General solutions for HJB equations in a special case. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:52:08Z http://mathoverflow.net/feeds/question/70379 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70379/general-solutions-for-hjb-equations-in-a-special-case General solutions for HJB equations in a special case. Steven 2011-07-14T21:59:38Z 2011-07-27T21:52:24Z <p>I am reading the book of <a href="http://www.amazon.com/Controlled-Processes-Viscosity-Stochastic-Probability/dp/0387260455" rel="nofollow">Wendell Flemming</a> in control theorem to learn the <a href="http://en.wikipedia.org/wiki/Hamilton-Jacobi-Bellman_equation" rel="nofollow">HJB equation</a></p> <p>Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\leq g_1(x)\leq C_1$ and $0\leq g_2(x)\leq e^{x}C_2$ for some FIXED constants $C_1,C_2$. If $f_i : i=1,2$ are $C^2$ functions then A FIXED $\lambda$, define</p> <p>$Af_i(x)=(\mu -\frac{\sigma^2}{2})\frac{\partial f_i(x)}{\partial x}+\frac{\sigma^2}{2}\frac{\partial^2 f_i(x)}{\partial x^2} +\lambda(g_i(x)-f_i(x))$. </p> <p>I am interested in finding two functions $f_1,f_2\in C^2$ such that for each $x\in\mathbb{R}$ :</p> <p>$\min \{ \rho f_1(x) - Af_1(x), f_1(x)- f_2(x) +e^x D_1\} =0$</p> <p>$\min \{ \rho f_2(x) - Af_2(x), f_2(x)- f_1(x) - e^x D_2\} =0$</p> <p>where $\mu,\sigma,\rho, D_1>D_2$ are fixed positive constants satisfying: $\rho&lt;\mu&lt;\rho+\lambda$</p> <p>If consider the equation $\rho f_i(x) - Af_i(x)=0$ ALONE, using Laplace transform,I came </p> <p>up with the following solutions for $\rho f_i(x) - Af_i(x)=0$ where $i=1,2$ </p> <p>$f_i(x)=B_1^i e^{\nu_1x}+B_2^ie^{\nu_2x} - \frac{2\lambda}{(\nu_2 - \nu_1)\sigma^2}\int_0^x g_i(x-t)(e^{\nu_2t} - e^{\nu_1 t})dt$</p> <p>with $\nu_{1,2} =(\mu/\sigma^2 -1/2)\pm\sqrt{(\mu/\sigma^2 -1/2)^2+ 2(\rho+\lambda)/\sigma^2}$ and $B^i_{1,2}$ are some real numbers </p> <p>My question( hope that it does not violate the forum's rules) is that: how can I use these solutions to solve the above HJB equations? Thanks so much for your interests. </p>