Exponentially growing solutions of parabolic Isaacs' equations

Strömberg, Thomas

doi:10.1016/j.jmaa.2008.07.038

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…By [5,Theorem 2.3.5] it follows that the function v is a usc subsolution of (2.10). Similary, v is an lsc supersolution of (2.10) By the mentioned comparison results of [3] or [16] we have v ≤ v. The opposite inequality is clear from the definition of v, v. It follows that the function v = v = v coincides with the unique viscosity solution of (2.5), (2.6), and the second equality (2.7) holds true: lim…”

Section: The Main Resultsmentioning

confidence: 73%

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Asymptotic sequential Rademacher complexity of a finite function class

Rokhlin

2016

Arch. Math.

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Abstract. For a finite function class we describe the large sample limit of the sequential Rademacher complexity in terms of the viscosity solution of a G-heat equation. In the language of Peng's sublinear expectation theory, the same quantity equals to the expected value of the largest order statistics of a multidimensional G-normal random variable. We illustrate this result by deriving upper and lower bounds for the asymptotic sequential Rademacher complexity.

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Section: The Main Resultsmentioning

confidence: 73%

“…Denote v + (resp., v − ) the viscosity solution of (2.5), satisfying the terminal condition v + (1, x) = g + (x) (resp., v − (1, x) = g − (x)), x ∈ R m , and the linear growth condition. By the comparison result of [3, Theorem 2.1] or [16,Theorem 5]…”

Section: Thusmentioning

confidence: 99%

Asymptotic sequential Rademacher complexity of a finite function class

Rokhlin

2016

Arch. Math.

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“…Peng [14] obtained a uniqueness theorem for a class of second order parabolic equations under the polynomial growth condition. Strömberg [15] considered the Cauchy problem for parabolic Isaacs's equations:…”

Section: Introductionmentioning

confidence: 99%

The Tychonoff uniqueness theorem for the G-heat equation

Lin

2011

Sci. China Math.

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“…Following the arguments developed, for example, in Strömberg [19] Theorem 5, we have the following comparison principle. Proposition 4.1.…”

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confidence: 99%

Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

Buckdahn

Quincampoix

2014

Ann. Probab.

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International audienceIn the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $\pi$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $\pi$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $\pi$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_j)$. It is shown that the associated lower and upper value functions $W^{\pi}$ and $U^{\pi}$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $\pi$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation

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Exponentially growing solutions of parabolic Isaacs' equations

Cited by 3 publications

References 22 publications

Asymptotic sequential Rademacher complexity of a finite function class

Asymptotic sequential Rademacher complexity of a finite function class

The Tychonoff uniqueness theorem for the G-heat equation

Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

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