Stochastic differential equations : an introduction with applications

By Bernt Øksendal. 1987

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Citations in the corpus

Maximum Likelihood Training of Score-Based Diffusion Models (2021, positive): “From the chain rule for the KL divergence, we also have DKLpμ } νq “ DKLppT } πq ` EpT pzqrDKLpμp ̈ | xpT q “ zq } νp ̈ | x̂θpT q “ zqqs, where the KL divergence in the final term can be computed by applying the Girsanov theorem [34] to Eq.”
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms (2015, methodology): “vergence in probability. In (27), bis known as the drift, and ˙is known as the diffusion matrix. When they satisfy Lipschitz conditions, one can show that (27) (or (29)) has a unique strong solution (Oksendal (2013), Chapter 5). For our purposes in this paper, we consider the special case where b;˙d”
Exact simulation of diffusions (2005, unknown): “See, for instance, [8], Chapter 8.”
Sampling the posterior: An approach to non-Gaussian data assimilation (2007, background): “The subject of Brownian motion and stochastic calculus is described in [19], whilst texts on SDEs include [11,29].”
Retrospective exact simulation of diffusion sample paths with applications (2006, methodology): “In fact, we will restrict our attention to SDEs of the type dX t ¼ Æ(X t) dt þ dBt, X0 ¼ x 2 R, t 2 [0, T ], (2) for some drift function Æ, since (1) can be transformed into an SDE of unit coefficient under mild additional conditions on , by applying the transformation X t !”
On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems (2007, methodology): “where T0;1 is the set of all (Ft)t‚0 adapted stopping times with values in [0;1] and g is the payofi function mapping R+ into R. Let us recall the basics of optimal stopping theory (we refer to El Karoui [8], Karatzas Shreve [14], Appendix D Theorem D9 and Theorem D12, see also Theorems 10.1.9 and 1”