Temporal Risk Processes

Abstract

This chapter provides an introduction to inter-temporal probability processes used commonly in modeling risk processes. The chapter begins with the questions “what is time,” “what is memory?” how are their definitions used to construct quantitative and temporal models. Elementary models such as probability models with the Markov property, random (binomial) walks, Poisson processes and continuous state and time stochastic processes are both presented intuitively and applied to many risk problems. These stochastic processes are then extended to more complex situations including long run memory models (fractal models), short memory models as well as to models departing from the basic Markov property and random walks models. While some of these models require a more advanced quantitative background than assumed for Chaps. 3 and 4, their applications are used to highlight both their importance and their implications to financial and risk models.

References and Additional Reading

Time has been and remains both a philosophical concept and the means to organize our thoughts in a tractable and temporal perspective. Aristotle, Saint Augustine, Einstein, Janet (see Fraisse 1957) and so many others have sought to grasp the meaning of time and reveal its future. In my paper 1978a, time is used in its modeling context and used to reconcile the passing of time, the memory of past events, and their interaction with its change and the unpredictable future. The simplest of all such models are of course stochastic processes, whether memory-less or well structured Markov memory models (see also Tapiero 1994c on stochastic modeling). The complexity of these factors leads mostly to intractable and chaotic evolutions (Lorenz 1966; May 1974, 1976) and to the evolution of mathematical techniques (chaos theory for example) that seek to reduce this intractability to a recognizable quantitative framework. In economics and finance, authors such as Grandmont and Malgrange (1986) on nonlinear dynamics, Medio (1992) on chaotic dynamics and an impressive number of research papers published under the banners of econophysics and financial physics (see the many publications in Physica A) are providing multiple avenues in stochastic and nonlinear modeling and making an appreciable contribution to risk engineering and to finance.

Stochastic models are used in many field. This is the case in physics, Einstein (1906, 1956); in engineering, Stratonovich (1963); in actuarial science and stochastic models in general such as for example, Feller (1951–1971), Cox and Miller (1965), Levy (1937), Lundberg (1909, 1940), Prabhu (1965, 1980), Seal (1969, 1978), Tetens (1786), Klugman et al. (2000) on Loss Models, Lax, Wei and Xu, on random processes in physics and finance, Shaked and Shantikumar (1994) on reliability and many others. In addition to an extensive literature on stochastic processes in insurance (for example, Embrechts 2000a, b; Embrechts et al. 1997, 2001, 2002, 2003, 2009). Optimization of such models are reached by applications of stochastic dynamic programming and control (e.g. Bensoussan 1982 on control, Bensoussan and Tapiero 1982 on impulse control, Bismut 1975, 1978 on the Growth and intertemporal allocation of risks based on a duality in optimal stochastic control (see also Bismut 1976, on Theorie Probabiliste du Controle des Diffusions).

Applications in Operations and risk engineering abound as well. Queuing Theory is a branch of discrete events stochastic processes (Saaty 1961; Gross and Harris 1985; Basawa and Prabhu 1981; Basawa and Rao 1980), on the statistical analysis of queuing processes and application of quality control techniques to queues operations, Hsu and Tapiero (1987a, b, 1988a, b, 1992, 1994), Hsu et al. (1993), Liu et al. (1990) on general queues, Tapiero (1982a, b) applications in advertising stochastic models and in marketing, including repeat purchase models and information advertising content (Tapiero 2000a), queue in environmental control problems (Tapiero 2005a), Barnett and Toby (1994), on Outliers in Statistical Data.

Stochastic models consist fundamentally of elementary probability events. In Chap. 3 we noted for example that the Bernoulli process underlies a broad number of probability models. Such a model, expanded to its multivariate form, and sequenced in some logical manner can lead to complex processes that may be analyzed either numerically or by simulation (Carreira-Perpinan and Reinals 2000 on mixtures of multivariate Bernoulli probability models, Neural computation de la Selva 1988 on correlated random walks, Galambos 1978 on extreme statistics, Leadbetter et al. 1983 on extremes in related sequences and Rubenstein 1981 on simulation, Eberlein (2001) on Levy processes). In physics texts such models abound and increasingly are applied to financial problems (Lax et al. 2006 on Random Processes in Physics and their Applications in Finance), Cartea and Howison (2003) on Limits of Levy-Stable Processes and their Applications to Option Pricing, Bouchaud and Potters (2003), on applications of stochastic models of physics to financial risks and derivatives pricing. Their applications are varied, we referred to Fujiwara (2004) on Zipf law in firms bankruptcy, Garlaschelli et al. (2005) on the scale free nature of market investment networks, Saito et al. (2007) interfirm relationships, etc.

Just as the binomial distribution and the negative binomial (Pascal) probability distribution were used to calculate the number of “success” events occurring in a given time interval and the random number of events to be repeated to obtain a given number of “success” events, in a temporal perspective, we also use run processes. For example, Mood (1940) provided some basic results on the distribution of runs, Chow et al. (1971a, b), Wolfowitz (1943), Wald (1947, 1961) on sequential analysis and decision function, Robbins and Siegmund (1971) on Optimal stopping time, Siegmund (1985), Fou ans Koutras (1994), Capocelli and Ricciardi (1972) on inverse first passage times in stochastic differential equations (with subsequent papers extending their results).

In economics ARCH and GARCH models (Engle 1987, 1995; Bollerslev et al. 1992, 1994, on ARCH and GARCH models, Nelson and Foster 1994 for example) have led to our constructing multivariate stochastic processes (essentially mean and the volatility stochastic variations—one dependent on the other). Parkinson and subsequently Yang and Zhang used hi and lows (the range process) in time series to assess this volatility. Stochastic models in finance are used extensively to assess the price of options. Other stochastic models based on Gamma processes were introduced by Baxter (2007) with their application in finance by Carr and Madan as well as a number of recent papers that suggest that we construct stochastic models of models correlations (see De la Selva and Lindenberg 1988 on correlated random walks for a discrete events approach to such models).

Range processes statistics complement variance statistics. Feller (1951) provided initial theoretical results on the range process (The asymptotic distribution of the range of sums of random variables) while Hurst (1951) study of the water level of the Nile based on records taken over thousands of years has led Mandelbrot to introduce fractal models and to the mathematical study of the Hurst index (a Range to Standard deviation statistic). Mandelbrot (1997, 2004), Mandelbrot and Taqqu (1979), Granger (1980), Beran (1994) on the statistics of long run memory model, Mandelbrot and Wallis (1968), Otway (1995) as well as Tapiero and Vallois with many subsequent applications beginning with Mandelbrot and his students. These applications include: Booth et al. (1982), Cheung (1983), Fung and Lo (1995), Green and Fielitz (1977, 1980), on applications of RS analysis in FX markets, Mandelbrot and Van Ness (1968), Rogers and Satchell (1991) on Hi and Los and variance estimates etc.

In economics, this index was used as “an index of chaos,” and to study various facets of economic time series. For example, Diebold and Rudenbusch (1989) use the R/S analysis in monetary economics, Allen (2000) in financial contagion, Martınez-Jaramillo et al. (2010) on contagion and systemic risks, Haubrich and Lo (1989) on long run dependence and business cycles, Helms et al. (1984) on memory in commodity markets, Hsieh (1991) on chaos in financial markets, Lo (1991, 1997) on fat tails in stock market, Anderson (1957) on contagions in collective risk theory etc. Booth et al. (1982, 1972, 1997, 2004), Cheung (1983), Diebold and Rudenbusch (1989), Lo (1991), Mandelbrot (1963, 1972, 1997, 2004) are some applications to FX markets, commodities and economic problems.

In continuous stochastic models, Imhof (1985) provided the first results for the range in a Brownian motion while Vallois (1995, 1996), Vallois and Tapiero (1995a, b, 1996, 1997, 2001), Tapiero and Vallois (1997a, b) provide explicit results on range processes (see also Siebenaler 1997; Siebenaler et al. 1997; Tapiero and Vallois 1996, 2000).

Stochastic models of Contagion and Persistence are also varied, some include a seminal book on random walks and persistence by Weiss and Rubin (1983), Domokos and Thoth (1984), Persistent random walks in a one dimensional and random environment, Balinth (1986), Mzasoliver et al. (1992), on the continuum limit of a two dimensional persistent random walk, Amaral et al. (1997) on persistence model in Physics, Claes and Van den Broeck (1987) the random walk with persistence, Stanley et al. (1996), Pottier (1996) persistence in biased random walks and in a random environment, Weiss (1994), seminal book on Aspects and Applications of the Random Walk, Weiss (2002), Some applications of persistent random walks and the telegrapher’s equation, Weiss and Rubin (1983), De la Selva et al. (1988), Correlated random walks (Pottier 1996; Patlak 1953).

My own research with Pierre Vallois (Vallois and Tapiero 1997a, b, 2009) and the recent seminal paper by Herman and Vallois (2010a), who provide both a summary and explicit convergence results for persistent stochastic processes leading to both limit Brownian motions and telegraphic processes (second order partial differential equations in both state and time). These models provide an important foundation to contagious stochastic processes. Additional and related studies include Tapiero, C.S. and P. Vallois on The Inter-Event Range Process and Testing for Chaos in Time Series, Vallois P. and C.S. Tapiero (1997a), on Range Reliability in Random Walks.