**Author details**

*that amounts to determining the value J*∗, *given by*

88 Dynamic Programming and Bayesian Inference, Concepts and Applications

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

inf *g*∈G*<sup>A</sup>*

> *<sup>t</sup>*<sup>1</sup> *t*0

result about optimal bank coverager ratios in the simplified case.

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

*Suppose that the following assumptions hold.*

<sup>+</sup> exp(−*r<sup>f</sup>*

*J*(*g*),

exp(−*r<sup>f</sup>*

the value function *v* satisfies (20) and (21).

*J* ∗ = inf *g*∈G*<sup>A</sup>*

*J*(*g*) = **E**

*and the optimal control law g*∗, *if it exists,*

*J* ∗ = inf *g*∈G*<sup>A</sup>*

*g*∗ = arg min

*g*∈G*<sup>A</sup>*

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

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

(*<sup>l</sup>* <sup>−</sup> *<sup>t</sup>*0))[*b*2(*u*<sup>2</sup>

(*t*<sup>1</sup> <sup>−</sup> *<sup>t</sup>*0))*b*1(*x*(*t*1))

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

*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*.*

*J*(*g*), (17)

*<sup>t</sup>*) + *<sup>b</sup>*3(*xt*)]*dt*

, (18)

*J*(*g*),

*J*(*g*) ∈ G*A*.

J. Mukuddem-Petersen1, M.A. Petersen1 and MP. Mulaudzi\*2

\*Address all corespodence to mulaump@unisa.ac.za

Faculty of Commerce and Administration, North West University, South Africa

Department of Decision Sciences, University of South Africa
