In addition, as mentioned in Multiple Streams via Leap-frog, using leap-frogging can, in some instances, change the statistical properties of the sequences being generated. For leap-frogging additional computation is required both at initialization and during the generation of the variates. Both methods required additional computation compared with generating a single sequence, but for skip-ahead this computation occurs only at initialization. Of these two, if possible, skip-ahead should be used in preference to leap-frogging. In contrast leap-frogging requires you to know the maximum number of streams required, prior to generating the first value. Skip-ahead requires no a-priori information on the number of streams required. Notice the lack of pattern at all scales. Leap-frogging requires no a-priori knowledge about the number of variates being generated, whereas skip-ahead requires you to know (approximately) the maximum number of variates required from each stream. We can visualize this random sequence by drawing a path that changes direction according to each number, known as a random walk. Of the remaining two methods, both skip-ahead and leap-frogging use the sequence from a single generator, both guarantee that the different sequences will not overlap and both can be scaled to an arbitrary number of streams. Brownian Bridge and Stochastic Differential Equations These remarks extend readily to the case of a non-free Wiener process. For further discussions in this regard we refer to Glasserman (2004). This also means that the search for new approaches and new construc. Indeed, one would most likely also ensure that time X 3 was one of the first bridge points that was constructed: ‘lower’ dimension values would be used to construct both the left and right bridge points used in (2) to interpolate X 3, so that the distribution of X 3 benefits as much as possible from the uniformity properties of the quasi-random sequence. written on pseudorandom sequences (we shall also write PR for pseudoran- domness). When constructing the sample paths, one would therefore ensure that time 3 was one of the interpolation points of the bridge, and that a ‘lower’ dimension value was used in (2) to construct the corresponding bridge point X 3. For example, consider a model which is particularly sensitive to the behaviour of the underlying process at time 3. Often the ‘lower’ dimension values ( Z 1 i, Z 2 i, etc.) display better uniformity properties than the ‘higher’ dimension values ( Z N + 1 i, Z N i, etc.) so that the ‘lower’ dimension values should be used to construct the most important sections of the sample path. The question is how to use the dimension values of each N + 1 dimensional quasi-random point. In other words, the interpolation interval t i, t k must not contain any other known points, otherwise the covariance structure of the process will be incorrect. Any method of choosing which t j ∈ t i, t k to interpolate next is equally valid, provided t i is the nearest known point to the left of t j and t k is the nearest known point to the right of t j. However when it comes to deciding how the successive interpolation times t j should be chosen, there is virtually no restriction. For X to behave like a usual (free) Wiener process we should set X t 0 equal to some value x ∈ ℝ d and then set X T = x + C T - t 0 Z where Z is any d-dimensional standard Normal random variable. All that is needed to complete the specification is to fix the start point X t 0 and end point X T, and to specify how successive interpolation times t j are chosen. Clearly this algorithm is iterative in nature. Where Z is a d-dimensional standard Normal random variable and C is any d × d matrix such that C C T is the desired covariance structure for the (free or non-free) Wiener process X. Produce random numbers from other distributions.X t j = X t i t k - t j + X t k t j - t i t k - t i + C Z t k - t j t j - t i t k - t i The methods described in this section detail how to Methods for generating pseudorandom numbers usually start with uniform random numbers, Common Pseudorandom Number Generation Methods Random number generators (RNGs) like those in MATLAB ® are algorithms for generating pseudorandom numbersįor more information on the GUI for generating random numbers from supportedĭistributions, see Explore the Random Number Generation UI.įor more complex probability distributions, you can use the methods described in Representing Sampling Distributions Using Markov Chain Samplers. They differ from true random numbers in that they are generated by an algorithm, rather than a That, on average, they pass statistical tests regarding their distribution and correlation. Pseudorandom numbers are generated by deterministic algorithms.
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