The value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the Lévy measure, this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite/infinite variation jumps is in $W^{2,1}_{p, loc}$ with $p\in(1, \infty)$. As a consequence, the smooth-fit property holds.
We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of…
We consider the optimal stopping of a class of spectrally negative jump diffusions. We state a set of conditions under which the value is shown to have a representation in terms of an ordinary nonlinear programming problem. We establish a connection between the considered problem and a stopping problem of an associated continuous diffusion process and demonstrate how this connection may be applied for characterizing the stopping policy and its value. We also establish a set of typically satisfied conditions under which increased volatility as well as higher jump-intensity decelerates rational …
We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is t…
We study the optimal stopping problem for a monotonous dynamic risk measure induced by a BSDE with jumps in the Markovian case. We show that the value function is a viscosity solution of an obstacle problem for a partial integro-differential variational inequality, and we provide an uniqueness result for this obstacle problem.
We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic ca…
We present a novel method for solving a class of time-inconsistent optimal stopping problems by reducing them to a family of standard stochastic optimal control problems. In particular, we convert an optimal stopping problem with a non-linear function of the expected stopping time in the objective into optimization over an auxiliary value function for a standard stochastic control problem with an additional state variable. This approach differs from the previous literature which primarily employs Lagrange multiplier methods or relies on exact solutions. In contrast, we characterize the auxilia…
We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set $\mathcal{O}$. The stopping horizon is either random, equal to the first exit from the set $\mathcal{O}$, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of $\mathcal{O}$. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or $ε$-opti…
We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift and volatility of a process in order to maximize a running reward based on that process, and the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. Our primary goal is to show that the minimal probability of ruin, whose stoc…
We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. Specifically, we obtain the viscosity property of value functions, the uniqueness of viscosity solutions, the regularity of value functions and the form of optimal stopping intervals. Finally, we provide an application of the results.
We analyze an optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities $\mathcal{P}$. The maturity is specified as the hitting time to level $0$ of some continuous index process at which the payoff process is even allowed to have a positive jump. When $\mathcal{P}$ is a collection of semimartingale measures, the optimal stopping problem can be viewed as a {\it discretionary} stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.
We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising for instance in finance and economics. The underlying process is a strong solution of one dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and it is based on a contradiction scheme inspired by the maximum principle in partial differential equations (PDE) theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the bound…
We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometr…
This paper studies regularity property of the value function for an infinite-horizon discounted cost impulse control problem, where the underlying controlled process is a multidimensional jump diffusion with possibly `infinite-activity' jumps. Surprisingly, despite these jumps, we obtain the same degree of regularity as for the diffusion case, at least when the jump satisfies certain integrability conditions.
We summarize the general results of El Karoui [1981] on optimal stopping problems for processes which are measurable with respect to Meyer-$σ$-fields. Meyer-$σ$-fields are due to Lenglart [1980] and include the optional and predictable $σ$-field as special cases. Novel contributions of our work are path regularity results for Meyer measurable processes and limit results for Meyer-projections. We will also clarify a minor issue in the proof of the optimality result in El Karoui [1981]. These extensions were inspired and needed for the proof of a stochastic representation theorem in Bank and Bes…
We study a practical optimization problems for venture capital investments and/or Research and Development (R&D) investments. The first problem is that, given the amount of the initial investment and the reward function at the initial public offering (IPO) market, the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent (if needed) investments. We describe this problem as a mixture of singular stochastic control and optimal stopping problems and give an explicit solution. The former corresponds to finding an optimal subsequent investment policy for th…
We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is $C^1$ across the optimal stopping boundary. Our proof, which only uses the classical theory of parabolic partial differential equations of [7,8], is an alternative to the proof that uses the the theory of vicosity solutions [14]. This new proof relies on constructing a monotonous sequence of functions, each of which is a value…
Let $X$ be a bounded càdlàg process with positive jumps defined on the canonical space of continuous paths. We consider the problem of optimal stopping the process $X$ under a nonlinear expectation operator $\cE$ defined as the supremum of expectations over a weakly compact family of nondominated measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an $\cE-$supermartingale, and an $\cE-$martingale up to its first hitting time of the obs…
We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $ξ$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $ξ$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $ξ$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American…
In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le τ\le 1}\mathsf{E}[\mathrm{e}^{X_τ}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we devel…
This paper analyses a stochastic differential game of control and stopping in which one of the players modifies a diffusion process using impulse controls, an adversary then chooses a stopping time to end the game. The paper firstly establishes the regularity and boundedness of the upper and lower value functions from which an appropriate variant of the dynamic programming principle (DPP) is derived. It is then proven that the upper and lower value functions coincide so that the game admits a value and that the value of the game is a unique viscosity solution to a HJBI equation described by a …
Regularity of the impulse control problem for a non-degenerate $n$-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined by Davis, Guo, and Wu (SICON 2010). Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function $u$ of the impulse control problem satisfies $u\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n})$.
A finite horizon optimal stopping problem for an infinite dimensional diffusion $X$ is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $\mathcal{H}$ with a non-linear diffusion coefficient $σ(X)$ and a generic unbounded operator $A$ in the drift term. When the gain function $Θ$ is time-dependent and fulfils mild regularity assumptions, the value function $\mathcal{U}$ of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $σ(X)$ is speci…
We study an infinite horizon optimal stopping problem which arises naturally in the optimal timing of a firm/project sale or in the valuation of natural resources: the functional to be maximised is a sum of a discounted running reward and a discounted final reward. The running and final rewards as well as the instantaneous interest rate (used for calculating discount factors) depend on a Feller-Markov process modelling the underlying randomness in the world, such as prices of stocks, prices of natural resources (gas, oil, etc.), or factors influencing the interest rate. For years, it had seeme…
We study the regularity of the stochastic representation of the solution of a class of initial-boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transitio…
We study an optimal stopping problem when the state process is governed by a general Feller process. In particular, we examine viscosity properties of the associated value function with no a priori assumption on the stochastic differential equation satisfied by the state process. Our approach relies on properties of the Feller semigroup. We present conditions on the state process under which the value function is the unique viscosity solution to an Hamilton-Jacobi-Bellman (HJB) equation associated with a particular operator. More specifically, assuming that the state process is a Feller proces…
We study a combined optimal control/stopping problem under a nonlinear expectation ${\cal E}^f$ induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function $u$ associated with this problem is generally irregular. We first establish a {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} $u^*$ (resp. $u_*$). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical ex…
We consider an investment problem in which an investor performs capital injections to increase the liquidity of a firm for it to maximise profit from market operations. Each time the investor performs an injection, the investor incurs a fixed transaction cost. In addition to maximising their terminal reward, the investor seeks to minimise the risk of loss of their investment (from a possible firm ruin) by exiting the market at some point in time. We show that the problem can be reformulated in terms of a new stochastic differential game of control and stopping in which one of the players modif…
In this paper we investigate a kind of optimal control problem of coupled forward-backward stochastic system with jumps whose cost functional is defined through a coupled forward-backward stochastic differential equation with Brownian motion and Poisson random measure. For this end, we first study the regularity of solutions for this kind of forward-backward stochastic differential equations. We obtain that the value function is a deterministic function and satisfies the dynamic programming principle for this kind of optimal control problem. Moreover, we prove that the value functions is a vis…
We provide an algorithm to find the value and an optimal strategy of the solitaire variant of the Ten Thousand dice game in the framework of Markov Control Processes. Once an optimal critical threshold is found, the set of non-stopping states of the game becomes finite, and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.
This paper analyzes a class of impulse control problems for multi-dimensional jump diffusions in the finite time horizon. Following the basic mathematical setup from Stroock and Varadhan \cite{StroockVaradhan06}, this paper first establishes rigorously an appropriate form of Dynamic Programming Principle (DPP). It then shows that the value function is a viscosity solution for the associated Hamilton-Jacobi-Belleman (HJB) equation involving integro-differential operators. Finally, under additional assumptions that the jumps are of infinite activity but are of finite variation and that the diffu…
We study a constrained stochastic control problem with jumps; the jump times of the controlled process are given by a Poisson process. The cost functional comprises quadratic components for an absolutely continuous control and the controlled process and an absolute value component for the control of the jump size of the process. We characterize the value function by a "polynomial" of degree two whose coefficients depend on the state of the system; these coefficients are given by a coupled system of ODEs. The problem hence reduces from solving the Hamilton Jacobi Bellman (HJB) equation (i.e., a…
