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Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach. (English) Zbl 0978.91039
The authors consider a financial market consisting of a stock and bond. The value of the stock \(S_{t}\) follows the equation \[ dS_{t}=\left(\mu+\int_{\mathbb{R}\setminus\{0\}}(e^{z}-1-z{\mathbb 1} _{|z|<1})\nu(dz)\right)S_{t} dt+S_{t-}\int_{\mathbb{R}\setminus\{0\}}(e^{z}-1)\widetilde N(dt,dz), \] where \(\mu\) is a constant, \(\widetilde N(dt,dz)=N(dt,dz)-dt\times\nu(dz)\) is the compensated Poisson random measure. The bond has the dynamics \(dB_{t}=rB_{t} dt\), where \(r>0\) is the interest rate, \[ r<\widehat\mu= \mu+\int_{\mathbb{R}\setminus\{0\}} (e^{z}-1-z{\mathbb 1} _{|z|<1})\nu(dz). \] Consider an investor who wants to put his money in the stock and the bond so as to maximize the utility. Let \(\pi_{t}\in [0,1]\) be the fraction of the wealth invested in the stock at time \(t\). Let us introduce the process \[ Y_{t}^{\pi,C}= ye^{-\beta t}+\beta e^{-\beta t}\int_{[0,t]}e^{\beta s} dC_{s} \] modeling the average past consumption. Here \(C_{t}\) is the cumulative consumption up to time \(t\). Let us define the value function as \[ V(x,y)=\sup_{\pi,C\in A_{x,y}}E\left[\int_0^{\infty}e^{-\delta t}U(Y_{t}^{\pi,C})dt\right], \] where \(\delta>0\) is the discount factor, \(A_{x,y}\) is a set of admissible controls, \(U\) is a utility function. The authors prove that the value function \(V\) is a unique constrained viscosity solution of the integro-differential variational inequality \[ \begin{split} \max \Biggl\{\beta v_{y}-v_{x};\;U(y)-\delta v-\beta yv_{y}+ \max_{\pi} \biggl[(r+(\widehat\mu-r)\pi)xv_{x}\\ +\int_{\mathbb{R}\setminus\{0\}} (v(x+\pi x(e^{z}-1),y)-v(x,y)-\pi xv_{x}(x,y)(e^{z}-1))\nu(dz) \biggr] \Biggr\}=0 \end{split} \] in \(D=\{(x,y)\in \mathbb{R}^2:x>0,y>0\}\). Also a new comparison result for the state constraint problem for a class of integro-differential variational inequalities is obtained.

91G10 Portfolio theory
93E20 Optimal stochastic control
49L20 Dynamic programming in optimal control and differential games
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