Vanishing central bank intervention in stochastic impulse control

Abstract

Stochastic control of exchange rates when a central bank employs anti-inflationary stochastic differential equation (SDE) monetary policy is the key topic of our paper. Despite low money growth SDE policy means exchange rates invariably violate the central bank’s targets. Monetary policy also incorporates interventions reflected by sudden money supply jumps that moderate deviations from targets. Controlling exchange rates involves minimizing target deviation and intervention costs. Restrictions on these costs ensure intervention vanishes under the optimal control, implying the central bank engineers freely floating exchange rates instead of managed floating or fixed exchange rates. Econometric evidence suggests discretionary interventions may be ineffective or generate excess volatility and speculation in currency markets. Our result demonstrates mathematically that such collateral damage discourages intervention.

Appendix

Proof of Lemma 4.1

To see that $E|\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s|<\infty$ first observe that since the $B(\cdot )$ signed measure of a singleton is zero we have $\int _{{\{\tau _{1}\}}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s) dB_s=0$. Recalling that $y_s=x_s$ for all $s<\tau _1$ it follows that almost surely we have

$$\begin{aligned}&\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s \nonumber \\&\quad =\int _{[0,\tau _{1})}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s+\int _{{\{\tau _{1}\}}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s \nonumber \\&\quad =\int _{[0,\tau _{1})}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s=\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s \end{aligned}$$

(6.1)

There exists a sequence ${\{\pi _n\}}_{n=1}^{\infty }$ where $\pi _n={\{t_0^n,t_1^n,\ldots ,t^n_{m_{n}}\}}$ for each $n\ge 1$ such that

• (1) $t_i^n$ is a stopping time for all $n\ge 1$ and all $i=0,\ldots , m_n$ such that $t_i^n\le \tau _1$.

• (2) for all $\omega \in \Omega$ and $n\ge 1$ the set $\pi _n(\omega )={\{t_0^n(\omega ),t_1^n(\omega ),\ldots ,t^n_{m_{n}}(\omega )\}}$ is a partition of $[0,\tau _1(\omega )]$; specifically $0=t_0^n(\omega )\le t_1^n(\omega )\le t_2^n(\omega )\le \cdots \le t_{m_{n}}^n(\omega )=\tau _1(\omega )$.

• (3) for all $\omega \in \Omega$ we have $|\pi _n(\omega )|\equiv \max _{1\le i\le m_{n}}(t_i^n(\omega )-t_{i-1}^n(\omega ))\rightarrow 0$ as $n\rightarrow \infty$.

• (4) given any $n\ge 1$ and $\omega \in \Omega$ there exists an integer $N(\omega )\ge 1$ with $N(\omega )\le m_n$ such that $t_{i-1}^n(\omega )<t^n_i(\omega )$ for all $i\le N(\omega )$ then $t_i^n(\omega )=\tau _1(\omega )$ when $N(\omega )\le i\le m_n$.

Stopping times satisfying these conditions must exist since $\tau _1\le h_1$ where $h_1<\infty$ is a constant. We may take $t_i^n=(\frac{i}{2^{n}})h_1\bigwedge \tau _1$ for $i=0,\ldots , 2^n$ for each $n\ge 1$, for instance.

Now since $(\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s), s\in [0,\tau _1])$ is continuous $\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s$ is a Riemann–Stieltjes integral and thus using ${\{\pi _n\}}_{n=1}^{\infty }$ almost surely

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x}\left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) =\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s \end{aligned}$$

Hence, almost surely

$$\begin{aligned} \lim _{n\rightarrow \infty }\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x}\left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| = \left| \int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s\right| \end{aligned}$$

Applying Fatou’s lemma yields

$$\begin{aligned}&E\left| \int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s\right| =E\left| \int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,y_s)\ dB_s\right| \nonumber \\&\quad \le \liminf _{n\rightarrow \infty }E\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x} \left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| \end{aligned}$$

(6.2)

We will show that $\sup _{n\ge 1}E|\sum _{i=1}^{m_{n}}\exp (-\lambda t_i^n)\frac{\partial \phi }{\partial x}(t_i^n,y_{t_{i}^{n}})(B_{t_{i}^{n}}-B_{t_{i-1}^{n}})|<\infty$ and employing (6.2) this will secure integrability of $\int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s$ since

$$\begin{aligned}&E\left| \int _0^{\tau _{1}}\exp (-\lambda s)\frac{\partial \phi }{\partial x}(s,x_s)\ dB_s\right| \nonumber \\&\quad \le \liminf _{n\rightarrow \infty }E\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x} \left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| \nonumber \\&\quad \le \sup _{n\ge 1}E\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x}\left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| \end{aligned}$$

(6.3)

Let $|b|,|\frac{\partial \phi }{\partial x}|\le L$ for some constant $L>0$. Theorem 3.1 implies there exists a constant $C>0$ such that $|B_{t_{i}^{n}}-B_{t_{i-1}^{n}}|\le C( |t_i^n-t^n_{i-1}|+|k_{t_{i}^{n}}-k_{t_{i-1}^{n}}|^2)$ for all $i=1,\ldots , m_n$ and all $n\ge 1$. Recall Theorem 3.1 can be applied to these oscillations of $(B_t,t\ge 0)$ since the points $t_i^n$ satisfy $0\le t_i^n\le \tau _1\le h_1$ for all $i=0,\ldots , m_n$ and all $n\ge 1$ where $h_1>0$ is a fixed real number; in fact $h_1$ generates C. Also let $M_t=\int _0^t \sigma (s,k_s)\ dW_s$ and observe that $|k_{t_{i}^{n}}-k_{t_{i-1}^{n}}|^2\le 2|\int _{t_{i-1}^{n}}^{t^{n}_{i}} b(s,k_s)\ ds|^2+2|M_{t^{n}_{i}}-M_{t_{i-1}^{n}}|^2\le 2L^2(t_i^n-t^n_{i-1})^2+2|M_{t^{n}_{i}}-M_{t_{i-1}^{n}}|^2$. Note that since $t_0^n=0$ we have $M_{t_{0}^{n}}=0$ for all $n\ge 1$. Combining inequalities yields

$$\begin{aligned}&\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x}\left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| \\&\quad \le \left( CL+2CL^3h_1\right) \sum _{i=1}^{m_{n}}\left( t^n_i-t_{i-1}^n\right) +2CL\sum _{i=1}^{m_{n}}\left| M_{t^{n}_{i}}-M_{t_{i-1}^{n}}\right| ^2 \\&\quad \le \left( CL+2CL^3h_1\right) \tau _1+2CL\sum _{i=1}^{m_{n}}\left| M_{t^{n}_{i}}-M_{t_{i-1}^{n}}\right| ^2 \\&\quad \le \left( CL+2CL^3h_1\right) h_1+2CL\sum _{i=1}^{m_{n}}\left| M_{t^{n}_{i}}-M_{t_{i-1}^{n}}\right| ^2 \end{aligned}$$

Since $(M_t,t\ge 0)$ is a continuous martingale and $t^n_i$ a bounded stopping time for all $i=0,\ldots ,m_n$ and $n\ge 1$, the optional sampling theorem implies that $(M_{t_{i}^{n}},\mathcal {F}_{t_{i}^{n}}, i=0,\ldots ,m_n)$ is a martingale where $\mathcal {F}_{t_{i}^{n}}$ is the stopping time $\sigma$-field generated by $t^{n}_{i}$. Moreover, by Doob’s maximal quadratic inequality $E(M_{t_{i}^{n}})^2\le E\sup _{t\in [0,h_{1}]}|M_t|^2\le 4E|M_{h_{1}}|^2<\infty$ for all $i=0,\ldots ,m_n$ and $n\ge 1$. This implies that $M_{t^{n}_{i}}\in L^2(\Omega ,\mathcal {F},P)$ for all $i=0,\ldots ,m_n$ and $n\ge 1$. Using the martingale property we have

$$\begin{aligned} E\sum _{i=1}^{m_{n}}\left| M_{t^{n}_{i}}-M_{t_{i-1}^{n}}\right| ^2= & {} \sum _{i=1}^{m_{n}} E\left( M_{t^{n}_{i}}-M_{t_{i-1}^{n}}\right) ^2 \\= & {} \sum _{i=1}^{m_{n}}EM_{t^{n}_{i}}^2-EM_{t_{i-1}^{n}}^2=EM_{t^{n}_{m_{n}}}^2 \end{aligned}$$

By the Burkholder–Davis–Gundy inequality there is a constant $J>0$ such that for every $n\ge 1$, $EM_{t^{n}_{m_{n}}}^2=E(\int _0^{t_{m_{n}}^{n}}\sigma (s,k_s)\ dW_s)^2\le JE\int _0^{\tau _{1}}\sigma ^2(s,k_s)\ ds\le JE\int _0^{h_{1}}\sigma ^2(s,k_s)\ ds<\infty$ since $\sigma$ is bounded. Hence for every $n\ge 1$ we have

$$\begin{aligned}&E\left| \sum _{i=1}^{m_{n}}\exp \left( -\lambda t_i^n\right) \frac{\partial \phi }{\partial x}\left( t_i^n,y_{t_{i}^{n}}\right) \left( B_{t_{i}^{n}}-B_{t_{i-1}^{n}}\right) \right| \\&\quad \le \left( CL+2CL^3h_1\right) h_1+2CLJE\int _0^{h_{1}}\sigma ^2\left( s,k_s\right) \ ds \end{aligned}$$

implying $\sup _{n\ge 1}E|\sum _{i=1}^{m_{n}}\exp (-\lambda t_i^n)\frac{\partial \phi }{\partial x}(t_i^n,y_{t_{i}^{n}})(B_{t_{i}^{n}}-B_{t_{i-1}^{n}})|<\infty$ as desired. $\square$