Related papers: The Expected Time to End the Tug-of-War in a Wedge

The objective is the interplay between stochastic processes and partial differential equations. To be more precise, we focus on the connection between the nonlinear p-Laplace equation, and the stochastic game called tug-of-war with noise.…

In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity…

We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized $p(x,t)$-Laplace operator. Our…

Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which…

Nonlinear PDEs, mean value properties, and stochastic differential games are intrinsically connected. In this short expository note, we will describe how the solutions to certain PDEs (of $p$-Laplacian type) can be interpreted as limits of…

We give a tug-of-war interpretation of the regularized $p$-Laplacian $\divgg\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $\Omega\subset\R^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this…

We study a tug-of-war game with varying probabilities. In particular, we show that the value of the game is locally asymptotically H\"{o}lder continuous. We also show the existence and uniqueness of values of the game. As an application, we…

In this paper we prove that a function $ u\in\mathcal{C}(\bar{\Omega})$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary…

In this paper, we study a certain type of noisy tug-of-war game which can be regarded as an interpretation of a certain type of boundary value problem for the normalized $p$-Laplace equation, where $1<p<2$. More precisely, we will…

We present a probabilistic approach to the obstacle problem for for the $p$-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the step size go to zero, not unlike…

In this paper we use probabilistic arguments (Tug-of-War games) to obtain existence of viscosity solutions to a parabolic problem of the form $$ {cases} K_{(x,t)}(D u)u_t (x,t)= \frac12 <D^2 u J_{(x,t)}(D u),J_{(x,t)}(D u) (x,t) &{in}…

In tug-of-war, two players compete by moving a counter along edges of a graph, each winning the right to move at a given turn according to the flip of a possibly biased coin. The game ends when the counter reaches the boundary, a fixed…

In this paper, we investigate a class of tug-of-war games that incorporate a constant payoff discount rate at each turn. The associated model problems are $p$-Laplace type partial differential equations with zeroth-order terms. We establish…

In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} \Delta_{p}^{N}u=0 & \textrm{in $ \Omega$,}\\ \langle \beta , Du \rangle…

This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized $p$-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder,…

We propose a new finite difference approximation to the Dirichlet problem for the homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensional domain, in connection with the Tug of War games with noise. Our game and the related…

We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multi-dimensional financial market. The issuer and the holder try to manipulate asset…

This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the…

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the…

We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case…