**3.1. The optimal bank LCR problem**

As in [10], in our contribution, we choose to determine a control law *g*(*t*, *xt*) that minimizes the cost function *<sup>J</sup>* : <sup>G</sup>*<sup>A</sup>* <sup>→</sup> <sup>R</sup>+, where <sup>G</sup>*<sup>A</sup>* is the class of admissible control laws

$$\mathcal{G}\_A = \{ \mathcal{g} : T \times \mathcal{X} \to \mathcal{U} | \mathcal{g} \text{ Borel measurable} \\ \text{\(\mathcal{G}\)-measurable} \\ \text{\(\mathcal{G}\)-measurable} \\ \text{\(\mathcal{G}\)-measurable} \\ \text{\(\mathcal{G}\)-measurable}$$

with the closed-loop system for *g* ∈ G*<sup>A</sup>* being given by

*d*(*x*<sup>2</sup>

the SDE (13), the term

*<sup>t</sup>*)−<sup>1</sup> <sup>=</sup> <sup>−</sup>(*x*<sup>2</sup>

*dxt* = *x*<sup>1</sup>

= [*r*

= [−(*x*2)−<sup>1</sup>

76 Dynamic Programming and Bayesian Inference, Concepts and Applications

<sup>−</sup>(*x*<sup>2</sup> *t*)−<sup>1</sup>

*<sup>t</sup> <sup>d</sup>*(*x*<sup>2</sup>

<sup>−</sup>*xt*[*r<sup>i</sup>*

+ *xt<sup>π</sup><sup>T</sup> t* Σ*y*

+[*u*<sup>1</sup> *<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup>

= *xt*[*r*

+ (*σ<sup>e</sup>*

the term <sup>−</sup>(*σe*)<sup>2</sup> <sup>+</sup> *xt*(*σe*)<sup>2</sup> = (*σe*)2(*xt* <sup>−</sup> <sup>1</sup>) appears.

discussion) have not been tested in the literature before.

**3. Optimal Basel III liquidity coverage ratios**

**3.1. The optimal bank LCR problem**

<sup>R</sup>(*t*)*xt* <sup>−</sup> *<sup>r</sup><sup>e</sup>*

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

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

*t*)−2*dx*<sup>2</sup> *<sup>t</sup>* + 1 2

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

*t*)−<sup>1</sup> + (*x*<sup>2</sup>

2(*x*2)−<sup>3</sup>

*t*)+(*x*<sup>2</sup>

 *dWt*,

*t*)−1*dx*<sup>1</sup>

*<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup>

*<sup>t</sup>* <sup>−</sup>*σe*(<sup>1</sup> <sup>−</sup> *xt*) <sup>−</sup>*σ<sup>i</sup>*

*<sup>t</sup>* + (*σ<sup>e</sup>*

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

<sup>0</sup> <sup>−</sup>*σ<sup>e</sup> <sup>σ</sup><sup>i</sup>*

*<sup>t</sup>* + *<sup>u</sup>*<sup>1</sup>

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

)2(<sup>1</sup> <sup>−</sup> *xt*)<sup>2</sup> + (*σ<sup>i</sup>*

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

−*re <sup>t</sup>* + *xtr<sup>e</sup>*

*<sup>t</sup>* ] + *xt*((*σ<sup>e</sup>*

*<sup>t</sup> <sup>d</sup> <sup>&</sup>lt; <sup>x</sup>*2, *<sup>x</sup>*<sup>2</sup> *<sup>&</sup>gt; <sup>t</sup>*

)<sup>2</sup> + (*σ<sup>i</sup>*

*<sup>t</sup>* + *<sup>d</sup> < <sup>x</sup>*1,(*x*2)−<sup>1</sup> *> <sup>t</sup>*

)2) <sup>−</sup> (*σ<sup>e</sup>*

)<sup>2</sup> + *r y t T*

)2(*xt*)<sup>2</sup> + (*xt*)2*<sup>π</sup><sup>T</sup>*

*<sup>t</sup>*(*xt* − 1),

*xt dWt* )2)]*dt*

)2]*dt*

*<sup>π</sup><sup>t</sup>*]*dt* <sup>+</sup>

*<sup>t</sup> <sup>C</sup><sup>t</sup><sup>π</sup><sup>t</sup>*

1/2

*dWt*,

*t*)−1((*σ<sup>e</sup>*

*<sup>t</sup>* <sup>+</sup> *xt<sup>r</sup> y t πt*

)<sup>2</sup> + (*σ<sup>i</sup>*

)2]*dt*

for stochastic *W* : Ω × *T* → R that is a standard Brownian motion. Note that in the drift of

*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*r<sup>e</sup>*

appears because it models the effect of depreciation of both HQLAs and NCOs. Similarly,

The predictions made by our previously constructed model are consistent with the empirical evidence in contributions such as [13]. For instance, in much the same way as we do, [13] describes how NCOs affect LCRs. On the other hand, to the best of our knowledge, the modeling related to collateral and LCR reference processes (see Section 3 for a comprehensive

In order to determine an optimal cash injection rate (seen as an adjustment to the normal provisioning rate) and HQLA allocation strategy, it is imperative that a well-defined objective function with appropriate constraints is considered. The choice has to be carefully made in order to avoid ambiguous solutions to our stochastic control problem (compare with [10]).

As in [10], in our contribution, we choose to determine a control law *g*(*t*, *xt*) that minimizes

the cost function *<sup>J</sup>* : <sup>G</sup>*<sup>A</sup>* <sup>→</sup> <sup>R</sup>+, where <sup>G</sup>*<sup>A</sup>* is the class of admissible control laws

)<sup>2</sup> + (*σ<sup>i</sup>*

$$d\mathbf{x}\_{l} = A\_{l}\mathbf{x}\_{l}dt + \sum\_{j=0}^{n} \mathbf{B}^{j} \mathbf{x}\_{l} \mathbf{g}^{j}(t, \mathbf{x}\_{l})dt + a\_{l}dt$$

$$+ \sum\_{j=1}^{3} \mathbf{M}^{j}(\mathbf{g}(t, \mathbf{x}\_{l})) \mathbf{x}\_{l} dW\_{l'}^{j} \quad \mathbf{x}(t\_{0}) = \mathbf{x}\_{0}.\tag{15}$$

Furthermore, the cost function, *<sup>J</sup>* : <sup>G</sup>*<sup>A</sup>* <sup>→</sup> <sup>R</sup>+, of the LCR problem is given by

$$J(\mathbf{g}) = \mathbf{E} \left[ \int\_{t\_0}^{t\_1} \exp(-r^f (l - t\_0)) b(l, \mathbf{x}\_{l\prime} \mathbf{g}(l, \mathbf{x}\_l)) dl \right]$$

$$+ \exp(-r^f (t\_1 - t\_0)) b^1(\mathbf{x}(t\_1)) \big], \tag{16}$$

where *<sup>g</sup>* ∈ G*A*, *<sup>T</sup>* = [*t*0, *<sup>t</sup>*1] and *<sup>b</sup>*<sup>1</sup> : X → <sup>R</sup><sup>+</sup> is a Borel measurable function (compare with [10]). Furthermore, *<sup>b</sup>* : *<sup>T</sup>* ×X ×U → <sup>R</sup><sup>+</sup> is formulated as

$$b(t, \mathbf{x}, \boldsymbol{\mu}) = b^2(\boldsymbol{\mu}^2) + b^3(\mathbf{x}^1/\mathbf{x}^2)\_{\boldsymbol{\mu}}$$

for *<sup>b</sup>*<sup>2</sup> : <sup>U</sup><sup>2</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>+. Also, *<sup>r</sup><sup>f</sup>* <sup>∈</sup> R is called the *NCO forecasting rate*, where *b*1, *b*<sup>2</sup> and *b*<sup>3</sup> are chosen below. In order to clarify the stochastic problem, the following assumption is made.

**Assumption 3.1. (Admissible Class of Control Laws)** *Assume that* G*<sup>A</sup>* = ∅.

We are now in a position to state the stochastic optimal control problem for a continuous-time LCR model that we solve (compare with [10]). The said problem may be formulated as follows.

**Problem 3.2. (Optimal Bank LCR Problem)** *Consider the stochastic system* (15) *for the LCR problem with the admissible class of control laws,* G*A*, *given by* (14) *and the cost function, J* : G*<sup>A</sup>* → R+, *given by* (16)*. Solve*

$$\inf\_{\mathcal{g}\in\mathcal{G}\_A} J(\mathcal{g})\_\prime$$

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

$$J^\* = \inf\_{\mathcal{g} \in \mathcal{G}\_A} J(\mathcal{g})\_{\prime\prime}$$

*1. The cost function is assumed to satisfy*

*by*

*<sup>b</sup>*2(*u*2) <sup>∈</sup> *<sup>C</sup>*2( <sup>R</sup>),

<sup>0</sup> <sup>=</sup> *Dtv*(*t*, *<sup>x</sup>*) + <sup>1</sup>

+*x*(*r*

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

+*u*<sup>2</sup> *t* ∗

*<sup>v</sup>*(*t*1, *<sup>x</sup>*) = exp(−*r<sup>f</sup>*

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

*g*2∗(*t*, *x*) = *u*2<sup>∗</sup>

*Furthermore, the value of the problem is*

*<sup>π</sup>*<sup>∗</sup> <sup>=</sup> <sup>−</sup> *Dxv*(*t*, *<sup>x</sup>*)

*xDxxv*(*t*, *x*)

*J*

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

*Then the optimal control law is*

<sup>+</sup> exp(−*r<sup>f</sup>*

*Du*<sup>2</sup>*u*<sup>2</sup> *<sup>b</sup>*2(*u*2) *<sup>&</sup>gt;* 0, <sup>∀</sup>*u*<sup>2</sup> <sup>∈</sup> R,

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

2 [(*σ<sup>e</sup>*

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

<sup>0</sup> <sup>=</sup> *Dxv*(*t*, *<sup>x</sup>*) + exp(−*r<sup>f</sup>*

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

*Dxv*(*t*, *<sup>x</sup>*) + exp(−*r<sup>f</sup>*

*<sup>t</sup>* + (*σ<sup>e</sup>*

)2]*Dxv*(*t*, *x*)

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

*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*

)2(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> + (*σ<sup>i</sup>*

(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*0))*b*3(*x*) <sup>−</sup> [*Dxv*(*t*, *<sup>x</sup>*)]<sup>2</sup>

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

*<sup>g</sup>*3,*k*∗(*t*, *<sup>x</sup>*) = min{1, max{0, *<sup>π</sup><sup>k</sup>*∗}}, *<sup>g</sup>*3,*k*<sup>∗</sup> : *<sup>T</sup>* ×X → R, (25)

)<sup>2</sup> + (*σ<sup>i</sup>*

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

2*Dxxv*(*t*, *x*)

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

, *<sup>g</sup>*2<sup>∗</sup> : *<sup>T</sup>* ×X → <sup>R</sup>+, (23)

∗ = *J*(*g*∗) = **E**[*v*(*t*, *x*0)]. (26)

*<sup>t</sup>* , (24)

)2(*xt*)2]*Dxxv*(*t*, *x*)

Optimizing Basel III Liquidity Coverage Ratios

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

79

)2)*Dxv*(*t*, *x*)

*t* ∗ )

(*t*<sup>1</sup> <sup>−</sup> *<sup>t</sup>*0))*b*1(*x*), (21)

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

*<sup>t</sup>* , (20)

*<sup>t</sup>*). (22)

*<sup>u</sup>*<sup>2</sup>→−<sup>∞</sup> *Du*<sup>2</sup> *<sup>b</sup>*2(*u*2) = <sup>−</sup>∞, lim *<sup>u</sup>*<sup>2</sup>→+<sup>∞</sup> *Du*<sup>2</sup> *<sup>b</sup>*2(*u*2)=+∞; (19)

lim

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

$$\mathcal{g}^\* = \arg\min\_{\mathcal{g}\in\mathcal{G}\_A} J(\mathcal{g}) \in \mathcal{G}\_A.$$
