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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.

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