Taxation  

• Loss-carry-forward: tax paid at the maximum
• What is the best tax rate to maximise revenue?

Risk process  

• Risk process: $X_t=X_0+\mathit{ct}+{\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{N_t}{\xi }_i,$ with $N_t$ Poisson process, ${\xi }_i$ i.i.d.
• Can include Brownian part and fast, small jumps : Lévy process
• Wealth of insurance company: $c$ is premium rate, ${\xi }_i$ are claims
• Intrinsic motivation: continuous equivalent of random walk

Reflected risk process  

$$\begin{array}{llll}\hfill Y_t& =X_t-\left({\bar{X}}_t-{\bar{X}}_0\right),& \hfill & \\ \hfill {\bar{X}}_t& =\underset{s\le t}{\sup <!--\; FUNCTION\; APPLICATION\; -->}{X}_s\vee {\bar{X}}_0& \hfill & \end{array}$$

Tax process – partially reflected  

$$V_t=X_t-{\int \; <!--\; nolimits\; -->}_0^t\delta \left(V_s\right)d{\bar{X}}_s,{\bar{X}}_t=\underset{s\le t}{\sup <!--\; FUNCTION\; APPLICATION\; -->}{X}_s\vee {\bar{X}}_0$$

• $\delta$ is the tax rate in loss-carry-forward taxation
• Introduces spatial dependence in a tractable way
• Questions: existence? computation of functionals? optimal $\delta$?

Types of tax rate  

Two ideas in literature:

• $V_t=X_t-{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^t\delta \left(V_s\right)d{\bar{X}}_s$ – not obvious it exists – $\left(V,\bar{V}\right)$ is Markov
• $V_t^{\prime }=X_t-{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^t\gamma \left(X_s\right)d{\bar{X}}_s$ – clear that it exists – $\left(V^{\prime },{\bar{V}}^{\prime }\right)$ is not Markov

Existence  

Theorem – Al Ghanim, Loeffen, W. (2020)

If the differential equation

$$y^{\prime }\left(t\right)=1-\delta \left(y\left(t\right)\right),y\left(0\right)=\bar{x},$$

has a unique solution, then the equation

$$V_t=X_t-{\int \; <!--\; nolimits\; -->}_0^t\delta \left(V_s\right)d{\bar{X}}_s,X_0=x,{\bar{X}}_0=\bar{x},$$

has a unique solution $V$.

• Let $\gamma \left(s\right)=\delta \left(y\left(s-\bar{x}\right)\right)$.
• $V_t^{\prime }=X_t-{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^t\gamma \left(X_s\right)d{\bar{X}}_s$ satisfies the equation for $V$: existence
• Converse direction: uniqueness
• Core idea: look at $V$ and $X$ while they are drifting at the maximum

Functionals  

$$\begin{array}{c}\text{Present value of tax collected until exiting }\left[0,a\right]\text{:}\\ w\left(x,\bar{x}\right)=𝔼\left[{\int \; <!--\; nolimits\; -->}_0^{{\tau }_0^a}{e}^{-\mathit{qs}}\delta \left(V_s\right)d{\bar{X}}_s<mfenced\; close=""\; separators=""\; open="|"><mrow></mrow></mfenced>X_0=x,{\bar{X}}_0=\bar{x}\right],\\ {\tau }_0^a=\inf <!--\; FUNCTION\; APPLICATION\; -->\left\{t\ge 0:V_t\notin \left[0,a\right]\right\},\end{array}$$

Functionals: computation  

$$w\left(x,\bar{x}\right)=𝔼\left[{\int \; <!--\; nolimits\; -->}_0^{{\tau }_0^a}{e}^{-\mathit{qs}}\delta \left(V_s\right)d{\bar{X}}_s<mfenced\; close=""\; separators=""\; open="|"><mrow></mrow></mfenced>X_0=x,{\bar{X}}_0=\bar{x}\right],$$

Theorem – Al Ghanim, Loeffen, W. (2020+)

Assume $\left(x,\bar{x}\right)\mapsto f\left(x,\bar{x}\right)$, for $x\le \bar{x}$, satisfies:

• $f\left(x,\bar{x}\right)=0$ if $x<0$ or $x\ge a$,
• $\left(\mathcal{L}-q\right)f\left(x,\bar{x}\right)=0$ if $0\le x\le a$,
• $\delta \left(\bar{x}\right){\partial }_{x}f\left(\bar{x},\bar{x}\right)+\left(\delta \left(\bar{x}\right)-1\right){\partial }_{\bar{x}}f\left(\bar{x},\bar{x}\right)=\delta \left(\bar{x}\right)$

where $\mathcal{L}$ is the generator of $X$ (acting on coordinate $x$). Then, $f=w$.

$\delta \cdot {\partial }_{x}g+\left(\delta -1\right)\cdot {\partial }_{\bar{x}}g=0$ for $x=\bar{x}$ is the condition for $g$ to be in the domain of the generator of $\left(V,\bar{V}\right)$

Functionals: discussion  

• Apply to get

  $$w\left(x,\bar{x}\right)=\frac{W^{\left(q\right)}\left(x\right)}{W^{\left(q\right)}\left(\bar{x}\right)}{\int \; <!--\; nolimits\; -->}_{\bar{x}}^a\exp <!--\; FUNCTION\; APPLICATION\; -->\left(-{\int \; <!--\; nolimits\; -->}_{\bar{x}}^y\frac{W^{\left(q\right)\prime }\left(r\right)}{W^{\left(q\right)}\left(r\right)\left(1-\delta \left(r\right)\right)}dr\right)\frac{\delta \left(y\right)}{1-\delta \left(y\right)}dy$$

  where $W^{\left(q\right)}$ are scale functions (with known Laplace transform).

• Not the only way, but:
  • Flexible
  • Can guess $w$ with some dodgy method, and verify with the theorem
  • Reduces need for excursion theory or approximation by bounded variation processes

Taxation with bail-outs  

Optimal control  

• What is the best tax rate to maximise revenue?
• For $H$ a predictable process, define ${\mathbb{P}}^H$, under which $V_t=X_t-{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^t{H}_{s}d{\bar{X}}_s$
• If $H_t=\delta \left(V_t\right)$, this is a tax process

$$H^{\ast }={\mathrm{argmax}<!--\; FUNCTION\; APPLICATION\; -->}_H{𝔼}^H\left[{\int \; <!--\; nolimits\; -->}_0^{{\tau }_0^{\infty }}{e}^{-\mathit{qs}}{H}_{s}d{\bar{X}}_s\right]?$$

Optimal control  

$$H^{\ast }={\mathrm{argmax}<!--\; FUNCTION\; APPLICATION\; -->}_H{𝔼}^H\left[{\int \; <!--\; nolimits\; -->}_0^{{\tau }_0^{\infty }}{e}^{-\mathit{qs}}{H}_{s}d{\bar{X}}_s\right]$$

• If totally free to choose: reflection above (at some level), i.e. $H^{\ast }=\bar{X}\vee b$
• If tax rate constrained to $H_t\in \left[\alpha ,\beta \right]$: switch from $\alpha$ to $\beta$ when $V$ crosses some level $b$, i.e. $H_t^{\ast }=\delta \left(V_t\right)$,

  $$\delta \left(x\right)=\left\{\begin{array}{cc}\alpha ,\hfill & x\le b,\hfill \\ \beta ,\hfill & x>b\hfill \end{array}\right.$$

• Extension to maximising tax revenue minus bail-out cost, etc. – many possibilities
• Solutions are explicit (in terms of scale functions)

Optimal control: explicit result  

• Assumption: the tail of the Lévy measure is log-convex
• Think of the Lévy measure as $\mathit{\lambda F}\left(dx\right)$, where $\lambda$ is the rate of the Poisson process of jumps, and $F$ is the distribution of the jumps
• Optimal taxation is of form

  $$\delta \left(x\right)=\left\{\begin{array}{cc}\alpha ,\hfill & x\le b,\hfill \\ \beta ,\hfill & x>b\hfill \end{array}\right.$$

• Let

  $$R\left(x\right)={\int \; <!--\; nolimits\; -->}_x^{\infty }\frac{W^{\left(q\right)}{\left(s\right)}^{1-\frac{1}{1-\beta }}{W}^{\left(q\right)\mathit{\prime \prime }}\left(s\right)}{W^{\left(q\right)\prime }{\left(s\right)}^2}ds$$

• $b=0$ iff $R\left(0\right)\ge 0$, otherwise $b$ is the unique root of $R\left(b\right)=0$

A quirk, or, didn’t someone do this 8 years ago?  

• Wang and Hu (IME, 2012): optimal control of tax rates $\gamma \left(X_s\right)$
• Their solution is equivalent to ours
• But it is expressed in terms of an optimal ${\gamma }^{\ast }\left(X_s\right)$
• Years later we showed $\gamma \left(X_s\right)$ and $\delta \left(V_s\right)$ tax rates are equivalent (1st theorem from today)
• We optimise over all tax rates (predictable $H_s$)

Optimal control: example  

• $X_t=ct+{\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{N_t}{\xi }_i,$
• $N_t$ a $\lambda$-Poisson process; $-{\xi }_i$ i.i.d. exponential with rate $\mu$ (mean $1∕\mu$)
• $c=\left(1+𝜃\right)\lambda ∕\mu$
• $𝜃$ is ‘loading factor’: $𝔼\left[X_1\mid X_0=0\right]=\mathit{𝜃\lambda 𝔼}\left[-{\xi }_1\right]=\mathit{𝜃\lambda }∕\mu$,
• $\mathbb{P}\left(X_t>0\text{ forever}\mid X_0=0\right)=1-\frac{1}{1+𝜃}$

Further reading  

References