alpha }$=(δ₁, δ₂, …, δ_m)，I(y, ${\vec \alpha }$)$ \buildrel \Delta \over = \sum\limits_{k = 1}^m {{I_k}} \left( {\frac{y}{{{\delta _k}}}} \right)$, 0≤y≤∞.易知I(·, ${\vec \alpha }$)存在反函数，用τ(·, ${\vec \alpha }$)来表示.

定理2   标准的完备的市场m是均衡的充分必要条件是

$ L\left( t \right) = {{\rm{e}}^{^{ - \int_0^t {\mathit{\beta }\left( u \right){\rm{d}}u} }}}B\left( t \right)\mathit{\tau }\left( {\varepsilon \left( t \right),\vec \alpha } \right),0 \le t \le T. $

(7)

其中${\vec \alpha }$满足

$ E\int_0^t {{{\rm{e}}^{^{ - \int_\mathit{0}^\mathit{t} {\mathit{\beta }\left( \mathit{u} \right)\mathit{du}} }}}} \mathit{\tau }\left( {\varepsilon \left( t \right),\vec \alpha } \right)\left[ {{I_k}\left( {\frac{1}{{{\mathit{\delta }_k}}}\mathit{\tau }\left( {\mathit{\varepsilon }\left( t \right),\vec \alpha } \right)} \right) - {\varepsilon _k}\left( t \right)} \right]{\rm{d}}\mathit{t = }{\rm{0}}\mathit{,}\left( {k = 1, \cdots ,m} \right). $

(8)

此时，投资者k的最优消费过程可表示为

$ {{\hat c}_k}\left( t \right) = {I_k}\left( {\frac{1}{{{\mathit{\delta }_k}}}\mathit{\tau }\left( {\varepsilon \left( t \right),\vec \alpha } \right)} \right),0 \le t \le T. $

(9)

证明若市场是均衡的，令δ_k=$\frac{1}{{{y_k}}}$，式(4) 和(5) 等价于

$ {{\hat c}_k}\left( t \right) = {I_k}\left( {\frac{1}{{{\mathit{\delta }_k}}}{{\rm{e}}^{\int_\mathit{0}^\mathit{t} {\mathit{\beta }\left( \mathit{u} \right)\mathit{du}} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right), $

(10)

$ E\int_0^t {\frac{{L\left( t \right)}}{{B\left( t \right)}}} \left[ {{I_k}\left( {\frac{1}{{{\mathit{\delta }_k}}}{{\rm{e}}^{\int_\mathit{0}^\mathit{t} {\mathit{\beta }\left( \mathit{u} \right)\mathit{du}} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right) - {\varepsilon _k}\left( t \right)} \right]{\rm{d}}\mathit{t = }{\rm{0}}. $

(11)

则由市场均衡的条件，可得

$ \cdots \cdots \cdots \cdots \varepsilon \left( t \right) = \sum\limits_{k = 1}^m {{{\hat c}_k}\left( t \right)} = \sum\limits_{k = 1}^m {{I_k}} \left( {\frac{1}{{{\mathit{\delta }_k}}}{{\rm{e}}^{\int_\mathit{0}^\mathit{t} {\mathit{\beta }\left( \mathit{u} \right)\mathit{du}} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right) = \\ I\left( {{{\rm{e}}^{\int_\mathit{0}^\mathit{t} {\mathit{\beta }\left( \mathit{u} \right)\mathit{du}} }}\frac{{L\left( t \right)}}{{B\left( t \right)}},\mathit{\vec \alpha }} \right),0 \le t \le T. $

(12)

由于τ(·, ${\vec \alpha }$)表示I(·, ${\vec \alpha }$)的反函数，则(12) 式可以推导出(7) 式.将(7) 式代入式(10) 和(11)，可得式(8) 和(9).

反之，对于标准的完备的市场m, 若满足式(11) 和(12)，则${{{\hat c}_k}}$(t)=I_k($\frac{1}{{{\delta _k}}}$τ(ε(t), ${\vec \alpha }$))=I_k($\left( {\frac{1}{{{\delta _k}}}{{\rm{e}}^{\int_0^t {\beta \left( u \right){\rm{d}}u} }} \cdot \frac{{L\left( t \right)}}{{B\left( t \right)}}} \right)$.从而有$\sum\limits_{k = 1}^m {{{\hat c}_k}} \left( t \right) = \sum\limits_{k = 1}^m {{I_k}} \left( {\frac{1}{{{\delta _k}}}{{\rm{e}}^{\int_0^t {\beta \left( u \right){\rm{d}}u} }}\frac{{L\left( t \right)}}{{B\left( t \right)}}} \right) = \varepsilon \left( t \right)$，则市场m是均衡的.

定理3  定义A={k|k=1, 2, …, m; ε_k(t)=c_k, 0≤t≤T}.设${\vec \alpha }$满足(8) 式，则k∈A当且仅当

$ {{\delta }_{k}}\le \frac{\mathit{\tau }\left( \varepsilon \left( t \right),\vec{\alpha } \right)}{U_{k}^{'}\left( {{{\bar{c}}}_{k}} \right)},0\le t\le T. $

(13)

定义 ${{\vec \alpha }^*}$=(δ^*, δ₂^*, …, δ_m^*)满足

$ \delta _k^ * \buildrel \Delta \over = \left\{ \begin{array}{l} \inf \left( {\mathop {\min }\limits_{0 \le t \le T} \frac{{\tau \left( {\varepsilon \left( t \right),\vec \alpha } \right)}}{{{{U'}_k}\left( {{{\bar c}_k}} \right)}}} \right), \cdots \cdots \cdots \cdots \cdots k \in A,\\ {\delta _k},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k \notin A. \end{array} \right. $

(14)

则f(${\vec \alpha }$=f(${{\vec \alpha }^*}$)，τ(ε(t), ${\vec \alpha }$)=τ(ε(t), ${{\vec \alpha }^*}$)，

$ {I_k}\left( {\frac{1}{{{\delta _k}}}\mathit{\tau }\left( {\varepsilon \left( t \right),\mathit{\vec \alpha }} \right)} \right) = {I_k}\left( {\frac{1}{{\delta _k^*}}\mathit{\tau }\left( {\varepsilon \left( t \right),{{\mathit{\vec \alpha }}^*}} \right)} \right),0 \le t \le T,\left( {k = 1, \cdots ,m} \right). $

(15)

证明  由(14) 式及I_k(y)≥c_k，可以看出k∈A, 当且仅当I_k($\frac{1}{{{\delta _k}}}$τ(ε(t), ${\vec \alpha }$))≥c_k，此不等式等价于(13) 式.由(14) 式可知，δ_kU_k′(c_k)≤δ_k^*U_k′(c_k)≤τ(ε(t), ${\vec \alpha }$)≤f(${\vec \alpha }$)，则f(${\vec \alpha }$)=f(${\vec \alpha }$^*).又δ_k^*U_k′(c_k)≤τ(ε(t), ${\vec \alpha }$)，则I(τ(ε(t), ${\vec \alpha }$), ${\vec \alpha }$^*)=$\sum\limits_{k \in A} {{I_k}} \left( {\frac{{\tau \left( {\varepsilon \left( t \right),\vec \alpha } \right)}}{{\delta _k^*}}} \right) + \sum\limits_{k \in A} {{I_k}} \left( {\frac{{\tau \left( {\varepsilon \left( t \right),\vec \alpha } \right)}}{{{\delta _k}}}} \right) = \sum\limits_{k = 1}^m {{I_k}} \left( {\frac{{\tau \left( {\varepsilon \left( t \right),\vec \alpha } \right)}}{{{\delta _k}}}} \right) = \varepsilon \left( t \right)$.即τ(ε(t), ${\vec \alpha }$)=τ(ε(t), ${\vec \alpha }$^*).如果k$ \notin $A，则δ_k^*=δ_k，τ(ε(t), ${\vec \alpha }$)=τ(ε(t), ${\vec \alpha }$^*)，(15) 式显然成立.如果k∈A，则有$\frac{1}{{{\delta _k}}}$τ(ε(t), ${\vec \alpha }$)≥$\frac{1}{{\delta _k^*}}$τ(ε(t), ${\vec \alpha }$)≥U′_k(c_k)，I_k($\frac{1}{{\delta _k^*}}$τ(ε(t), ${\vec \alpha }$))=I_k($\frac{1}{{\delta _k^*}}$τ(ε(t), ${\vec \alpha }$^*)).即${\vec \alpha }$^*=(δ^*, δ₂^*, …, δ_m^*)是(8) 式的解.此定理表明(8) 式的解不唯一，但最优消费过程${{{\hat c}_k}}$(t)(k=1, …, m)是唯一确定的.

定理4  优化问题

$ U\left( {c,\mathit{\vec \alpha }} \right) \buildrel \Delta \over = \mathop {\max }\limits_{\begin{array}{*{20}{c}} {{c_1} \ge {{\bar c}_1}, \cdots ,{c_m} \ge {{\bar c}_m}}\\ {{c_{_1}}{\rm{ + }}{c_{_2}}{\rm{ + }} \cdots + {c_m} = c} \end{array}} \sum\limits_{k = 1}^m {{\mathit{\delta }_k}{U_k}\left( {{c_k}\left( t \right)} \right)} $

(16)

的解为(9) 式，且U′(c, ${\vec \alpha }$)=τ(c, ${\vec \alpha }$).

证明  依据效用函数的定义, 可得

$ \begin{align} & \sum\limits_{k=1}^{m}{{{\delta }_{k}}{{U}_{k}}}\left( {{c}_{k}} \right)\le \sum\limits_{k=1}^{m}{{{\delta }_{k}}\left[ {{U}_{k}}\left( {{{\hat{c}}}_{k}} \right)+\left( {{c}_{k}}-{{{\hat{c}}}_{k}} \right)U_{k}^{'}\left( {{{\hat{c}}}_{k}} \right) \right]}=\\ &\sum\limits_{k=1}^{m}{{{\delta }_{k}}{{U}_{k}}}\left( {{{\hat{c}}}_{k}} \right)+\sum\limits_{\left\{ k\left| {{{\hat{c}}}_{k}} \right.={{{\bar{c}}}_{k}} \right\}}^{m}{{{\delta }_{k}}U_{k}^{'}}\left( {{{\hat{c}}}_{k}} \right)\left( {{c}_{k}}-{{{\hat{c}}}_{k}} \right)+ \\ & \mathit{\tau }\left( c,\vec{\alpha } \right)\sum\limits_{\left\{ k\left| {{{\hat{c}}}_{k}} \right.>{{{\bar{c}}}_{k}} \right\}}^{m}{\left( {{c}_{k}}-{{{\hat{c}}}_{k}} \right)}\le \sum\limits_{k=1}^{m}{{{\delta }_{k}}{{U}_{k}}}\left( {{{\hat{c}}}_{k}} \right)+\mathit{\tau }\left( c,\vec{\alpha } \right)\sum\limits_{k=1}^{m}{\left( {{c}_{k}}-{{{\hat{c}}}_{k}} \right)}=\sum\limits_{k=1}^{m}{{{\delta }_{k}}{{U}_{k}}}\left( {{{\hat{c}}}_{k}} \right). \\ \end{align} $

因此(9) 式是优化问题(16) 的解, 且有

$ U\left( {c,\mathit{\vec \alpha }} \right) = \sum\limits_{k = 1}^m {{\mathit{\delta }_k}{U_k}({I_k}(\frac{1}{{{\mathit{\delta }_k}}}\mathit{\tau }{\rm{(}}c,\mathit{\vec \alpha })))} . $

(17)

由(17) 式, 可推导出U′(c, ${\vec \alpha }$)=$\frac{{\rm{d}}}{{{\rm{d}}c}}\sum\limits_{k = 1}^m {{\delta _k}} {U_k}\left( {{I_k}\left( {\frac{1}{{{\delta _k}}}\tau \left( {c,\vec \alpha } \right)} \right)} \right) = $$\sum\limits_{k = 1}^m {\frac{1}{{{\delta _k}}}\tau \left( {c,\vec \alpha } \right)} {{I'}_k}\left( {\frac{1}{{{\delta _k}}}\tau \left( {c,\vec \alpha } \right)} \right)\tau '\left( {c,\vec \alpha } \right) = $$\sum\limits_{k = 1}^m {\tau \left( {c,\vec \alpha } \right)} \frac{{\rm{d}}}{{{\rm{d}}c}}{I_k}\left( {\frac{1}{{{\delta _k}}}\tau \left( {c,\vec \alpha } \right)} \right) = \tau \left( {c,\vec \alpha } \right)\frac{{\rm{d}}}{{{\rm{d}}c}}I\left( {\tau \left( {c,\vec \alpha } \right),\vec \alpha } \right) = \tau \left( {c,\vec \alpha } \right)$.

定理5  标准的完备的市场m=(r(t), ρ_i(t), μ_i(t), σ_i(t), φ_i(t), S_i(t), i=1, 2) 是均衡的充分必要条件是

$ \begin{gathered} r\left( t \right) = \beta \left( t \right) - \frac{1}{{U'\left( {\varepsilon \left( t \right),\vec \alpha } \right)}} \hfill \\ \left[ {U''\left( {\varepsilon \left( t \right),\vec \alpha } \right)\left( {{\mu _\varepsilon }\left( t \right) - {\varphi _\varepsilon }\left( t \right)\lambda \left( t \right)} \right) + } \right.\frac{1}{2}U'''\left( {\varepsilon \left( t \right),\vec \alpha } \right)\sigma _\varepsilon ^2\left( t \right) \hfill \\ \left. { + \left( {U'\left( {\varepsilon \left( {{t^ - }} \right) + {\varphi _\varepsilon }\left( t \right),\vec \alpha } \right) - U'\left( {\varepsilon \left( {{t^ - }} \right),\vec \alpha } \right)} \right)} \right]\lambda \left( t \right), \hfill \\ \end{gathered} $

(18)

$ {{\theta }_{1}}\left( t \right)=-\frac{{{U}^{''}}\left( \varepsilon \left( t \right),\vec{\alpha } \right)}{{{U}^{'}}\left( \varepsilon \left( t \right),\vec{\alpha } \right)}{{\mathit{\sigma }}_{\varepsilon }}\left( t \right);{{\theta }_{2}}\left( t \right)=-\frac{-1}{{{U}^{'}}\left( \varepsilon \left( t \right),\vec{\alpha } \right)}\left( {{U}^{'}}\left( \varepsilon \left( {{t}^{-}} \right)+{{\mathit{\varphi }}_{\varepsilon }}\left( t \right),\vec{\alpha } \right)-\\ {{U}^{'}}\left( \varepsilon \left( {{t}^{-}} \right),\vec{\alpha } \right) \right)\lambda \left( t \right). $

(19)

证明  由(11) 式可以得到τ(ε(t), ${\vec \alpha }$)=${{{\rm{e}}^{\int_0^t {\beta \left( u \right){\rm{d}}u} }} \cdot \frac{{L\left( t \right)}}{{B\left( t \right)}}}$, 0≤t≤T，τ(ε(0), ${\vec \alpha }$)=1，由Ito′s引理可得

$ \mathit{\tau }\left( {\varepsilon \left( t \right),\vec \alpha } \right) = 1 + \int_0^t {\mathit{\tau }\left( {\varepsilon \left( u \right),\vec \alpha } \right)} \left( {\mathit{\beta }\left( u \right) - r\left( u \right)} \right){\rm{d}}\mathit{u} - \int_0^t {\mathit{\tau }\left( {\varepsilon \left( u \right),\vec \alpha } \right){\theta _1}} \left( u \right){\rm{d}}\mathit{W}\left( u \right) - \\ \int_0^t {\mathit{\tau }\left( {\varepsilon \left( u \right),\vec \alpha } \right){\theta _2}} \left( u \right){\rm{d}}\mathit{M}\left( u \right). $

(20)

由定理4知τ(ε(t), ${\vec \alpha }$)=U′(ε(t), ${\vec \alpha }$), 0≤t≤T.则有

$ \begin{array}{l} \tau \left( {\varepsilon \left( t \right),\vec \alpha } \right) = 1 + \int_0^t {\left[ {U''\left( {\varepsilon \left( u \right),\vec \alpha } \right)\left( {{\mu _\varepsilon }\left( u \right) - {\varphi _\varepsilon }\left( u \right)\lambda \left( u \right)} \right) + \\ \frac{1}{2}U'''\left( {\varepsilon \left( u \right),\vec \alpha } \right)\sigma _\varepsilon ^2\left( u \right)} \right]{\rm{d}}u} + \\ \int_0^t {\left[ {U'\left( {\varepsilon \left( {{u^ - }} \right) + {\varphi _\varepsilon }\left( u \right),\vec \alpha } \right) - U'\left( {\varepsilon \left( {{u^ - }} \right),\vec \alpha } \right)} \right]\lambda \left( u \right){\rm{d}}u} + \\ \int_0^t {U''\left( {\varepsilon \left( u \right),\vec \alpha } \right){\sigma _\varepsilon }\left( u \right){\rm{d}}W\left( u \right) + } \\ \int_0^t {\left[ {U'\left( {\varepsilon \left( {{u^ - }} \right) + {\varphi _\varepsilon }\left( u \right),\vec \alpha } \right) - U'\left( {\varepsilon \left( {{u^ - }} \right),\vec \alpha } \right)} \right]\lambda \left( u \right){\rm{d}}M\left( u \right)} . \end{array} $

(21)

比较式(20) 和(21) 可以得到式(18) 和(19).

3 结论

在所建立的跳扩散模型下的金融市场m=(r(t), ρ_i(t), μ_i(t), σ_i(t), φ_i(t), S_i(t), i=1, 2) 中，市场的均衡是存在的.根据所得到的结果，只需选取${\vec \alpha }$^*=(δ^*, δ₂^*, …, δ_m^*)满足式(7) 和(8)，r(t), θ₁(t), θ₂(t)由式(18)、(19) 和(20) 所确定，μ₁(t), μ₂(t)满足(3) 式，则金融市场m是均衡市场.

市场的均衡是存在的且是唯一的.由(7) 式和τ(ε(t), ${\vec \alpha }$)=τ(ε(t), ${\vec \alpha }$^*)知，L(t)是唯一确定的，由(6) 式和定理2知，最优投资组合及最优消费过程(${{\hat c}_k},{{\hat \pi }_k}$)∈A_k是唯一的.