**3.2. Optimal bank LCRs in the simplified case**

In this section, we determine a solution to Problem 3.2 in the case where the term [*t*0, *t*1] is fixed. In order to find the optimal control processes, we use the dynamic programming algorithm for stochastic optimization where we consider an appropriate Hamilton-Jacobi-Bellman equation (HJBE). In the sequel, we assume that the optimal control laws exist, with the objective function, *J*, given by (16) being continuous twice-differentiable. Then a combination of integral calculus and Itô's formula (see, for instance, [15]) shows that the value function *v* satisfies (20) and (21).

Consider the simplified system (13) for the LCR problem with the admissible class of control laws, G*A*, given by (14) but with X = R (compare with [10]). In this section, we have to solve

$$\begin{aligned} \inf\_{\mathcal{G}\in\mathcal{G}\_{\mathcal{A}}} J(\mathbf{g}),\\ f^\* = \inf\_{\mathcal{G}\in\mathcal{G}\_{\mathcal{A}}} J(\mathbf{g}), \end{aligned} \tag{17}$$

$$\begin{aligned} J(\mathbf{g}) = \mathbf{E}\left[\int\_{t\_0}^{t\_1} \exp(-r^f(l-t\_0))[b^2(u\_t^2) + b^3(\mathbf{x}\_t)]dt \\ + \exp(-r^f(t\_1-t\_0))b^1(\mathbf{x}(t\_1))\big], \end{aligned} \tag{18}$$

where *<sup>b</sup>*<sup>1</sup> : <sup>R</sup> <sup>→</sup> <sup>R</sup>+, *<sup>b</sup>*<sup>2</sup> : <sup>R</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> and *<sup>b</sup>*<sup>3</sup> : <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> are all Borel measurable functions. For the simplified case, the optimal cost function (17) is determined with the simplified cost function, *J*(*g*), given by (18). In this case, assumptions have to be made in order to find a solution for the optimal cost function, *J*∗ (compare with [10]). Next, we state an important result about optimal bank coverager ratios in the simplified case.

### **Theorem 3.3. (Optimal Bank LCRs in the Simplified Case)**

*Suppose that g*2<sup>∗</sup> *and g*3<sup>∗</sup> *are the components of the optimal control law, g*∗, *that deal with the optimal cash injection rate, u*2∗, *and optimal HQLA allocation, πk*∗, *respectively. Consider the nonlinear optimal stochastic control problem for the simplified LCR system* (13) *formulated in Problem* 3.2*. Suppose that the following assumptions hold.*

availability of more suitable data of sufficient granularity as well as improved extrapolation and interpolation techniques.

We have already made several contributions in support of the endeavors outlined in the previous paragraph. For instance, our journal article [12] deals with issues related to liquidity risk and the financial crisis. Also, the role of information asymmetry in a subprime context is related to the main hypothesis of the book [14].
