<p>I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution. 
If there exists an analytical solution (for a very basic model with a one-state variable), it is possible to derive the initial conditions for your control and state variable from your differential equations.</p>

<p>However, in systems where there does not exist an analytical solution, you are supposed to solve it numerically, in which case you must give a numerical value for one of your variables (or more until you find steady state values for all your variables.)</p>

<p>After, you can find the steady-state values of your variables in which case, your initial conditions should not be so far from the steady state level (otherwise, there would be some convergence problems in your model if you don't choose the appropriate initial values, close to steady state. It is another issue.)</p>

CC BY-SA 3.0
2015-09-11T06:27:15.520Z
But in the case of numerical solution you are stuck assuming the boundary conditions?  Or deducing them from some aspect of the problem, but they are not a result of the derivation?
pdevar
CC BY-SA 3.0
2015-09-11T09:50:36.510Z
In many cases, you give an arbitrary value. This is the case in many macro papers with dynamic optimization.
The reason behind is that, in some cases, for example, you have 4 unknowns and your system reduces to three dimensional equation system. So, in this case, you are obliged to give a numerical value for one the variables. After, you will find the numerical value for other three variables according to the value that you have given for one of them. For sure, the numerical values that you will find will be derived in your system of equation according to the value that you choose.
optimal control
CC BY-SA 3.0
2015-09-09T19:07:42.763Z
2015-09-09T19:12:46.903Z
2015-09-09T19:12:46.903Z
optimal control
<p>Here's some information with regard to the intuition of each. </p>

<p>The <strong>Kolmogorov
forward equation</strong> is often referred to as the Fokker-Planck equation.
It is a partial differential equation (PDE) that describes
the time evolution of the probability density function of a variable
over a state. That is, suppose we have information about the state
<span class="math-container">$x$</span> at a time <span class="math-container">$t$</span>. The forward equation tells us the distribution
of <span class="math-container">$x$</span> at a time <span class="math-container">$s&gt;t$</span>. </p>

<p>The <strong>Kolmogorov backward equation</strong> on the other hand
is used to understand the probability of a state
ending up in a set <span class="math-container">$B$</span> at some time <span class="math-container">$s$</span>. Define a function of the state,
<span class="math-container">$u_s(x)$</span>. At the final time <span class="math-container">$s$</span>, <span class="math-container">$u_s(x) = 1$</span> if <span class="math-container">$x \in B$</span>. Otherwise,
it is zero. Then, for any time <span class="math-container">$t &lt; s$</span>, <span class="math-container">$u_t(x)$</span> describes
the probability that <span class="math-container">$X_s$</span> given <span class="math-container">$X_t = x_t$</span> will end
up in the set <span class="math-container">$B$</span>. The Kolmogorov
backward equation described the evolution of the probability function. 
<span class="math-container">$u_s(x)$</span> then is the final condition of a
this PDE.</p>

<p>I've taken this information mostly from Wikipedia:</p>

<ul>
<li><a href="https://en.wikipedia.org/wiki/Kolmogorov_equations" rel="nofollow noreferrer">Kolmogorov equations</a> </li>
<li><a href="https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation" rel="nofollow noreferrer">Fokker–Planck equation</a> </li>
<li><a href="https://en.wikipedia.org/wiki/Kolmogorov_backward_equations_(diffusion)" rel="nofollow noreferrer">Kolmogorov backward equations</a> </li>
</ul>

CC BY-SA 4.0
2017-07-25T20:46:56.673Z
2018-11-18T04:12:05.417Z
2018-11-18T04:12:05.417Z
jmbejara
<p>So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition.  Here is from Stokey, 2008:</p>

<p>"The backward equation involves time $t$ and the initial condition $x$, with
the current state $y$ held fixed. A similar PDE, the Kolmogorov forward
equation (KFE), involves $t$ and $y$, with the initial state $x$ fixed. The forward
equation is useful for characterizing the limiting distribution, if one exists."</p>

<p>In what situations do the two arrise?  For instance, if I know the current state and am interested in a probability distribution over possible initial states, I use the backward equation.  If I know the current state and am interested in the probability distribution over the state in the future, I use the forward equation.  Is this correct?</p>

<p>On a more technical note, how does one define boundary conditions for these PDE's?  Are the boundary conditions a result of the derivation?  Perhaps Stokey is not the best reference on this topic...</p>

CC BY-SA 3.0
2015-09-09T18:56:39.107Z
Could you please share the link of your reference ?
optimal control
CC BY-SA 3.0
2015-09-11T06:27:37.473Z
http://press.princeton.edu/titles/8766.html
pdevar
<continuous-time><stochastic-calculus>
Intuition of the Kolmogorov Equations
pdevar
# Intuition of the Kolmogorov Equations

(Asked by: username_0 on 2015-09-09)

So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:

"The backward equation involves time $t$ and the initial condition $x$, with
the current state $y$ held fixed. A similar PDE, the Kolmogorov forward
equation (KFE), involves $t$ and $y$, with the initial state $x$ fixed. The forward
equation is useful for characterizing the limiting distribution, if one exists."

In what situations do the two arrise? For instance, if I know the current state and am interested in a probability distribution over possible initial states, I use the backward equation. If I know the current state and am interested in the probability distribution over the state in the future, I use the forward equation. Is this correct?

On a more technical note, how does one define boundary conditions for these PDE's? Are the boundary conditions a result of the derivation? Perhaps Stokey is not the best reference on this topic...

**Tags:** continuous-time, stochastic-calculus

> **Comments:**
> **username_1**: Could you please share the link of your reference ?  
> **username_0**: http://press.princeton.edu/titles/8766.html  

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## 2 answers

### Answer by username_1 on 2015-09-09. Score: 1 | Accepted answer

I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution.
If there exists an analytical solution (for a very basic model with a one-state variable), it is possible to derive the initial conditions for your control and state variable from your differential equations.

However, in systems where there does not exist an analytical solution, you are supposed to solve it numerically, in which case you must give a numerical value for one of your variables (or more until you find steady state values for all your variables.)

After, you can find the steady-state values of your variables in which case, your initial conditions should not be so far from the steady state level (otherwise, there would be some convergence problems in your model if you don't choose the appropriate initial values, close to steady state. It is another issue.)

> **Comments:**
> **username_0**: But in the case of numerical solution you are stuck assuming the boundary conditions?  Or deducing them from some aspect of the problem, but they are not a result of the derivation?  
> **username_1**: In many cases, you give an arbitrary value. This is the case in many macro papers with dynamic optimization. The reason behind is that, in some cases, for example, you have 4 unknowns and your system reduces to three dimensional equation system. So, in this case, you are obliged to give a numerical value for one the variables. After, you will find the numerical value for other three variables according to the value that you have given for one of them. For sure, the numerical values that you will find will be derived in your system of equation according to the value that you choose.  

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### Answer by username_2 on 2017-07-25. Score: 0

Here's some information with regard to the intuition of each.

The **Kolmogorov
forward equation** is often referred to as the Fokker-Planck equation.
It is a partial differential equation (PDE) that describes
the time evolution of the probability density function of a variable
over a state. That is, suppose we have information about the state
$x$ at a time $t$. The forward equation tells us the distribution
of $x$ at a time $s>t$.

The **Kolmogorov backward equation** on the other hand
is used to understand the probability of a state
ending up in a set $B$ at some time $s$. Define a function of the state,
$u\_s(x)$. At the final time $s$, $u\_s(x) = 1$ if $x \in B$. Otherwise,
it is zero. Then, for any time $t < s$, $u\_t(x)$ describes
the probability that $X\_s$ given $X\_t = x\_t$ will end
up in the set $B$. The Kolmogorov
backward equation described the evolution of the probability function.
$u\_s(x)$ then is the final condition of a
this PDE.

I've taken this information mostly from Wikipedia:

* [Kolmogorov equations](https://en.wikipedia.org/wiki/Kolmogorov_equations)
* [Fokker–Planck equation](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation)
* [Kolmogorov backward equations](https://en.wikipedia.org/wiki/Kolmogorov_backward_equations_(diffusion))


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