Optimal Stopping Time Strategy for Paying Tax

This paper attempts to study the optimal stopping time for semi-Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the expected discounted cost functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and ε-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end.

In this paper, we study the optimal multiple stopping problem under the filtration-consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions, rather than a process that is right-continuous with left limits. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.

In this thesis we investigate optimal stopping problems with expectation cost constraints. We focus on reducing the set of stopping times as well as on deriving a partial differential equation for the value function. If the process to stop is a time-homogeneous Ito-process, we show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. In addition, we prove a classical verification theorem and apply it to several examples. Furthermore, we consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures. We apply the results to analyze a sequential testing problem and show that in this problem the optimal stopping times are given by at most two consecutive exit times of intervals. Finally, using the theory of Tchebycheff systems we examine when we can reduce the set of stopping times in the constrained problem to first exit times of intervals. In this case, the law of the process at the stopping time is a weighted sum of at most 2 Dirac measures.

Optimal stopping problems for multiarmed bandit processes with arms' independence

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $\phi$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$. This new reward $\phi$ is not a right-continuous adapted process as in the classical case, but a family of random variables. For such a reward, we prove a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).

Optimal double stopping time problem

Let $B=(B_t)_{0\le t\le 1}$ be a standard Brownian motion and $\theta$ be the moment at which B attains its maximal value, i.e., $B_\theta=\max_{0\le t\le 1}B_t$. Denote by $({\cal F}^B_t)_{0\le t\le 1}$ the filtration generated by B. We prove that for any $({\cal F}^B_t)$-stopping time $\tau$ $(0\le\tau\le 1)$, the following equality holds: $$ {\bf E}(B_\theta-B_\tau)^2={\bf E}|\theta-\tau|+{\frac{1}{2}}. $$ Together with the results of [S. E. Graversen, G. Peskir, and A. N. Shiryaev, {\em Theory Probab. Appl.}, 45 (2000), pp. 41--50] this implies that the optimal stopping time $\tau_*$ in the problem $$ \inf_\tau{\bf E}|\theta-\tau| $$ has the form $$ \tau_*=\inf\big\{0\le t\le 1: S_t-B_t\ge z_*\sqrt{1-t}\,\big\}, $$ where $S_t=\max_{0\le s\le t}B_s$, $z_*$ is a unique positive root of the equation $4\Phi(z)-2z\phi(z)-3=0$, and $\phi(z)$ and $\Phi(z)$ are the density and the distribution function of a standard Gaussian random variable. Similarly, we solve the optimal stopping problems $$ \inf_{\tau\in{\mathfrak{M}}_\alpha}{\bf E}(\tau-\theta)^+ \quad\mbox{and}\quad \inf_{\tau\in{\mathfrak{N}}_\alpha}{\bf E}(\tau-\theta)^-, $$ where ${\mathfrak{M}}_\alpha=\{\tau :\,{\bf E}(\tau-\theta)^-\le \alpha\}$, and ${\mathfrak{N}}_\alpha=\{\tau :\,{\bf E}(\tau-\theta)^+\le\alpha\}$. The corresponding optimal stopping times are of the same form as above (with other $z_*$'s).

Partial differential equation filters are well-known for their good processing results, but we must first solve an important question on optimal parameters selection because it affects the denoising results and the stability of the equation directly, which include the selection of diffusivity and optimal stopping time. This paper gets a right optimal stopping time in linear model by analyzing its analytical solution, and also decides the selection of diffusivity and optimal stopping time in nonlinear model from the aspects of both theory and application. The practical experiment results show that the selected parameters are best.

We study the optimal stopping time problem $v(S)={\rm ess}\sup_{\theta \geq S} E[\phi(\theta)|\mathcal {F}_S]$, for any stopping time $S$, where the reward is given by a family $(\phi(\theta),\theta\in\mathcal{T}_0)$ \emph{of non negative random variables} indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose $v(0) < + \infty$ and $ (\phi(\theta),\theta\in \mathcal{T}_0)$ upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory.

PurposeThe purpose of this paper is to investigate how to determine optimal investing stopping time in a stochastic environment, such as with stochastic returns, stochastic interest rate and stochastic expected growth rate.Design/methodology/approachTransformation method was used for solving optimal stopping problem by providing a way to transform path‐dependent problem into a path‐independent one. Based on option pricing theory, optimal investing stopping time was thought of as an optimal executed timing problem of American‐style option.FindingsFirst, the authors transform a path‐dependent stop timing problem to a path‐independent one with transformation under very general conditions, to directly use the existing conclusion of optimal stopping time literature. Second, when dynamics of capital growth is homogeneous, the authors changed the two dimensional optimal stop timing problem into a single dimension problem based on the assumption of zero exercise costs. Third, the authors investigated the comparative dynamics about asset selling boundary on asset value, state variable and return predictability. With constant discount rate and growth rate, the optimal selling timing depends on the simple comparison between capital cost and growth rate.Originality/valueThe paper's contributions to analysis method may be as follows. The authors demonstrate how to transform a path‐dependent stopping problem into a path‐independent one under general conditions. The transform method in this article can be applied to other path‐dependent optimal stopping problems. In particular, a Riccati ordinary differential equation for the transformation is set up. In most examples commonly met in finance, the equation can be solved explicitly.

In this paper, we consider the optimal stopping problems on semi-Markov processes (sMPs) with finite horizon and aim to establish the existence and algorithm of optimal stopping times. The key method is the equivalence between optimal stopping problems on sMPs and a special class of semi-Markov decision processes (sMDPs). We first introduce the optimality equation and show the existence of the optimal policies of finite-horizon sMDPs with additional terminal costs. Based on the optimal stopping problems on sMPs, we give an explicit construction of sMDPs such that the optimal stopping times of sMPs are equivalent to the optimal policies of the constructed sMDPs. Then, using the results of sMDPs developed here, we not only prove the existence of the optimal stopping times of sMPs, but also provide an algorithm for computing the optimal stopping times of sMPs. Moreover, we show that the optimal and \(\varepsilon \)-optimal stopping time can be characterized by the hitting time of some special sets. Finally, we give an example to illustrate the effectiveness of our conclusions.

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE(max0≤t≤τXt−cτ), whereX= (Xt)t≥0is geometric Brownian motion with drift μ and volatility σ &gt; 0, and the supremum is taken over all stopping times forX. The payoff is shown to be finite, if and only if μ &lt; 0. The optimal stopping time is given by τ*= inf {t&gt; 0 |Xt=g*(max0≤t≤sXs)} wheres↦g*(s) is themaximalsolution of the (nonlinear) differential equationunder the condition 0 &lt;g(s) &lt;s, where Δ = 1 − 2μ / σ2andK= Δ σ2/ 2c. The estimate is establishedg*(s) ∼ ((Δ − 1) /KΔ)1 / Δs1−1/Δass→ ∞. Applying these results we prove the following maximal inequality:where τ may be any stopping time forX. This extends the well-known identityE(supt&gt;0Xt) = 1 − (σ2/ 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ &amp;gt; 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ &amp;lt; 0. The optimal stopping time is given by τ* = inf {t &amp;gt; 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 &amp;lt; g(s) &amp;lt; s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t&amp;gt;0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Edge Computing is a new computing paradigm that aims to enhance the Quality of Service (QoS) of applications running close to end users. However, edge nodes can only host a subset of all the available services and collected data due to their limited storage and processing capacity. As a result, the management of edge nodes faces multiple challenges. One significant challenge is the management of the services present at the edge nodes especially when the demand for them may change over time. The execution of services is requested by incoming tasks, however, services may be absent on an edge node, which is not so rare in real edge environments, e.g., in a smart cities setting. Therefore, edge nodes should deal with the timely and wisely decision on whether to perform a service replication (pull-action) or tasks offloading (push-action) to peer nodes when the requested services are not locally present. In this paper, we address this decision-making challenge by introducing an intelligent mechanism formulated upon the principles of optimal stopping theory and applying our time-optimized scheme in different scenarios of services management. A performance evaluation that includes two different models and a comparative assessment that includes one model are provided found in the respective literature to expose the behavior and the advantages of our approach which is the OST. Our methodology (OST) showcases the achieved optimized decisions given specific objective functions over services demand as demonstrated by our experimental results.

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.