Mathematically, LQ control problems are closely related to {doc}`the Kalman filter <kalman>`

* Recursive formulations of linear-quadratic control problems and Kalman filtering problems both involve matrix **Riccati equations**.

* Classical formulations of linear control and linear filtering problems make use of similar matrix decompositions (see for example [this lecture](https://python-advanced.quantecon.org/lu_tricks.html) and [this lecture](https://python-advanced.quantecon.org/classical_filtering.html)).

* Recursive formulations of linear-quadratic control problems and Kalman filtering problems both involve matrix [Riccati equations](https://en.wikipedia.org/wiki/Riccati_equation).

* Classical formulations of linear control and linear filtering problems make use of similar matrix decompositions (see for example[Classical Control with Linear Algebra](https://python-advanced.quantecon.org/lu_tricks.html) and [Classical Prediction and Filtering With Linear Algebra](https://python-advanced.quantecon.org/classical_filtering.html)).

In reading what follows, it will be useful to have some familiarity with

* matrix manipulations

* vectors of random variables

* dynamic programming and the Bellman equation (see for example {doc}`this lecture <intro:short_path>` and {doc}`this lecture <optgrowth>`)

* dynamic programming and the Bellman equation (see for example {doc}`Shortest Paths <intro:short_path>` and {doc}`Optimal Growth <optgrowth>`)

For additional reading on LQ control, see, for example,

Here

* $u_t$ is a "control" vector, incorporating choices available to a decision-maker confronting the current state $x_t$

* $\{w_t\}$ is an uncorrelated zero mean shock process satisfying $\mathbb E w_t w_t' = I$, where the right-hand side is the identity matrix

* $\{w_t\}$ is a sequence of uncorrelated zero mean shock process satisfying $\mathbb E w_t w_t' = I$, where the right-hand side is the identity matrix

Regarding the dimensions

Here $a_t$ is assets, $r$ is a fixed interest rate, $c_t$ is

current consumption, and $y_t$ is current non-financial income.

If we suppose that $\{ y_t \}$ is serially uncorrelated and $N(0,

\sigma^2)$, then, taking $\{ w_t \}$ to be standard normal, we can write

If we suppose that $\{y_t\}$ is serially uncorrelated and $N(0,

\sigma^2)$, then, taking $\{w_t\}$ to be standard normal, we can write

the system as

$$

x_t' R x_t + u_t' Q u_t

```

Where the entries in matrices $R$ and $Q$ are chosen to match the specific problem that is being studied.

Here

* $R$ is assumed to be $n \times n$, symmetric and nonnegative definite.

* $Q$ is assumed to be $k \times k$, symmetric and positive definite.

```{note}

In fact, for many economic problems, the definiteness conditions on $R$ and $Q$ can be relaxed. It is sufficient that certain submatrices of $R$ and $Q$ be nonnegative definite. See {cite}`HansenSargent2008` for details.

```

```{note}

The use of $R$ and $Q$ notations may differ depending on the source. Some authors use $Q$ to be the matrix associated with the state variables and $R$ with the control variables.

```

#### Example 1

* We assume $R_f$ to be $n \times n$, symmetric and nonnegative definite.

* We allow $\beta = 1$, and hence include the undiscounted case.

* $x_0$ may itself be random, in which case we require it to be independent of the shock sequence $w_1, \ldots, w_T$.

* $x_0$ may itself be random, in which case we require it to be independent of the shock sequence $\{w_1, \ldots, w_T\}$.

(lq_cp)=

### Information

### Solution

To solve the finite horizon LQ problem we can use a dynamic programming

strategy based on backward induction that is conceptually similar to the approach adopted in {doc}`this lecture <intro:short_path>`.

strategy based on backward induction that is conceptually similar to the approach adopted in {doc}`Shortest Paths <intro:short_path>`.

For reasons that will soon become clear, we first introduce the notation $J_T(x) = x' R_f x$.

\}

$$

We use $u$ instead of $u_{T-1}$ here to simplify notation.

At this stage, it is convenient to define the function

```{math}

(The algebra is a good exercise --- we'll leave it up to you.)

If we continue working backwards in this manner, it soon becomes clear that $J_t (x) = x' P_t x + d_t$ as claimed, where $\{P_t\}$ and $\{d_t\}$ satisfy the recursions

If we continue working backwards in this manner, it soon becomes clear that $J_t (x) = x' P_t x + d_t$ as claimed, where $P_t$ and $d_t$ satisfy the recursions

```{math}

:label: lq_pr

In particular, the sequence of controls given by {eq}`lq_oc` and {eq}`lq_lom` solves our finite horizon LQ problem.

Rephrasing this more precisely, the sequence $u_0, \ldots, u_{T-1}$ given by

Rephrasing this more precisely, the sequence $\{u_0, \ldots, u_{T-1}\}$ given by

```{math}

:label: lq_xud

Now that the problem is expressed in LQ form, we can proceed to the solution

by applying {eq}`lq_pr` and {eq}`lq_oc`.

After generating shocks $w_1, \ldots, w_T$, the dynamics for assets and

After generating shocks $\{w_1, \ldots, w_T\}$, the dynamics for assets and

consumption can be simulated via {eq}`lq_xud`.

The following figure was computed using $r = 0.05, \beta = 1 / (1+ r),