One problem of the investment, taking into account the risks.
We study the problem of optimal investment the initial capital into two kinds of investments in order to get the most benefit. Change in share capital is described by system of two ordinary linear nonhomogeneous differential equations with the random coefficients
$\frac{\mathrm{d}x_{1} }{\mathrm{d} t}=\varepsilon _{1}\left ( t,\omega \right )x_{1}+\varepsilon _{3}\left ( t,\omega \right )$,
$\frac{\mathrm{d}x_{2} }{\mathrm{d} t}=\varepsilon _{2}\left ( t,\omega \right )x_{2}+\varepsilon _{4}\left ( t,\omega \right )$.
Here $x_{1}$, $x_{2}$ - change in share capital, $\varepsilon _{j}$,$j$= 1, 2, 3, 4 - stochastic processes $\omega $ - random event, $x_{1}\left ( 0 \right )+x_{2}\left ( 0 \right )=1$ - the initial share capital. You want $x_{1}\left ( 0 \right )$, $x_{2}\left ( 0 \right )$ to choose so the value $I=M\left ( x_{1}\left ( 1 \right )+x_{2}\left ( 1 \right ) \right )$ in was the greatest for a given level of risk $0\leq r=D\left ( x_{1}\left ( 1 \right )+x_{2}\left ( 1 \right ) \right )$. Here $M\left ( x_{1}\left ( 1 \right )+x_{2}\left ( 1 \right ) \right )$ denotes the mathematical expectation ànd $D\left ( x_{1}\left ( 1 \right )+x_{2}\left ( 1 \right ) \right )$ - dispersion of the $ x_{1}\left ( 1 \right )+x_{2}\left ( 1 \right )$.
It is assumed that stochastic processes $\varepsilon _{j}$,$j$= 1, 2, 3, 4 specified characteristic functional
$\psi \left ( u_{1},u_{2},u_{3},u_{4} \right )=M\exp\left ( i\int_{0}^{1}\sum_{1}^{4}\varepsilon _{j}\left ( s \right )u_{j}\left ( s \right ) ds\right )$.
Here $u_{1}$,$u_{2}$,$u_{3}$,$u_{4}$ - functions of set $L_{1}\left ( 0,1 \right )$ integrable on the segment $\left [ 0,1 \right ]$ functions, $i$ - the imaginary unit.
Let
$y_{j}=y_{j}\left ( t,u_{1},u_{2},u_{3},u_{4} \right )=M\left ( x_{j}\left ( t \right )e\left ( u_{1},u_{2},u_{3},u_{4} \right ) \right )$, $j$=1,2.
Note that $y\left ( t,0,0,0,0 \right )=Mx_{j}\left ( t \right )$, $j$=1,2. For $y_{j}$, $j$=1,2 received the deterministic equations
$\frac{\partial y_{1}}{\partial t}=-i\frac{\delta _{p}y_{1}}{\delta u_{1}\left ( t \right )}-i\frac{\delta _{p}\psi }{\delta u_{3}\left ( t \right )}$,
$\frac{\partial y_{2}}{\partial t}=-i\frac{\delta _{p}y_{2}}{\delta u_{2}\left ( t \right )}-i\frac{\delta _{p}\psi }{\delta u_{4}\left ( t \right )}$,
and initial conditions
$y_{j}\left ( 0,u_{1},u_{2},u_{3},u_{4} \right )=x_{j0}\psi \left ( u_{1},u_{2},u_{3},u_{4} \right )$, $j$=1,2.
Here $\frac{\delta_{p} y_{1}}{\delta u_{1}\left ( t \right )}$ is the partial variational derivative.
The solution of this problem is given by
$y_{1}=x_{10}\psi \left ( u_{1} - i\chi \left ( 0,t \right ),u_{2},u_{3},u_{4} \right ) - i\int_{0}^{t}\frac{\delta _{p}\psi \left ( u_{1} - i\chi \left ( s,t \right ),u_{2},u_{3},u_{4}\right )}{\delta u_{3}\left ( s \right )}ds$,
$y_{2}=x_{20}\psi \left ( u_{1}, u_{2}- i\chi \left ( 0,t \right ),u_{3},u_{4} \right ) - i\int_{0}^{t}\frac{\delta _{p}\psi \left ( u_{1},u_{2} - i\chi \left ( s,t \right ),u_{3},u_{4}\right )}{\delta u_{4}\left ( s \right )}ds$.
The mathematical expectation od the capital for $t$ = 1 is given by
$M\left ( x_{1}\left ( 1 \right )+ x_{2}\left ( 1 \right )\right )=x_{10}\psi \left (- i\chi \left ( 0,1 \right ),0,0,0 \right ) +\left ( 1-x_{10} \right )\psi \left ( 0,-i\chi \left ( 0,1 \right ),0,0 \right ) - i\int_{0}^{1}\frac{\delta _{p}\psi \left ( - i\chi \left ( s,1 \right ),0,0,0\right )}{\delta u_{3}\left ( s \right )}+\frac{\delta _{p}\psi \left ( 0, - i\chi \left ( s,1 \right ),0,0\right )}{\delta u_{4}\left ( s \right )}ds$.
The first two moment functions of solutions are find. The resulting algorithm for determining the optimal allocation of the initial capital for a given risk. Bibl. 6.
Keywords: Problem about investments, risks, differential equations with random coefficients, characteristic functional, variational derivative.