# Solving a nonlinear equation with a Markov process and RVs

Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later.

Update

$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\gamma}*[r_k(A_{t+1},a)-r_h(A_{t+1})+\eta)] \right]=0$$

where $$R=(1-\theta_1-\theta_2)*r^f + \theta_1*r_k(A,a)+\theta_2 *[r_h(A)+\eta]$$

Also consider that the number $$b(A)$$ is pinned down recursively by the following equation:

$$b_t = G[E_t(b_{t+1}*R_{t+1}^{1-\gamma})]$$, where $$G(\cdot)$$ is a known function

Main Assumptions

1. $$A_t$$ is a random variable that follows a Markov process, and we know its value at time $$t$$, but not at time $$t+1$$.

2. $$a$$ is an $$IID$$ random variable lognormally distributed with some mean $$\mu_a$$ and variance $$\sigma^2_a$$.

3. $$\eta$$ is another $$IID$$ random variable normally distributed with some mean $$\mu_{\eta}$$ and variance $$\sigma^2_\eta$$.

4. The RVs are independent of each other. That is, $$a$$ is independent of $$\eta$$, while these two can be correlated with $$A$$, thought are still independently drawn from $$A$$.

5. $$b(A)$$ is a number that depends on $$A$$ and I do not know their functional relationship.

6. $$r^f$$ is a known number.

7. by $$r_k(A,a)$$ and respectively $$r_h(A)$$ I mean a functional relationship (which I know) between these two that depend on $$A, a$$ and $$A$$ respectively.

To sum up, the expectations operator is about the unknowns in the next period $$A,a$$ and $$\eta$$. To fix the ideas. Let's suppose a simple case, where $$a,\eta$$ take two values and $$A$$ follows a two-state Markov process and also take two values. Let's also suppose that at time $$t$$ we are in one of these states. In that case, the Expectations operator is defined over the next possible state for $$A$$, and the possible values for the $$IID$$ RVs.

THE COMPUTATIONAL TASK: Solve for $$\theta_1$$ and $$\theta_2$$

Apparently, the solution for each $$\theta$$ will depend on the possible values of $$A$$, so I expect to be a vector.

Any guidance on how to solve this equation will be appreciated.

• That sounds like a very challenging problem; I'm not sure you have much hope if you can't even evaluate the random function (since you do not know $b$). But I don't think $\theta$ will be a vector (unless you set it up as such). – Christian Clason Oct 21 '14 at 12:11
• You might want to search for "parameter estimation" or "calibration" for Markov processes. – Christian Clason Oct 21 '14 at 12:13
• @ChristianClason thanks for the response. Let's say that I can calibrate the markov process this is somehow straight forward to me. Can you still believe solve it in the way you described in my other post ? But is not totally clear how I will define the Gaussian-Hermite Quadrature. – user17880 Oct 21 '14 at 14:27
• I was thinking to solve the problem, for some given value of $b$, then I can employ some other equations from the model to pin down $b$ for the given values found for $\theta's$ – user17880 Oct 21 '14 at 14:30
• In principle, that might work. Your expectation is now a two-dimensional integral $E(f(a,\nu)) = \int\int (t,s)d\phi_a(t)d\phi_\nu(s)$, which you can approximate with tensor-product quadrature $\sum_i\sum_j w_i^a w_j^\nu f(t_i^a, t_j^\nu)$, where $w_i^a$ and $t_i^a$ are the Gauss-Hermite weights and nodes for a lognormal distribution and $w_j^\nu$ and $t_j^\nu$ are those for a normal distribution (as in my answer to your other question). – Christian Clason Oct 21 '14 at 16:09