**2.2. Description of the simplified LCR model**

**Definition 2.4. (Stochastic System for the LCR Model)** *Define the* stochastic system for the

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

0 *r<sup>i</sup> <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup> t* ;

0 0

*t*

<sup>0</sup> <sup>−</sup>*x*<sup>2</sup>

*ut* = *u*2 *t πt* 

*At* = 

> *x*2 *<sup>t</sup> <sup>x</sup>*<sup>1</sup> *t r y t T*

 *x*1 *t πT t* Σ*y <sup>t</sup>* <sup>−</sup>*x*<sup>2</sup>

 

> *x*2 *t* 0 *u*2 *<sup>t</sup>* + *x*1 *t r y t T*

:= *B*0*xtu*<sup>0</sup>

*<sup>t</sup>* + *n* ∑ *j*=1

:= 0 1 0 0 *xtu*<sup>3</sup> *<sup>t</sup>* + *n* ∑ *j*=1

:= *n* ∑ *j*=0 [*Bj xt*]*u<sup>j</sup> t*;

*Wy t W<sup>e</sup> t W<sup>i</sup> t*

 ,

*N*(*xt*) =

*S*(*xt*, *ut*) =

*<sup>t</sup> >* 0, *<sup>σ</sup><sup>i</sup>*

*N*(*xt*)*ut* :=

*Wt* =

*dxt* = *Atxtdt* + *N*(*xt*)*utdt* + *atdt* + *S*(*xt*, *ut*)*dWt*, (11)

, *<sup>u</sup>* : <sup>Ω</sup> <sup>×</sup> *<sup>T</sup>* <sup>→</sup> <sup>R</sup>*n*<sup>+</sup>1,

; *at* =

*<sup>t</sup> <sup>σ</sup><sup>e</sup>* <sup>0</sup>

*<sup>t</sup> <sup>σ</sup><sup>e</sup> <sup>x</sup>*<sup>2</sup> *t σi* ;

*<sup>t</sup> are mutually (stochastically) independent standard Brownian motions. It is*

 *πt*

> *xtπj t*

 *πj t*

*<sup>t</sup> >* 0 *and C<sup>t</sup> >* 0 *(compare with [10]).*

0

 *r y*,*j <sup>t</sup>* 0 0 0

 *x*1 *t r y*,*j t* 0

 *x*2 *t u*1 *t* 0

 ;

LCR model *as*

*where W<sup>y</sup>*

and

*<sup>t</sup>* , *<sup>W</sup><sup>e</sup>*

*assumed that for all t* <sup>∈</sup> *<sup>T</sup>*, *<sup>σ</sup><sup>e</sup>*

We can rewrite (11) as follows.

*<sup>t</sup> and W<sup>i</sup>*

*with the various terms in this SDE being*

74 Dynamic Programming and Bayesian Inference, Concepts and Applications

The model can be simplified if attention is restricted to the system with the LCR, as stated earlier, denoted in this section by *lt* = *x*<sup>1</sup> *<sup>t</sup>* .(*x*<sup>2</sup> *<sup>t</sup>*)−<sup>1</sup> (compare with [10]).

**Definition 2.5. (Stochastic Model for a Simplified LCR)** *Define the* simplified LCR system *by the SDE*

$$\begin{split} d\mathbf{x}\_{t} &= \mathbf{x}\_{t} [r^{\mathbf{R}}(t) + r\_{t}^{\varepsilon} - r\_{t}^{i} + (\sigma^{\varepsilon})^{2} + (\sigma^{i})^{2} + \widetilde{r}\_{t}^{\overline{y}^{T}} \widetilde{\boldsymbol{\pi}}\_{t}] dt \\ &+ [\boldsymbol{u}\_{t}^{1} + \boldsymbol{u}\_{t}^{2} - \boldsymbol{r}\_{t}^{\varepsilon} - (\sigma^{\varepsilon})^{2}] dt \\ &+ [(\sigma^{\varepsilon})^{2} (1 - \mathbf{x}\_{t})^{2} + (\sigma^{i})^{2} \mathbf{x}\_{t}^{2} + \mathbf{x}\_{t}^{2} \widetilde{\boldsymbol{\pi}}\_{t}^{T} \widetilde{\boldsymbol{\mathcal{C}}}\_{t} \widetilde{\boldsymbol{\pi}}\_{t}]^{1/2} d\overline{\boldsymbol{W}}\_{t} \quad \mathbf{x}(t\_{0}) = \mathbf{x}\_{0}. \end{split} \tag{13}$$

The model is derived as follows. The starting point is the two-dimensional SDE for *x* = (*x*1, *x*2)*<sup>T</sup>* as in the equations (9) and (10). Next, we use the Itô's formula (see, for instance, [15]) to determine

*d*(*x*<sup>2</sup> *<sup>t</sup>*)−<sup>1</sup> <sup>=</sup> <sup>−</sup>(*x*<sup>2</sup> *t*)−2*dx*<sup>2</sup> *<sup>t</sup>* + 1 2 2(*x*2)−<sup>3</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>* = [−(*x*2)−<sup>1</sup> *<sup>t</sup>* (*r<sup>i</sup> <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup> t*)+(*x*<sup>2</sup> *t*)−1((*σ<sup>e</sup>* )<sup>2</sup> + (*σ<sup>i</sup>* )2)]*dt* <sup>−</sup>(*x*<sup>2</sup> *t*)−<sup>1</sup> <sup>0</sup> <sup>−</sup>*σ<sup>e</sup> <sup>σ</sup><sup>i</sup> dWt*, *dxt* = *x*<sup>1</sup> *<sup>t</sup> <sup>d</sup>*(*x*<sup>2</sup> *t*)−<sup>1</sup> + (*x*<sup>2</sup> *t*)−1*dx*<sup>1</sup> *<sup>t</sup>* + *<sup>d</sup> < <sup>x</sup>*1,(*x*2)−<sup>1</sup> *> <sup>t</sup>* = [*r* <sup>R</sup>(*t*)*xt* <sup>−</sup> *<sup>r</sup><sup>e</sup> <sup>t</sup>* + *<sup>u</sup>*<sup>1</sup> *<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup> *<sup>t</sup>* <sup>+</sup> *xt<sup>r</sup> y t πt* <sup>−</sup>*xt*[*r<sup>i</sup> <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup> <sup>t</sup>* ] + *xt*((*σ<sup>e</sup>* )<sup>2</sup> + (*σ<sup>i</sup>* )2) <sup>−</sup> (*σ<sup>e</sup>* )2]*dt* + *xt<sup>π</sup><sup>T</sup> t* Σ*y <sup>t</sup>* <sup>−</sup>*σe*(<sup>1</sup> <sup>−</sup> *xt*) <sup>−</sup>*σ<sup>i</sup> xt dWt* = *xt*[*r* <sup>R</sup>(*t*) + *r<sup>e</sup> <sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>i</sup> <sup>t</sup>* + (*σ<sup>e</sup>* )<sup>2</sup> + (*σ<sup>i</sup>* )<sup>2</sup> + *r y t T <sup>π</sup><sup>t</sup>*]*dt* <sup>+</sup> +[*u*<sup>1</sup> *<sup>t</sup>* + *<sup>u</sup>*<sup>2</sup> *<sup>t</sup>* <sup>−</sup> *<sup>r</sup><sup>e</sup> <sup>t</sup>* <sup>−</sup> (*σ<sup>e</sup>* )2]*dt* + (*σ<sup>e</sup>* )2(<sup>1</sup> <sup>−</sup> *xt*)<sup>2</sup> + (*σ<sup>i</sup>* )2(*xt*)<sup>2</sup> + (*xt*)2*<sup>π</sup><sup>T</sup> <sup>t</sup> <sup>C</sup><sup>t</sup><sup>π</sup><sup>t</sup>* 1/2 *dWt*,

G*<sup>A</sup>* = {*g* : *T* ×X → U|*g* Borel measurable (14) and there exists an unique solution to the closed-loop system},

(*t*, *xt*)*dt* + *atdt*

(*l* − *t*0))*b*(*l*, *xl*, *g*(*l*, *xl*))*dl*

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

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

*b*(*t*, *x*, *u*) = *b*2(*u*2) + *b*3(*x*1/*x*2),

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

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

**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>* →

> inf *g*∈G*<sup>A</sup>*

*J*(*g*),

*<sup>t</sup>* , *x*(*t*0) = *x*0. (15)

Optimizing Basel III Liquidity Coverage Ratios

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

77

, (16)

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

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

[10]). Furthermore, *<sup>b</sup>* : *<sup>T</sup>* ×X ×U → <sup>R</sup><sup>+</sup> is formulated as

assumption is made.

R+, *given by* (16)*. Solve*

follows.

*dxt* = *Atxtdt* +

+ 3 ∑ *j*=1 *Mj*

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

<sup>+</sup> exp(−*r<sup>f</sup>*

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

*n* ∑ *j*=0 *Bj xtg<sup>j</sup>*

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

exp(−*r<sup>f</sup>*

(*g*(*t*, *xt*))*xtdW<sup>j</sup>*

for stochastic *W* : Ω × *T* → R that is a standard Brownian motion. Note that in the drift of the SDE (13), the term

$$-r\_t^\varepsilon + \mathfrak{x}\_t r\_t^\varepsilon = -r\_t^\varepsilon (\mathfrak{x}\_t - 1)\_\prime$$

appears because it models the effect of depreciation of both HQLAs and NCOs. Similarly, the term <sup>−</sup>(*σe*)<sup>2</sup> <sup>+</sup> *xt*(*σe*)<sup>2</sup> = (*σe*)2(*xt* <sup>−</sup> <sup>1</sup>) appears.

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 discussion) have not been tested in the literature before.
