Stochastic Integration and Differential Equations

Protter, Philip

doi:10.1007/978-3-662-02619-9

Cited by 2,174 publications

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“…Applying the Ito chain rule to log[X t ] in equation (1) [12,13] gives, as a consequence of the Ito correction factor, the long-time limits…”

Section: Pacs Numbersmentioning

confidence: 99%

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Noise and metabolic free energy in high-order biocognition

Wallace

2015

Preprint

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We examine the possible role of 'noise' as a control signal for large-scale cognitive biological phenomena that recruit simpler cognitive modules into temporary, dynamic working coalitions. Noise color, as well as magnitude, may convey essential control information, a possibly important extension of the Data Rate Theorem. An adaptation of the Black-Scholes model suggests the availability of metabolic free energy can determine rates of coalition biocognition in the presence of noise. Evolutionary process may have exapted colored noise as a subtle tool for the regulation of biological phenomena, supplementing direct molecular signals. Experimental verification of this conjecture may be similarly subtle. PACS numbers:The life course of a higher organism is dominated by a spectrum of sophisticated systems that recruit lower level cognitive modules into dynamic coalitions to meet changing patterns of threat and affordance [1][2][3]. These systems include gene expression, wound healing, immune response, animal consciousness, and so on, and operate at vastly different rates. For a large animal, these particular examples typically have time constants of years, months, hours, and fractions of a second.Most notably, neural phenomena are strongly dominated by signals that are difficult to interpret as anything but 'noise', making determination of any 'neural code' a challenging enterprise indeed. As Koch [4] puts the matter, it has recently been realized that cortical networks are primarily driven by stochastic internal fluctuations of inhibition rather than by excitatory feed-forward inputspikes are preceded by a momentary reduction in inhibition rather than by an increase in excitation. For microscopic models of individual neurons, the phenomenon has been characterized as 'inverse stochastic resonance ' [5,6]. Here, we will provide a model of macroscopic stochastic logic gates similarly triggered by a reduction in 'noise'.That is, we suggest 'noise', tinted in various colors, can be a powerful control signal for such large-scale recruitment processes. We use methods similar to those applied to the study of noise-driven phase transitions [7]. A further noise argument, via a Black-Scholes model, illuminates the influence of metabolic free energy supply rate on the rate of coalition cognition.The Baars model of animal consciousness [3,8,9] posits the phenomenon as the assembly of unconscious cognitive modules into shifting working arrangements that address sensory or internal signals exceeding a dynamic threshold. Wallace [3] argues that the underlying mechanism is general, a consequence and evolutionary exaptation of the inevitable information crosstalk between cognitive processes. In contrast to the simple on-off threshold operation of neurons that may involve noise through stochastic resonance (e.g., [10]), the assembly of such working coalitions is usually a large-scale explosive 'reentrant' autocatalytic phenomenon in which signal boosts signal until a regulated maximum is reached [11]. We model that dynamic to include...

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“…Applying the Ito chain rule to log[X t ] in equation (1) [12,13] gives, as a consequence of the Ito correction factor, the long-time limits…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Other forms of noise, for example fractional Brownian motion, produce more complicated results, as do geographic extensions of the model [15,16]. The most complete treatment currently available appears to be via the Doleans-Dade exponential ( [12] Theorem 36).…”

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confidence: 99%

Noise and metabolic free energy in high-order biocognition

Wallace

2015

Preprint

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“…As soon as one wants to go beyond this situation, measurability and convergence become an issue. This observation prompted K. Itō to use martingale convergence theorems to tackle the convergence for a large class of integrators [56]. To do so we must make sure that I(t) is a martingale whenever X(t) is.…”

Section: A Definitionsmentioning

confidence: 99%

Stochastic calculus for uncoupled continuous-time random walks

et al. 2009

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The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy α-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional FokkerPlanck equation, that generalize the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo.

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“…We are given a …ltered probability space ( ; F; (F t ) t2[0;1) ; P ) satisfying the usual conditions (see Protter [23]). We denote the recorded default date as .…”

Section: Set Upmentioning

confidence: 99%

Distressed debt prices and recovery rate estimation

2008

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This paper has two purposes. First, it uses distressed debt prices to estimate recovery rates at default. In this regard, estimates are obtained for three recovery rate models: recovery of face value, recovery of Treasury, and recovery of market value. We show that identifying the "economic" default date, as distinct from the recorded default date, is crucial to obtaining unbiased estimates. The economic default date is de…ned to be that date when the market prices the …rm's debt as if it has defaulted. An implication is that the standard industry practice of using 30-day post default prices to compute recovery rate yields biased estimates. Second, we construct and estimate a distressed debt pricing model. We use this model to implicitly estimate the parameters of the embedded recovery rate process and to price distressed debt. Our distressed debt pricing model …ts market prices well, with an average pricing error of less than one basis point.

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Stochastic Integration and Differential Equations

Cited by 2,174 publications

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Noise and metabolic free energy in high-order biocognition

Noise and metabolic free energy in high-order biocognition

Stochastic calculus for uncoupled continuous-time random walks

Distressed debt prices and recovery rate estimation

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