**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.*

## *1. The cost function is assumed to satisfy*

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

78 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*.

$$\begin{aligned} b^2(u^2) &\in \mathbb{C}^2(\mathbb{R}),\\ \lim\_{\substack{\boldsymbol{u}^2 \to -\infty \\ \boldsymbol{u}^2 \to +\infty}} D\_{\boldsymbol{u}^2} b^2(\boldsymbol{u}^2) &= -\infty, \quad \lim\_{\boldsymbol{u}^2 \to +\infty} D\_{\boldsymbol{u}^2} b^2(\boldsymbol{u}^2) = +\infty; \\ D\_{\boldsymbol{u}^2 \boldsymbol{u}^2} b^2(\boldsymbol{u}^2) &> 0, \; \forall \boldsymbol{u}^2 \in \mathbb{R}, \end{aligned} \tag{19}$$

*with the differential operator, D, that is applied in this case to function b*2*.*

*2. There exists a function v* : *<sup>T</sup>* <sup>×</sup> <sup>R</sup> <sup>→</sup> R, *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*1,2(*<sup>T</sup>* × X ), *that is a solution of the HJBE given by*

$$\begin{aligned} 0 &= D\_l v(t, \mathbf{x}) + \frac{1}{2} [(\sigma^\epsilon)^2 (1 - \mathbf{x})^2 + (\sigma^i)^2 (\mathbf{x}\_l)^2] D\_{\mathbf{x}\mathbf{x}} v(t, \mathbf{x}) \\ &+ \mathbf{x} (r^\mathbf{F}(t) + r\_t^\epsilon - r\_t^i + (\sigma^\epsilon)^2 + (\sigma^i)^2) D\_{\mathbf{x}} v(t, \mathbf{x}) \\ &+ [u\_t^1 - r\_t^\epsilon - (\sigma^\epsilon)^2] D\_{\mathbf{x}} v(t, \mathbf{x}) \\ &+ u\_t^{2^\*} D\_{\mathbf{x}} v(t, \mathbf{x}) + \exp(-r^f (t - t\_0)) b^2 (u\_t^{2^\*}) \\ &+ \exp(-r^f (t - t\_0)) b^3 (\mathbf{x}) - \frac{[D\_{\mathbf{x}} v(t, \mathbf{x})]^2}{2D\_{\mathbf{x}\mathbf{x}} v(t, \mathbf{x})} \tilde{r}\_t^T \tilde{C}\_t^{-1} \tilde{r}\_{t'}^y \end{aligned} \tag{20}$$

$$v(t\_1, \mathbf{x}) = \exp(-r^f(t\_1 - t\_0))b^1(\mathbf{x}),\tag{21}$$

*where u*2<sup>∗</sup> *is the unique solution of the equation*

$$0 = D\_x v(t, \mathbf{x}) + \exp(-r^f(t - t\_0)) D\_{\mathbf{u}^2} b^2(\mathbf{u}\_t^2). \tag{22}$$

*Then the optimal control law is*

$$\mathbf{g}^{2\*}(t, \mathbf{x}) = \mathbf{u}^{2\*}, \mathbf{g}^{2\*} : T \times \mathcal{X} \to \mathbb{R}^+, \tag{23}$$

*with u*2<sup>∗</sup> ∈ U<sup>2</sup> *the unique solution of the equation (22)*

$$
\tilde{\pi}^\* = -\frac{D\_\mathbf{x} v(t, \mathbf{x})}{\mathbf{x} D\_\mathbf{x} v(t, \mathbf{x})} \tilde{\mathbf{C}}\_t^{-1} \tilde{r}\_{t, \mathbf{y}}^y \tag{24}
$$

$$\mathcal{g}^{3,k\*}(t,\mathbf{x}) = \min\{1, \max\{0, \tilde{\pi}^{k\*}\}\}, \mathcal{g}^{3,k\*}: T \times \mathcal{X} \to \mathbb{R},\tag{25}$$

*Furthermore, the value of the problem is*

$$J^\* = J(\emptyset^\*) = \mathbf{E}[v(t, \mathfrak{x}\_0)].\tag{26}$$

Next, we choose a particular cost functions for which an analytic solution can be obtained for the value function and control laws (compare with [10]). The following theorem provides the optimal control laws for quadratic cost functions.

**Theorem 3.4. (Optimal Bank LCRs with Quadratic Cost Functions):** *Consider the nonlinear optimal stochastic control problem for the simplified LCR system* (13) *formulated in Problem* 3.2*. Consider the cost function*

$$J(\mathbf{g}) = \mathbf{E}[\int\_{t\_0}^{t\_1} \exp(-r^f(l - t\_0)) [\frac{1}{2} \mathbf{c}^2((u^2)^2(l)) + \frac{1}{2} \mathbf{c}^3(\mathbf{x}\_l - l^r)^2] dl}$$

$$+ \frac{1}{2} \mathbf{c}^1(\mathbf{x}(t\_1) - l^r)^2 \exp(-r^f(t\_1 - t\_0)) \, ]. \tag{27}$$

*(b) The optimal control laws are*

with [10]).

*b*3(*x*) = <sup>1</sup>

The reference process, *l*

<sup>2</sup> *<sup>c</sup>*3(*<sup>x</sup>* <sup>−</sup> *<sup>l</sup>*

other hand, *x < l*

is strictly larger than a set value *l*

it is reasonable that costs with *x > l*

**4. Numerical results for LCRs**

*u*2<sup>∗</sup>

*π*∗

*(c) The value function and the value of the problem are*

(*t*, *x*) = *u*2<sup>∗</sup>

*g*2<sup>∗</sup>

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

*<sup>v</sup>*(*t*, *<sup>x</sup>*) = exp(−*r<sup>f</sup>*

*J*

more details). Another cost function that we can consider is

cost function considered is to keep *b*3(*x*) = 0 for *x < l*

*<sup>b</sup>*3(*x*) = *<sup>c</sup>*3[exp(*<sup>x</sup>* <sup>−</sup> *<sup>l</sup>*

that is strictly convex and asymmetric in *x* with respect to the value *l*

statistics for Class I and II banks for the sample period 2002 to 2012.

, *<sup>g</sup>*2<sup>∗</sup> : *<sup>T</sup>* ×X → <sup>R</sup>+,

*<sup>C</sup>*<sup>−</sup><sup>1</sup> *<sup>t</sup> <sup>r</sup> y*

*<sup>g</sup>*3,*k*∗(*t*, *<sup>x</sup>*) = *<sup>π</sup><sup>k</sup>*∗, *if <sup>π</sup><sup>k</sup>*<sup>∗</sup> <sup>∈</sup> [0, 1], min{1, max{0, *<sup>π</sup><sup>k</sup>*<sup>∗</sup>

(*t*<sup>1</sup> <sup>−</sup> *<sup>t</sup>*0))[ <sup>1</sup>

For the cost on the cash injection, the function (28) is considered, where the input variable *u*<sup>2</sup> is restricted to the set R+. If *u*<sup>2</sup> *>* 0 then the banks should acquire additional HQLAs. The cost function should be such that cash injections are maximized, hence *u*<sup>2</sup> *>* 0 should imply

by (28). Here, both positive and negative values of *u*<sup>2</sup> are penalized equally. An important reason for this is that an analytic solution of the value function can be determined (compare

meeting liquidity provisioning will be encoded in a cost on the LCR. If the LCR, *x > l*

solution of the value function and that case by itself is interesting (see, for instance, [10] for

*r* )+(*l*

In this section, we provide numerical-quantitative results about LCRs and their connections with HQLAs and NCOs to supplement the theoretical-quantitative treatment in Sections 2 and 3 (see [13] for more details). More precisely, we describe the LCR data and descriptive

that *b*2(*u*2) *>* 0. For Theorem 3.4 we have selected the cost function *b*2(*u*2) = <sup>1</sup>

2 (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

*t*) *x*

*<sup>t</sup>*)*qt*/*c*2, (33)

*<sup>t</sup>*)2*qt* +

∗ = *J*(*g*∗) = **E**[*v*(*t*0, *x*0)]. (37)

*<sup>r</sup>*, may be 1 that is the threshold for the LCR standard. The cost on

*<sup>r</sup>* <sup>−</sup> *<sup>x</sup>*) <sup>−</sup> <sup>1</sup>],

*<sup>r</sup>* are penalized lower than those with *x < l*

*<sup>r</sup>* (see, for instance, [10]).

*<sup>r</sup>*, then there may be a positive cost. We have selected the cost function

*<sup>r</sup>*)<sup>2</sup> in Theorem 3.4 given by (29). This is also done to obtain an analytic

*<sup>r</sup>*, then there should be a strictly negative cost. If, on the

*<sup>t</sup>* , *<sup>g</sup>*3<sup>∗</sup> : *<sup>T</sup>* ×X → <sup>R</sup>*k*, (34)

1 2

*lt*], (36)

<sup>2</sup> *<sup>c</sup>*2(*u*2)2, given

*<sup>r</sup>*. For this cost function,

*<sup>r</sup>*. Another

*r*,

*<sup>t</sup>* }}, *otherwise*, <sup>∀</sup>*<sup>k</sup>* <sup>∈</sup> <sup>Z</sup>*n*. (35)

Optimizing Basel III Liquidity Coverage Ratios

http://dx.doi.org/10.5772/58395

81

*We assume that the cost functions satisfy*

$$b^1(\mathbf{x}) = \frac{1}{2}c^1(\mathbf{x} - l^r)^2, \ c^1 \in (0, \infty);$$

$$b^2(\mathbf{u}^2) = \frac{1}{2}c^2(\mathbf{u}^2)^2, \ c^2 \in (0, \infty);\tag{28}$$

$$b^3(\mathbf{x}) = \frac{1}{2}c^3(\mathbf{x} - l^r)^2, \ c^3 \in (0, \infty), \tag{29}$$

$$l^r \in \mathbb{R}, \text{ called the reference value of the LCR}$$

*Define the first-order ODE*

$$\begin{aligned} -\dot{q}\_{t} &= -(q\_{t})^{2}/c^{2} + c^{3} + q\_{l}2(r^{\mathbb{R}}(t) + r\_{t}^{\varepsilon} - r\_{t}^{i} + (\sigma^{\varepsilon})^{2} + (\sigma^{i})^{2}) \\ &+ q\_{l}[-r^{f} - \widetilde{r}\_{t}^{\overline{y}^{T}}\widetilde{C}\_{t}^{-1}\vec{r}\_{t}^{y} + (\sigma^{\varepsilon})^{2} + (\sigma^{i})^{2}], \quad q\_{l\_{1}} = \mathfrak{c}^{1}; \end{aligned} \tag{30}$$

$$\begin{aligned}-\dot{\mathbf{x}}^{r}t &= -\mathbf{c}^{3}(\mathbf{x}^{r}\_{t} - l^{r})/q\_{t} - \mathbf{x}^{r}\_{t}[r^{\mathbf{R}}(t) + r^{\varepsilon}\_{t} - r^{i}\_{t} + (\sigma^{\varepsilon})^{2} + (\sigma^{i})^{2}] \\ &- [u^{1}\_{t} - r^{\varepsilon}\_{t} - (\sigma^{\varepsilon})^{2}] - (\mathbf{x}^{r}\_{t} - 1)((\sigma^{\varepsilon})^{2} + (\sigma^{i})^{2}) - (\sigma^{i})^{2}, \ \mathbf{x}^{r}(t\_{1}) = l^{r}; \end{aligned} \tag{31}$$

$$-\dot{l}\_{t} = -r^{f}l\_{t} + c^{3}(\mathbf{x}\_{t}^{r} - l^{r})^{2} - q\_{t}(\sigma^{\varepsilon})^{2}(\mathbf{x}\_{t}^{r} - 1)^{2} - q\_{t}(\sigma^{\varepsilon})^{2}(\mathbf{x}\_{t}^{r} - 1)^{2} - q\_{t}(\sigma^{i})^{2}(\mathbf{x}\_{t}^{r})^{2},\tag{14}$$

$$l(t\_1) = 0.\tag{32}$$

*The function x<sup>r</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup> *will be called the* LCR reference (process) function*. Then we have that the following hold.*

*(a) There exist solutions to the ordinary differential equations* (30)*,* (31) *and* (32)*. Moreover, for all t* ∈ *T, qt >* 0.

*(b) The optimal control laws are*

Next, we choose a particular cost functions for which an analytic solution can be obtained for the value function and control laws (compare with [10]). The following theorem provides

**Theorem 3.4. (Optimal Bank LCRs with Quadratic Cost Functions):** *Consider the nonlinear optimal stochastic control problem for the simplified LCR system* (13) *formulated in Problem* 3.2*.*

2

)2, *<sup>c</sup>*<sup>1</sup> <sup>∈</sup> (0, <sup>∞</sup>);

*<sup>r</sup>* <sup>∈</sup> R, *called the reference value of the LCR*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>*

*<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup>*

)<sup>2</sup> + (*σ<sup>i</sup>*

*<sup>t</sup>* + (*σ<sup>e</sup>*

*<sup>t</sup>* + (*σ<sup>e</sup>*

)<sup>2</sup> + (*σ<sup>i</sup>*

*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)<sup>2</sup> <sup>−</sup> *qt*(*σ<sup>i</sup>*

*l*(*t*1) = 0. (32)

<sup>R</sup>(*t*) + *r<sup>e</sup>*

<sup>R</sup>(*t*) + *r<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> <sup>1</sup>)((*σ<sup>e</sup>*

*The function x<sup>r</sup>* : *<sup>T</sup>* <sup>→</sup> <sup>R</sup> *will be called the* LCR reference (process) function*. Then we have that*

*(a) There exist solutions to the ordinary differential equations* (30)*,* (31) *and* (32)*. Moreover, for all*

)2(*x<sup>r</sup>*

*<sup>c</sup>*2((*u*2)2(*l*)) + <sup>1</sup>

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

2

*<sup>c</sup>*2(*u*2)2, *<sup>c</sup>*<sup>2</sup> <sup>∈</sup> (0, <sup>∞</sup>); (28)

)<sup>2</sup> + (*σ<sup>i</sup>*

)<sup>2</sup> + (*σ<sup>i</sup>*

)2) <sup>−</sup> (*σ<sup>i</sup>*

)2)

)2]

)2(*x<sup>r</sup> <sup>t</sup>*)2,

)2, *x<sup>r</sup>*

)2], *qt*<sup>1</sup> = *<sup>c</sup>*1; (30)

(*t*1) = *l r*

; (31)

)2, *<sup>c</sup>*<sup>3</sup> <sup>∈</sup> (0, <sup>∞</sup>), (29)

*<sup>c</sup>*3(*xt* <sup>−</sup> *<sup>l</sup> r* )2]*dl*

. (27)

(*<sup>l</sup>* <sup>−</sup> *<sup>t</sup>*0))[ <sup>1</sup>

)<sup>2</sup> exp(−*r<sup>f</sup>*

the optimal control laws for quadratic cost functions.

80 Dynamic Programming and Bayesian Inference, Concepts and Applications

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

*<sup>b</sup>*1(*x*) = <sup>1</sup>

*<sup>b</sup>*2(*u*2) = <sup>1</sup>

*<sup>b</sup>*3(*x*) = <sup>1</sup>

<sup>−</sup> *<sup>q</sup>*˙*<sup>t</sup>* <sup>=</sup> <sup>−</sup>(*qt*)2/*c*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*<sup>3</sup> <sup>+</sup> *qt*2(*<sup>r</sup>*

*<sup>t</sup>* − *l r*

*lt* + *c*3(*x<sup>r</sup>*

*r y t T <sup>C</sup>*<sup>−</sup><sup>1</sup> *<sup>t</sup> <sup>r</sup> y <sup>t</sup>* + (*σ<sup>e</sup>*

*<sup>t</sup>* <sup>−</sup> (*σ<sup>e</sup>*

)/*qt* <sup>−</sup> *<sup>x</sup><sup>r</sup>*

*<sup>t</sup>* − *l r* *t* [*r*

)<sup>2</sup> <sup>−</sup> *qt*(*σ<sup>e</sup>*

)2] <sup>−</sup> (*x<sup>r</sup>*

<sup>+</sup>*qt*[−*r<sup>f</sup>* <sup>−</sup>

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*c*3(*x<sup>r</sup>*

<sup>−</sup>[*u*<sup>1</sup> *<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup>*

+ 1 2 exp(−*r<sup>f</sup>*

*r*

*<sup>c</sup>*1(*x*(*t*1) <sup>−</sup> *<sup>l</sup>*

2

2

2

*l*

*<sup>c</sup>*1(*<sup>x</sup>* <sup>−</sup> *<sup>l</sup> r*

*<sup>c</sup>*3(*<sup>x</sup>* <sup>−</sup> *<sup>l</sup> r*

*Consider the cost function*

*Define the first-order ODE*

<sup>−</sup>*x*˙*<sup>r</sup>*

−˙ *lt* <sup>=</sup> <sup>−</sup>*r<sup>f</sup>*

*the following hold.*

*t* ∈ *T, qt >* 0.

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

*We assume that the cost functions satisfy*

$$\begin{array}{ccccc}\mathbf{u}^{2^\*}\_{t} = -(\mathbf{x} - \mathbf{x}\_{t}^{r})q\_{t}/c^2\\\mathbf{x}^\* = \dots \quad \mathbf{x}^\* \quad \mathbf{x}^\* = \dots \quad \dots \end{array} \tag{33}$$

$$\begin{aligned} \text{g}^{2^{\star}}(t, \mathbf{x}) &= \text{u}^{2^{\star}} \text{ . g}^{2^{\star}} : T \times \mathcal{X} \to \mathbb{R}^{+},\\ \tilde{\pi}\_{t}^{\*} &= -\frac{(\mathbf{x} - \mathbf{x}\_{t}^{\prime})}{\mathbf{x}} \tilde{\mathsf{C}}\_{t}^{-1} \tilde{r}\_{t}^{\prime} \text{ . g}^{3^{\star}} : T \times \mathcal{X} \to \mathbb{R}^{k}, \end{aligned} \tag{34}$$

$$\mathcal{g}^{3,k\*}(t,\mathbf{x}) = \begin{cases} \tilde{\pi}^{k\*} , & \text{if } \tilde{\pi}^{k\*} \in [0,1] , \\ \min\{1, \max\{0, \tilde{\pi}^{k\*}\_{\mathbf{t}}\}\} , & \forall k \in \mathbb{Z}^n. \end{cases} \quad \forall k \in \mathbb{Z}^n. \tag{35}$$

*(c) The value function and the value of the problem are*

$$v(t, \mathbf{x}) = \exp(-r^f(t\_1 - t\_0))[\frac{1}{2}(\mathbf{x} - \mathbf{x}\_t^r)^2 q\_t + \frac{1}{2}l\_t].\tag{36}$$

$$J^\* = J(\mathfrak{g}^\*) = \mathbf{E}[\mathfrak{v}(t\_0, \mathfrak{x}\_0)].\tag{37}$$

For the cost on the cash injection, the function (28) is considered, where the input variable *u*<sup>2</sup> is restricted to the set R+. If *u*<sup>2</sup> *>* 0 then the banks should acquire additional HQLAs. The cost function should be such that cash injections are maximized, hence *u*<sup>2</sup> *>* 0 should imply that *b*2(*u*2) *>* 0. For Theorem 3.4 we have selected the cost function *b*2(*u*2) = <sup>1</sup> <sup>2</sup> *<sup>c</sup>*2(*u*2)2, given by (28). Here, both positive and negative values of *u*<sup>2</sup> are penalized equally. An important reason for this is that an analytic solution of the value function can be determined (compare with [10]).

The reference process, *l <sup>r</sup>*, may be 1 that is the threshold for the LCR standard. The cost on meeting liquidity provisioning will be encoded in a cost on the LCR. If the LCR, *x > l r*, is strictly larger than a set value *l <sup>r</sup>*, then there should be a strictly negative cost. If, on the other hand, *x < l <sup>r</sup>*, then there may be a positive cost. We have selected the cost function *b*3(*x*) = <sup>1</sup> <sup>2</sup> *<sup>c</sup>*3(*<sup>x</sup>* <sup>−</sup> *<sup>l</sup> <sup>r</sup>*)<sup>2</sup> in Theorem 3.4 given by (29). This is also done to obtain an analytic solution of the value function and that case by itself is interesting (see, for instance, [10] for more details). Another cost function that we can consider is

$$b^3(\mathbf{x}) = c^3[\exp(\mathbf{x} - l^r) + (l^r - \mathbf{x}) - 1]\nu$$

that is strictly convex and asymmetric in *x* with respect to the value *l <sup>r</sup>*. For this cost function, it is reasonable that costs with *x > l <sup>r</sup>* are penalized lower than those with *x < l <sup>r</sup>*. Another cost function considered is to keep *b*3(*x*) = 0 for *x < l <sup>r</sup>* (see, for instance, [10]).
