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## Importance sampling, large deviations, and differential games

Venue: | Stoch. and Stoch. Reports |

Citations: | 69 - 18 self |

### Citations

5411 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...on H(α) . = log Z Rd exphα, yiµ(dy),α ∈ Rd, (1.1) and its Legendre transform L(β) . = sup α∈Rd [hα, βi−H(α)] , β ∈ Rd. (1.2) It is well known that L is a nonnegative, proper, strictly convex function =-=[30]-=-, [13, Lemma 6.2.3], and that L(β) = 0 if and only if β = R Rd yµ(dy). Suppose the set A has the property that inf β∈A◦ L(β) = inf β∈Ā L(β), (1.3) where A◦ and Ā denote the interior and closure of A... |

532 |
Optimal Control and Viscosity Solutions of Hamilton-JacobiBellman Equations. Birkhäuser,
- Bardi, Capuzzo-Dolcetta
- 2008
(Show Context)
Citation Context ...UF (x, t) = inf φ ·Z 1 t L( úφ(s)) ds+ F (φ(1)) ¸ , (3.4) where the inÞmum is over all absolutely continuous φ which satisfy φ(t) = x. A standard dynamic programming argument implies that UF satisÞes =-=[4, 21]-=- the DPE 0 = (UF )t + inf a∈Rd [L(a) + ha, (UF )xi] = (UF )t −H(−(UF )x), (3.5) with terminal condition UF (x, 1) = F (x). Comparing the PDE (3.3) with (3.5), and noting WF (x, 1) = 2UF (x, 1), we con... |

442 |
Numerical Methods for Stochastic Control Problems in Continuous Time,
- Kushner, Dupuis
- 2001
(Show Context)
Citation Context ...that UF (or, WF ) does not usually have an explicit solution, and so in many cases a numerical approximation is required. Convergent numerical approximations have been studied extensively (see, e.g., =-=[27]-=-). Practical experience and some theory (e.g., [16]), have shown that the numerical schemes which construct (provably) convergent approximations to UF (or, WF ) also yield nearly optimal feedback cont... |

353 |
A Weak Convergence Approach to the Theory of Large Deviations
- Dupuis, Ellis
- 1997
(Show Context)
Citation Context ...ltaneously. The use of stochastic control and logarithmic transforms to prove large deviation type results goes back to Fleming [20], and has been investigated in many different situations since then =-=[13, 15, 18, 22, 36]-=-. In this approach, the quantity of interest [e.g., −(logP{Sn/n ∈ A})/n in Cramérs Theorem] is represented as the minimal cost Un for a stochastic control problem. The large deviation rate also has ... |

258 |
Stochastic Optimal Control : The Discrete Time Case,
- Bertsekas, Shreve
- 1978
(Show Context)
Citation Context ...αnj (S̄ni,j/n),Ȳ ni,ji+2H(αni,j(S̄ni,j/n))) ¸ ≥ E · e Pn−1 j=i (−hαnj (S̄ni,j/n),Ȳ ni,ji+H(αni,j(S̄ni,j/n))) ¸2 = 1. Thus the left hand side is bounded below by the right hand side. It follows from =-=[5]-=- that V nF satisÞes the Bellman equation V nF (x, i) = infα Z e−2hα,yi+2H(α)V nF µ x+ 1 n y, i+ 1 ¶ ehα,yi−H(α)µ(dy) = inf α Z e−hα,yi+H(α)V nF µ x+ 1 n y, i+ 1 ¶ µ(dy), together with the terminal con... |

224 |
Large Deviation Techniques in Decision, Simulation, and Estimation.
- Bucklew
- 1990
(Show Context)
Citation Context ...ing. The Þrst result of this type was given by Siegmund [37]. The basic idea was subsequently investigated in many contexts, and a small selection of the literally hundreds of papers on the topics is =-=[1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]-=-. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in [10, 33, 34], while [26] gives a survey of rare-event simulation. However, more recent ... |

211 |
Fast Simulation of Rare Events in Queueing and Reliability Models,
- Heidelberger
- 1995
(Show Context)
Citation Context ...opics is [1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in [10, 33, 34], while =-=[26]-=- gives a survey of rare-event simulation. However, more recent work has challenged the validity of the heuristic. For example, in [24] it is asserted that if one uses the change of measure suggested b... |

208 |
On general minimax theorems.
- Sion
- 1958
(Show Context)
Citation Context ... The functional on the right-hand side is concave with respect to α and convex with respect to γ, and the sets Rd and C are both convex. Since the set C is in general non-compact, the min/max theorem =-=[38]-=- cannot be applied directly. However, Lemma 2.2 shows that the interchange of inf and sup is still valid in this case. In other words, WnF (x, i) = inf γ∈C sup α∈Rd ·Z W nF µ x+ 1 n y, i+ 1 ¶ γ(dy) + ... |

189 |
Large deviations and Applications.
- Varadhan
- 1984
(Show Context)
Citation Context ...and that L(β) = 0 if and only if β = R Rd yµ(dy). Suppose the set A has the property that inf β∈A◦ L(β) = inf β∈Ā L(β), (1.3) where A◦ and Ā denote the interior and closure of A, respectively. Then =-=[39]-=- we have the large deviation approximation lim n→∞ 1 n logP {Sn/n ∈ A} = − inf β∈A L(β). Now the distribution of X is rather complicated, and so it makes sense to consider the change of measure with r... |

74 |
Importance sampling in the Monte Carlo study of sequential tests.
- Siegmund
- 1976
(Show Context)
Citation Context ...ure used to prove the large deviation lower bound should be a good (perhaps nearly optimal) distribution to use for purposes of importance sampling. The Þrst result of this type was given by Siegmund =-=[37]-=-. The basic idea was subsequently investigated in many contexts, and a small selection of the literally hundreds of papers on the topics is [1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]. ... |

71 | Analysis of an importance sampling estimator for tandem queues.
- Glasserman, Kou
- 1995
(Show Context)
Citation Context ...ing. The Þrst result of this type was given by Siegmund [37]. The basic idea was subsequently investigated in many contexts, and a small selection of the literally hundreds of papers on the topics is =-=[1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]-=-. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in [10, 33, 34], while [26] gives a survey of rare-event simulation. However, more recent ... |

67 | Large deviations for stochastic processes,
- Feng, Kurtz
- 2006
(Show Context)
Citation Context ...ltaneously. The use of stochastic control and logarithmic transforms to prove large deviation type results goes back to Fleming [20], and has been investigated in many different situations since then =-=[13, 15, 18, 22, 36]-=-. In this approach, the quantity of interest [e.g., −(logP{Sn/n ∈ A})/n in Cramérs Theorem] is represented as the minimal cost Un for a stochastic control problem. The large deviation rate also has ... |

61 |
On large deviations theory and asymptotically efficient monte carlo estimation.
- Sadowsky, Bucklew
- 1990
(Show Context)
Citation Context ...s of papers on the topics is [1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in =-=[10, 33, 34]-=-, while [26] gives a survey of rare-event simulation. However, more recent work has challenged the validity of the heuristic. For example, in [24] it is asserted that if one uses the change of measure... |

55 |
Risk theory in a Markovian environment’,
- Asmussen
- 1989
(Show Context)
Citation Context ...ing. The Þrst result of this type was given by Siegmund [37]. The basic idea was subsequently investigated in many contexts, and a small selection of the literally hundreds of papers on the topics is =-=[1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]-=-. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in [10, 33, 34], while [26] gives a survey of rare-event simulation. However, more recent ... |

54 |
Counterexamples in importance sampling for large deviations probabilities.
- Glasserman, Wang
- 1997
(Show Context)
Citation Context ...me is asymptotically optimal are discussed in [10, 33, 34], while [26] gives a survey of rare-event simulation. However, more recent work has challenged the validity of the heuristic. For example, in =-=[24]-=- it is asserted that if one uses the change of measure suggested by large deviations, then in certain situations the corresponding importance sampling scheme has very poor properties. Examples are giv... |

43 |
Exit probabilities and optimal stochastic control
- FLEMING
- 1978
(Show Context)
Citation Context ...control methodology in which both upper and lower bounds are proved simultaneously. The use of stochastic control and logarithmic transforms to prove large deviation type results goes back to Fleming =-=[20]-=-, and has been investigated in many different situations since then [13, 15, 18, 22, 36]. In this approach, the quantity of interest [e.g., −(logP{Sn/n ∈ A})/n in Cramérs Theorem] is represented as ... |

36 |
Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains
- Bucklew, Ney, et al.
- 1990
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29 |
Asymptotic series and the method of vanishing viscosity,
- Fleming, Souganidis
- 1986
(Show Context)
Citation Context ...ltaneously. The use of stochastic control and logarithmic transforms to prove large deviation type results goes back to Fleming [20], and has been investigated in many different situations since then =-=[13, 15, 18, 22, 36]-=-. In this approach, the quantity of interest [e.g., −(logP{Sn/n ∈ A})/n in Cramérs Theorem] is represented as the minimal cost Un for a stochastic control problem. The large deviation rate also has ... |

29 |
On the optimality and stability of exponential twisting in Monte Carlo estimation
- Sadowsky
- 1993
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26 |
Simulating level-crossing probabilities by importance sampling
- Lehtohnen, Nyrhinen
- 1992
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25 | Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors
- Collamore
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24 |
Stochastic control for small noise intensities
- Fleming
- 1971
(Show Context)
Citation Context ... case the partial derivative (UF )x may not be deÞned for all times and spatial points. Under additional conditions one can show that the set of points where α∗(x, t) is well deÞned is open and dense =-=[19]-=-. At points where the gradient is not deÞned there are often superdifferentials, and a natural replacement for the DPE suggests feedback controls deÞned in terms of extreme superdifferentials rather t... |

21 |
On Monte Carlo estimation of large deviation probabilities.
- Sadowsky
- 1996
(Show Context)
Citation Context ...s of papers on the topics is [1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in =-=[10, 33, 34]-=-, while [26] gives a survey of rare-event simulation. However, more recent work has challenged the validity of the heuristic. For example, in [24] it is asserted that if one uses the change of measure... |

19 |
Large deviations and efficient simulation of excessive backlogs in a GI/G/m queue
- Sadowsky
- 1991
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16 |
On asymptotically efficient simulating of ruin probabilities in a Markovian environment
- Lehtonen, Nyhrinen
- 1992
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11 |
Probability theory,
- Breiman
- 1968
(Show Context)
Citation Context ...can extract a weakly convergent subsubsequence, still denoted by {γn}, such that γn ⇒ γ for some stochastic kernel γ whose second marginal is Lebesgue measure. We utilize the Skorokhod representation =-=[6]-=-, which allows us to assume (when calculating the limits of the integrals) that the convergence is actually w.p.1. It follows 15 from the lower semicontinuity of R, the convergence γn ⇒ γ, and the uni... |

10 |
On importance sampling in digital communications. Part I: Fundamentals
- Chen, Lu, et al.
- 1993
(Show Context)
Citation Context ...s of papers on the topics is [1, 2, 3, 7, 8, 9, 11, 12, 14, 23, 25, 28, 29, 31, 32, 35]. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in =-=[10, 33, 34]-=-, while [26] gives a survey of rare-event simulation. However, more recent work has challenged the validity of the heuristic. For example, in [24] it is asserted that if one uses the change of measure... |

10 |
Stochastic systems with small noise, analysis and simulation; a phase locked loop example
- Dupuis, Kushner
- 1987
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9 |
Stochastic control and exit probabilities of jump processes
- Sheu
- 1985
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8 |
1983]. Large Deviations and Rare Events in the Study of Stochastic Algorithms
- Cotrell, Fort, et al.
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8 |
The Existence of Value in
- Elliott
- 1976
(Show Context)
Citation Context ..., thus giving it the information advantage. Thus in the limit n → ∞ we expect to obtain a differential game with the advantage given to the minimizing player, the so-called lower game (see, e.g., =-=[17]-=-). We refer to such games as games in the small, since in the 22 discrete time prelimit game the players selections of controls are interleaved and sequential. In the case of the standard heuristic... |

7 |
Regenerative rare event simulation via likelihood ratios
- Asmussen, Rubinstein, et al.
- 1994
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7 |
Asymptotic expansions for Markov processes with Levy generators.
- Fleming, Soner
- 1989
(Show Context)
Citation Context ...UF (x, t) = inf φ ·Z 1 t L( úφ(s)) ds+ F (φ(1)) ¸ , (3.4) where the inÞmum is over all absolutely continuous φ which satisfy φ(t) = x. A standard dynamic programming argument implies that UF satisÞes =-=[4, 21]-=- the DPE 0 = (UF )t + inf a∈Rd [L(a) + ha, (UF )xi] = (UF )t −H(−(UF )x), (3.5) with terminal condition UF (x, 1) = F (x). Comparing the PDE (3.3) with (3.5), and noting WF (x, 1) = 2UF (x, 1), we con... |

6 |
The blind simulation problem and regenerative processes
- Bucklew
- 1998
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6 |
Convergence of the optimal feedback policies in a numerical method for a class of deterministic optimal control problems,
- Dupuis, Szpiro
- 2001
(Show Context)
Citation Context ... solution, and so in many cases a numerical approximation is required. Convergent numerical approximations have been studied extensively (see, e.g., [27]). Practical experience and some theory (e.g., =-=[16]-=-), have shown that the numerical schemes which construct (provably) convergent approximations to UF (or, WF ) also yield nearly optimal feedback controls, though they are usually limited by practical ... |

4 |
Conjugate processes and the simulation of ruin probability. Stochastic Process
- Asmussen
- 1985
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4 |
Importance sampling for the simulation of highly reliable Markovian systems
- Shahsbuddin
- 1994
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1 |
Minimizing exit probabilities; a large deviations approach
- Dupuis, Kushner
- 1989
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1 |
Large deviations for the inÞnite server queue in heavy traffic
- Glynn
- 1995
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