Techniques for Continuous Time Stochastic Processes

Time Stochastic Process

Stochastic Processes

Yûichirô Kakihara , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.E White Noise

A discrete or continuous time stochastic process { X(t)} is called a white noise if E{X(t)}   ≡   0 and K(s, t)   =   δ(s  − t), the Kronecker delta function, i.e.,=1 for s  = t AND =0 for s  ≠ t. A white noise does not exist in a real world. But as a mathematical model it is used in many fields. It is known that the derivative of a Wiener process as a generalized stochastic process is a white noise (cf. Section VIII).

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Principles and Methods for Data Science

David A. Spade , in Handbook of Statistics, 2020

3 Markov chain Monte Carlo background

This section introduces the background on Markov chains that is required for an understanding of the concepts related to MCMC methods. This introduction includes ideas related to the limiting behavior of Markov chains. We begin with a discussion of Markov chains on discrete-state spaces in order to develop comfort with Markov chains before extending these concepts to general state spaces. First, we define a Markov chain.

Definition 9

A Markov chain (X _t ) _t≥0 is a discrete-time stochastic process $\left\{X_0,X_1,\dots \right\}$ with the property that, given X ₀, X ₁,…, X _t−1, the distribution of X _t depends only on X _t−1. Formally, (X _t ) _t≥0 is a Markov chain if for all $A\subset \mathcal{S}$ , where $\mathcal{S}$ is the state space,

$\begin{array}{l}\hfill \text{P}(X_t\in A|X_0,\dots ,X_{t-1})=\text{P}(X_t\in A|X_{t-1}).\end{array}$

3.1 Discrete-state Markov chains

Let $\mathcal{S}=\left\{x_1,x_2,\dots \right\}$ be a discrete-state space. Then transition probabilities are of the form P _ij (t) = P[X _t = j|X _t−1 = i]. If (X _t ) _t≥0 is to converge to a stationary distribution, (X _t ) _t≥0 has to satisfy three conditions. First, the chain must be irreducible, which means that any state j can be reached from any state i in a finite number of steps. The chain must be positive recurrent, meaning that, on average, the chain starting in state i returns to state i in a finite number of steps for all $i\in \mathcal{S}$ . The chain must also be aperiodic, which means it is not expected to make regular oscillations between states. These terms are formalized below.

Definition 10

(X _t ) _t≥0 is irreducible if for all i, j, there exists an integer t > 0 such that P _ij (t) > 0.

Definition 11

An irreducible Markov chain (X _t ) _t≥0 is recurrent if the first return time τ _ii = $min\left\{t>0:X_t=i|X_0=i\right\}$ to state i has the property that for all i, $\text{P}({\tau }_{ii}<\infty )=1$ .

Definition 12

An irreducible, recurrent Markov chain is positive recurrent if for all i, $\mathbb{E}[{\tau }_{ii}]<\infty$ .

Definition 13

A Markov chain (X _t ) _t≥0 has stationary distribution π(⋅) if for all j and for all t ≥ 0,

$\begin{array}{l}\hfill \sum _i\pi (i){\text{P}}_{ij}(t)=\pi (j).\end{array}$

The existence of a stationary distribution for the chain is equivalent to that chain being positive recurrent.

Definition 14

An irreducible Markov chain is aperiodic if for all i,

$\begin{array}{l}\hfill gcd\left\{t>0:{\text{P}}_{ii}(t)>0\right\}=1.\end{array}$

Definition 15

(X _t ) _t≥0 is reversible if it is positive recurrent with stationary distribution π(⋅) if for all i, j, π(i)P _ij = π(j)P _ji .

The discrete-state Markov chain (X _t ) _t≥0 has a unique stationary distribution if it is irreducible, aperiodic, and reversible. The next example illustrates some of these properties.

Example 9

Consider the Markov chain (X _t ) _t≥0 with state space $\mathcal{S}=\left\{0,1,2\right\}$ . Let the transition probability matrix be given by

$\begin{array}{l}\hfill \mathbf{P}=\left[\begin{array}{lll}\frac{1}{8}& \frac{3}{8}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{8}& \frac{3}{8}\\ \frac{3}{8}& \frac{1}{2}& \frac{1}{8}\end{array}\right],\end{array}$

so that P _ij = P(X _t = j|X _t−1 = i). Since P₀₀ =P₁₁ =P₂₂ = $\frac{1}{8}$ , the chain is aperiodic, and since all transition probabilities are positive, the chain is clearly irreducible. Now we try to find a stationary distribution. To do this, we solve the following system of equations.

$\begin{array}{ll}\hfill \pi (0)& =\pi (0){\text{P}}_{00}+\pi (1){\text{P}}_{10}+\pi (2){\text{P}}_{20}\hfill \\ \hfill & =\frac{1}{8}\pi (0)+\frac{1}{2}\pi (1)+\frac{3}{8}\pi (2)\hfill \\ \hfill \pi (1)& =\pi (0){\text{P}}_{01}+\pi (1){\text{P}}_{11}+\pi (2){\text{P}}_{21}\hfill \\ \hfill & =\frac{3}{8}\pi (0)+\frac{1}{8}\pi (1)+\frac{1}{2}\pi (2)\hfill \\ \hfill \pi (2)& =\pi (0){\text{P}}_{02}+\pi (1){\text{P}}_{12}+\pi (2){\text{P}}_{22}\hfill \\ \hfill & =\frac{1}{2}\pi (0)+\frac{3}{8}\pi (1)+\frac{1}{8}\pi (2).\hfill \end{array}$

Solving this system gives

$\begin{array}{l}\hfill \pi (0)=\pi (1)=\pi (2)=\frac{1}{3}.\end{array}$

Since P is symmetric, P _ij π(i) =P _ji π(j) for all i, j. Thus, (X _t ) _t≥0 is reversible, so this stationary distribution is the chain's unique stationary distribution. Furthermore, the existence of the stationary distribution ensures that the chain is positive recurrent.

3.2 General state space Markov chain theory

Here, we generalize the concepts presented in Section 3.1 to cover a Markov chain that explores a general state space Ω. We begin with the notion of a transition kernel.

Definition 16

A function $K:[\mathrm{\Omega },\mathcal{B}(\mathrm{\Omega })]\mapsto [0,1]$ is a transition kernel if

1.

For all $A\in \mathcal{B}(\mathrm{\Omega })$ , K(⋅, A) is a nonnegative measurable function on Ω, and

2.

For all x ∈ Ω, K(x, ⋅) is a probability measure on $\mathcal{B}(\mathrm{\Omega })$ .

Definition 17

If (X _t ) _t≥0 is a Markov chain on a state space Ω, its transition kernel is given by

$\begin{array}{l}\hfill K(\mathbf{x},A)=\text{P}(X_{t+1}\in A|X_t=\mathbf{x})\end{array}$

for x ∈ Ω and $A\in \mathcal{B}(\mathrm{\Omega })$ . The k-step transition kernel for (X _t ) _t≥0 is

$\begin{array}{l}\hfill K^k(\mathbf{x},A)=\text{P}(X_{t+k}\in A|X_t=\mathbf{x}).\end{array}$

Now we address the notion of ϕ-irreducibility.

Definition 18

A Markov chain (X _t ) _t≥0 with transition kernel K(⋅, ⋅) on a state space Ω is ϕ-irreducible if there exists a nontrivial measure ϕ on $\mathcal{B}(\mathrm{\Omega })$ such that if ϕ(A) > 0, there exists an integer k _x for all x ∈ Ω such that $K^{k_{\mathbf{x}}}(\mathbf{x},A)>0$ .

Next, we discuss the idea of stationary distributions.

Definition 19

A probability measure π(⋅) on $\mathcal{B}(\mathrm{\Omega })$ is called a stationary measure for the Markov chain (X _t ) _t≥0 having state space Ω and transition kernel K(⋅, ⋅) if for all $A\in \mathcal{B}(\mathrm{\Omega })$ ,

$\begin{array}{l}\hfill \pi (A)={\int }_{\mathrm{\Omega }}\pi (\text{d}\mathbf{x})K(\mathbf{x},A).\end{array}$

In order to address uniqueness of the stationary measure, we need the concepts of recurrence and aperiodicity. In order to develop these, let η _A denote the expected number of visits the chain makes to the set $A\in \mathcal{B}(\mathrm{\Omega })$ . Then the set A is called a recurrent set if ${\eta }_A=\infty$ .

Definition 20

The Markov chain (X _t ) _t≥0 with state space Ω is recurrent if for all $A\in \mathcal{B}(\mathrm{\Omega })$ , A is recurrent.

In order to discuss aperiodicity, we need the notion of small sets.

Definition 21

A set $C\in \mathcal{B}(\mathrm{\Omega })$ is a small set if there exists a nontrivial measure ν _m on $\mathcal{B}(\mathrm{\Omega })$ and an integer m > 0 such that for all x ∈ C and $A\in \mathcal{B}(\mathrm{\Omega })$ , K ^m (x, A) ≥ ν _m (A).

Now we examine d-cycles. Let

$\begin{array}{l}\hfill E_C=\left\{n\ge 1:\text{the set}C\text{is small with respect to}{\nu }_n\text{, with}{\nu }_n={\delta }_n\nu \text{for some}{\delta }_n>0\right\}.\end{array}$

Definition 22

Let (X _t ) _t≥0 be a ϕ-irreducible Markov chain on Ω, and let $C\in \mathcal{B}(\mathrm{\Omega })$ , where C is a small set. Let $\left\{D_i\right\}$ , i = 1, …, n, partition Ω. If for each i, i = 1, …, n − 1, for x ∈ D _i , K(x, D _i+1) = 1 and for x ∈ D _n , K(x, D ₁) = 1, $\left\{D_i\right\}$ is a d-cycle. The largest value of d for which a d-cycle for (X _t ) _t≥0 occurs is the period of (X _t ) _t≥0. If d = 1, the chain is aperiodic, and if there exists a nontrivial set A that is small with respect to ν ₁, the chain is strongly aperiodic.

If a ϕ-irreducible Markov chain is recurrent and strongly aperiodic, its stationary distribution is unique (Meyn and Tweedie, 2005). One final property to discuss is that of reversibility.

Definition 23

A Markov chain (X _t ) _t≥0 on a state space Ω with transition kernel K that admits a density k(⋅, ⋅) and stationary measure π is reversible if for all x, y ∈ Ω,

$\begin{array}{l}\hfill k(\mathbf{x},\mathbf{y})\pi (\mathbf{x})=k(\mathbf{y},\mathbf{x})\pi (\mathbf{y}).\end{array}$

Reversibility is helpful because strong aperiodicity and recurrence can be difficult to verify. Uniqueness of the stationary distribution can also be verified by showing that (X _t ) _t≥0 is ϕ-irreducible, aperiodic, and reversible.

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Basic Concepts in Stochastic Processes

Oliver C. Ibe , in Markov Processes for Stochastic Modeling (Second Edition), 2013

2.2 Classification of Stochastic Processes

A stochastic process can be classified according to the nature of the time parameter and the values that $X(t,w)$ can assume. As discussed earlier, T is called the parameter set of the stochastic process and is usually a set of times. If T is an interval of real numbers and hence is continuous, the process is called a continuous-time stochastic process. Similarly, if T is a countable set and hence is discrete, the process is called a discrete-time stochastic process. A discrete-time stochastic process is also called a random sequence, which is denoted by $\left\{X\right[n]:n=1,2,\dots \}$ .

The values that $X(t,w)$ assumes are called the states of the stochastic process. The set of all possible values of $X(t,w)$ forms the state space, E, of the stochastic process. If E is continuous, the process is called a continuous-state stochastic process. Similarly, if E is discrete, the process is called a discrete-state stochastic process.

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Markov Renewal Processes

Oliver C. Ibe , in Markov Processes for Stochastic Modeling (Second Edition), 2013

6.7 Markov Regenerative Process

Markov regenerative processes (MRGPs) constitute a more general class of stochastic processes than traditional Markov processes. Markovian dependency, the first-order dependency, is the simplest and most important dependency in stochastic processes. The past history of a Markov chain is summarized in the current state and the behavior of the system thereafter only depends on the current state. The sojourn time of a homogeneous CTMC is exponentially distributed. However, nonexponentially distributed transitions are common in real-life systems. SMPs have generally distributed sojourn times but lack the ability to capture local behaviors during the intervals between successive regenerative points. MRGPs are discrete-state continuous-time stochastic processes with embedded regenerative time points at which the process enjoys the Markov property. MRGPs provide a natural generalization of SMPs with local behavior accounted for. Thus, the SMP, the discrete-time Markov chain, and the CTMC are special cases of the MRGP.

A stochastic process $\{Z(t),t\ge 0\}$ is called an MRGP, if there exists a Markov renewal sequence $\{(X_n,T_n),n\ge 0\}$ of random variables such that all the conditional finite dimensional distributions of $\{Z(T_n+t),t\ge 0\}$ given $\{Z(u),0\le u\le T_n,X_n=i\}$ are the same as those of $\{Z(t),t\ge 0\}$ given $X_0=i$ . The above definition implies that

(6.64) $P[Z(T_n+t)=j|Z(u),0\le u\le T_n,X_n=i]=P[Z(t)=j|X_0=i]$

The MRGP does not have the Markov property in general, but there is a sequence of time points $\{T_0=0,T_1,T_2,\dots \}$ at which Markov property holds. From the above definition, it is obvious that every SMP is an MRGP. The difference between SMP and MRGP is that in an SMP every state transition is a regeneration point, which is not necessarily true for the MRGP. The requirement of regeneration at every state transition makes the SMP of limited interest for transient analysis of systems that involve deterministic parameters as in a communication system. An example of the sample path of MRGP is shown in Figure 6.8, where it is shown that not every arrival epoch is a regeneration point.

Figure 6.8. An example of the sample path of an MRGP.

As discussed earlier, MRGPs are a generalization of many stochastic processes including Markov processes. The difference between SMP and MRGP can be seen by comparing the sequence $T_n$ with the sequence obtained from the state transition instants $K_n$ . In an SMP, every state transition is a regeneration point, which means that $K_n=T_n$ . For MRGP, every transition point is not a regeneration point and as such $T_n$ is a subsequence of $K_n$ . The fact that every transition is a regeneration point in the SMP limits its use in applications that involve deterministic parameters.

The time instants $T_n$ at which transitions take place are called regeneration points. The behavior of the process can be determined by the latest regeneration point, which can specify the elapsed time and the latest state visited. A regeneration point is independent of the elapsed time up to the instant of jump instant.

For an SMP, no state change occurs between successive regeneration points and the sample paths are piecewise constant and $T_n$ is when the nth jump occurs. But for an MRGP, the stochastic process between $T_n$ and $T_{n+1}$ can be any continuous-time stochastic process, including CTMC, SMP, or another MRGP. This means that the sample paths are no longer piecewise constant, because local behaviors exist between consecutive Markov regeneration points. The jumps do not necessarily have to be at the $T_n$ . More information on MRGPs can be found in Kulkarni (2010).

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Markov Chain Monte Carlo

A.M. Johansen , in International Encyclopedia of Education (Third Edition), 2010

Markov Chains

A discrete-time Markov chain is, roughly speaking, some collection of random variables with a temporal ordering which have the property that conditional upon the present, the future does not depend upon the past. This concept, which can be viewed as a form of something known as the Markov property, can be formalized by saying that a collection of random variables X ₁,X ₂,… forms a Markov chain if, and only if, the joint pdf of the first n elements of the sequence may be decomposed in the following manner for any value of n:

$p\left(x_1,\dots ,x_n\right)=p\left(x_1\right)p\left(x_2|x_1\right)\dots p\left(x_n|x_{n-1}\right)$

Although a class of continuous-time stochastic processes with a similar lack-of-memory property can be defined, these are rarely used in an MCMC context.

The basic idea behind MCMC is that, if it is possible to construct a Markov chain such that a sequence of draws from that chain has similar statistical properties (in some sense) to a collection of draws from a distribution of interest, then it is also possible to estimate expectations with respect to that distribution by using the standard Monte Carlo estimator but using the dependent collection of random variables obtained by simulating a Markov chain rather than an independent collection.

A few concepts are required to understand how a suitable chain can be constructed. The conditional probability densities p(x _n |x _n−1) are often termed transition kernels, as they can be thought of as the probability density associated with a movement from x _n−1 to x _n . If p(x _n |x _n−1) does not depend directly upon the value of n, then the associated Markov chain is termed time homogeneous (as its transitions have the same distribution at all times). A time homogeneous Markov chain with transition kernel k, is said to have a probability density f as an invariant or stationary distribution if

$\int f\left(x\right)k\left(y\right|x)\text{d}x=f(y)$

If a Markov chain satisfies a condition known as detailed balance with respect to a distribution, then it is reversible (in the sense that the statistics of the time-reversed process match those of the original process) and hence invariant with respect to that distribution. The detailed balance condition states, simply, that the probability of starting at x and moving to y is equal to the probability of starting at y and moving to x. Formally, given a distribution f and a kernel k, one requires that f  (x)k( y|x) = f  ( y)k(x|y) and simple integration of both sides with respect to x proves invariance with respect to f under this condition.

The principle of most MCMC algorithms is that, if a Markov chain has an invariant distribution, f, and (in some suitable sense) forgets where it has been, then using its sample path to approximate integrals with respect to f is a reasonable thing to do. This can be formalized under technical conditions to provide an analog of the law of large numbers (often termed the ergodic theorem) and the central limit theorem. The first of these results tells us that we can expect the sample average to converge to the appropriate expectation with probability one as the number of samples becomes large enough; the second tells us that the estimator we obtain is asymptotically normal with a particular variance (which depends upon the covariance of the samples obtained, demonstrating that it is important that the Markov chain forgets where it has been reasonably fast). These conditions are not always easy to verify in practice, but they are important: it is easy to construct examples which violate these conditions and have entirely incorrect behavior.

In order to use this strategy to estimate expectations of interest, it is necessary to construct Markov chains with the correct invariant distribution. There are two common approaches to this problem.

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Special Volume: Mathematical Modeling and Numerical Methods in Finance

Olivier Pironneau , Yves Achdou , in Handbook of Numerical Analysis, 2009

9.1 Limitation of the Black—Scholes model: the need for calibration

Consider a European option on a given stock with a maturity T and a payoff function P ₀ and assume that this option is on the market. Call p its present price. Also, assume the risk-free interest rate is the constant r. One may associate with p the so-called implied volatility, the volatility σ_imp, such that the price given by formula (1.6) at time t = 0 with σ = σ_imp coincides with p. If the Black—Scholes model was sharp, then the implied volatility would not depend on the payoff function P ₀. Unfortunately, for example, vanilla European puts or calls, it is often observed that the implied volatility is far from constant. Rather, it is often a convex function of the strike price. This phenomenon is known as the volatility smile. A possible explanation for the volatility smile is that the deeply out-of-the-money options are less liquid, thus relatively more expensive than the options in the money.

This shows that the critical parameter in the Black—Scholes model is the volatility σ. Assuming σ to be constant and using (1.3) often lead to poor predictions of the options prices. The volatility smile is the price paid for the too great simplicity of Black—Scholes assumptions. It is, thus, necessary to use more involved models that must be calibrated.

Let us first explain what the term calibration means: consider an arbitrage-free market described by a probability measure P on a scenario space (Ω, A). There is a risk-free asset whose price at time τ is e ^rt , r ≥ 0 and a risky asset whose price at time τ is S _τ. Specifying an arbitrage-free option pricing model necessitates the choice of a risk-neutral measure, that is, a probability P* equivalent to P such that the discounted price (e^-rt S _τ)_τ∈[0,T ] is a martingale under P*. Such a probability measure P* allows for the pricing of European options; consider a European option with payoff $P_{\circ }$ at maturity t ≤ T: its price at time τ ≤ t is P _τ = e^-r(t-τ) E ^P* ( $P_{\circ }$ (S _t )|ℱ_τ), where (ℱ_τ)_τ∈[0,T] is the natural filtration.

The pricing model P* must be compatible with the prices of the options observed on the market, whose number may be large. Model calibration consists of finding P* such that the discounted price (e^-rt S _τ)_τ∈[0,T] is a martingale and such that the option prices computed with the model coincide with the observed option prices. This is an inverse problem.

Popular extensions to the Black—Scholes model are

•

local volatility models: the volatility is a function of time and the spot price, that is, σ _t = σ(S _t , t). With suitable assumptions on the regularity and the behavior at infinity of the function σ, (1.6) holds and P _t = p(S _t , t), where p satisfies the final value problem (1.3), in which σ varies with t and S. Calibration of local volatility has been much studied (see Achdou and Pironneau [2002], Andersen and Brotherton-Ratcliffe [1998], Dupire [1997], Jackson, Süli and Howison [1998] for volatility calibration with European options and Achdou [2005], Achdou and Pironneau [2005] with American options).

•

• stochastic volatility models: one assumes that σ _t = f(y _t ), where y _t is a continuous time stochastic process, correlated or not to the process driving S _t (see Section 7). Stochastic volatility calibration has been performed by Nayak and Papanicolaou [2006].

•

• LéAvy driven spot price: one may generalize the Black—Scholes model by assuming that the spot price is driven by a more general stochastic process, a LéAvy process Cont and Tankov [2003], Merton [1976]. LéAvy processes are processes with stationary and independent increments that are continuous in probability. For a LéAvy process X _τ on a filtered probability space with probability P, the LéAvy—Khintchine formula says that there exists a function χ: R → ℂ such that

$\begin{array}{l}E({\text{e}}^{iuX\tau })={\text{e}}^{\tau \chi (u)},\\ \chi (u)=-\frac{{\sigma }^2{u}^2}{2}+i\beta u+{\int }_{|z|<1}({\text{e}}^{iuz}-1-iuz)v(\text{d}z)+{\int }_{|z|>1}({\text{e}}^{iuz}-1)v(\text{d}z),\end{array}$

for σ ≥ 0, β ∈ R and a positive measure ν on R\{0} such that ∫ _R min(1, z ²)ν(dz) < +∞. The measure ν is called the LéAvy measure of X. We focus on LéAvy measure with a density, ν(dz) = k(z)dz. Assume that the discounted price of the risky asset is a square-integrable martingale under P and that it is represented as the exponential of a LéAvy process:

${\text{e}}^{-r\tau }{S}_{\tau }=S_0{\text{e}}^{X\tau }.$

The martingale property is that E(e ^{X _τ} ) = 1, that is,

${\int }_{|z|>1}{\text{e}}^{z}v(\text{d}z)<\infty \text{and}\beta =-\frac{{\sigma }^2}{2}-{\int }_R({\text{e}}^z-1-z{1}_{|z|\le 1})k(z)(\text{d}z),$

and the square integrability comes from the condition ∫_|z|>1 e^2z k(z)dz < ∞.

With such models, the pricing function for a European option is obtained by solving a PIDE, with a nonlocal term (see Cont and Tankov [2003], Pham [1998] for the analysis of this equation and Achdou and Pironneau [2005], Cont and Voltchkova [2003], Matache, Nitsche and Schwab [2003], Matache, von Petersdoff and Schwab [2004] for numerical methods based on the PIDE). Calibration of LéAvy models with European options has been discussed by Cont and Tankov [2004, 2006].

In this paragraph, we assume that the model is characterized by parameters Θ in a suitable class Θ.

The last two classes of models (stochastic volatility and LéAvy driven assets) describe incomplete markets (see Cont and Tankov [2004]): the knowledge of the historical price process alone does not allow to compute the option prices in a unique manner. When the option prices do not determine the model completely, additional information may be introduced by specifying a prior model. If the historical price process has been estimated statistically from the time series of the underlying asset, this knowledge has to be applied in the inverse problem; calling P ₀ the prior probability measure obtained as an estimation of P, the inverse problem may be cast in a least-square formulation of the type: find Θ ∈ Θ that minimizes

(9.1) $\underset{i\in I}{\Sigma }{\omega }_i{\left(P\Theta (0,S_{\circ },t_i,x_i)-\overline{p}i\right)}^2+\rho J_2(P^{\Theta },P_0),$

where

•

ω _i are suitable positive weights,

•

S _∘ is the price of the underlying asset today,

•

P ^Θ(0, S _∘, t _i , x _i ) is the price of the option with maturity t _i strike x _i , computed with the pricing model associated with Θ,

•

ρJ ₂(P ^Θ, P ₀) is a regularization term that measures the closeness of the model P ^Θ to the prior. The number ρ > 0 is called the regularization parameter. This functional has two roles: (1) it stabilizes the inverse problem, and for that, ρ should be large enough and J ₂ should be convex or at least convex in a large enough region; (2) it guarantees that P ^Θ remains close to P ₀ in some sense. The choice of J ₂ is very important: J ₂(P ^Θ, P ₀) is often chosen as the relative entropy of the pricing measure P ^Θ with respect to the prior model P ₀ (see Avellaneda [1998]), because the relative entropy becomes infinite if P ^Θ is not equivalent to P ₀. Some authors have argued that such a choice may be too conservative in some cases, for two reasons: (a) the historical data that determine the prior may be missing or partially available and (b) in the context of volatility calibration, once the volatility is specified under P ₀, then the volatility under P ^Θ must be the same for the relative entropy to be finite. A different approach was considered that allowed for volatility calibration (see Avellaneda, Friedman, Holmes and Samperi [1997]).

Note that local volatility models describe complete markets; however, an additional regularization cost functional is necessary, as explained in the paragraph below.

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Integrated Population Biology and Modeling, Part B

Anuj Mubayi , ... Carlos Castillo-Chavez , in Handbook of Statistics, 2019

1.2 Examples of Basic Stochastic Processes

Often analysis of complex stochastic processes involves systematic simplification of stochastic model to any basic stochastic process. We provide some basic stochastic processes that are referred in the analysis of complex stochastic models introduced in this chapter. A flow chart of stochastic process that includes these basic stochastic processes is shown in Fig. 2.

Fig. 2. A flowchart showing different types of stochastic processes. Processes lower in the chart are special cases of the processes hierarchical above them. DTMC, CTMC, and SDE stands for discrete-time Markov chain, continuous-time Markov chain, and stochastic differential equations, respectively. Example of processes that are provided in this chapter are shown in blue color.

•

A random walk is an example of a stochastic process in discrete time, where a particle starts at the origin at time 0 and moves one distance left with probability p or one distance unit right with probability 1 − p at each time unit. The random variable X _n then denotes the position of the particle at time step n.

•

The Wiener process (or Brownian Motion) is an example of a stochastic process in continuous time. In 1828 the Scottish botanist Robert Brown observed that pollen grains suspended in water moved in an apparently random way, changing direction continuously, which was due to the pollen grains being bombarded by water molecules. The precise mathematical formulation of this phenomenon was later given by Norbert Wiener in 1923. A Wiener process is defined as a stochastic process W(t) _t≥0 which satisfies the following characteristics:

(i)

W(0) = 0

(ii)

W(t) _t≥0 has independent increments (i.e., $W_{t_1}$ , $W_{t_2}-W_{t_1}$ , …, $W_{t_2}-W_{t_1}$ are independent random variables for all 0 ≤ t ₁ < t ₂ … < t _k )

(iii)

W _s+t − W _s ∼ N(0, t) for all t > 0

Remark 1

This stochastic process is a building block of many of the models in the literature. It can be obtained as the limit of a random walk when the time steps and the jump sizes go to 0 in a suitable way.

Remark 2

The Wiener process is often called standard Brownian motion process or Brownian motion due to its connection with the physical process known as Brownian movement or Brownian motion. It is one of the best known Lévy processes (stochastic processes with stationary independent increments).

A Gaussian process is a simple stochastic process which satisfies the following property: For any finite set of indices t ₁, …, t _k the vector of random variables (X(t ₁), … X(t _k )) follows a k-dimensional normal distribution.

Remark 3

Any continuous-time stochastic process with independent increments and finite second moments ( $E(X^2(t))<\infty$ for all t) is a Gaussian process provided that X(t ₀) is Gaussian for some t ₀.

Remark 4

The Wiener process is a Gaussian process. The Wiener process is continuous with mean zero (E(W(t)) = 0) and variance proportional to the elapsed time (V ar(W(t)) = t).

A stationary stochastic process is a specific type of process in which X(t) has the same distribution as X(t + h) for all h > 0.

Remark 5

The Wiener process cannot be stationary since the variance increases with t. The auto-covariance function is given by Cov(W _t , W _s ) = min(s, t).

Remark 6

The sample paths of a Wiener process are nowhere differentiable as their total variation on a closed interval of the real line is infinite.

•

A stochastic differential equation can be defined as a stochastic process, X _t = X(t), satisfying the following equation:

(1) $\begin{array}{l}\hfill d{X}_t=f(X_t,t)dt+g(X_t,t)d{W}_t\end{array}$

which can be also written as

(2) $\begin{array}{l}\hfill X_t=X_{t_0}+{\int }_{t_0}^{t}f(X_s,s)ds+{\int }_{t_0}^{t}g(X_s,s)d{W}_s,\end{array}$

where W _t = W(t) is the Wiener process, $X_{t_0}$ is a random variable (independent of the Wiener process), f(⋅), the drift part or the deterministic component, and g(⋅) is the diffusion part or the stochastic component.

Remark 7

If f(⋅) and g(⋅) do not depend on t, the process is called time-homogeneous.

Remark 8

Suppose g(X _t , t) = c (i.e., constant function) then the second integral becomes c(W(t) − W(t ₀)), which is a random variable with expectation 0 since the increments of a Wiener process have expectation 0.

•

The limiting diffusion processes can be obtained as solutions of stochastic differential equations of the form

$\begin{array}{l}\hfill dX(t)=\mu (X(t))dt+\sigma (X(t))dW(t),\end{array}$

where W denotes a standard Wiener process. X(t) then has the well-known properties

(3) $\begin{array}{l}\hfill \underset{h\to 0}{lim}\frac{E_x\{X(t+h)-X(t)|X(t)=x\}}{h}=\mu (x),\end{array}$

(4) $\begin{array}{l}\hfill \underset{h\to 0}{lim}\frac{E_x\{{[X(t+h)-X(t)]}^2|X(t)=x\}}{h}=\sigma (x),\end{array}$

where E _x denotes expectation given X(t) = x. These properties show that the mean increment of the stochastic process X(t) during a small time interval (h), given that the process has reached x, is the same as the increment of the solution of the deterministic ordinary differential equation

(5) $\begin{array}{l}\hfill \frac{dX(t)}{dt}=\mu (X),\end{array}$

and the variability of the process is completely described by a function σ ².

•

A Wiener process with drift is a stochastic process, X _t , which satisfies the following equation:

$\begin{array}{l}\hfill d{X}_t=\mu dt+\sigma d{W}_t,\end{array}$

and its solution is

$\begin{array}{l}\hfill X_t=\mu t+\sigma W_t.\end{array}$

Hence, this X _t process is normally distributed with mean μt and variance σ ² t.

•

A Geometric Brownian motion (stochastic version of Malthusian model or Black-scholes model) is a stochastic process which satisfies

$\begin{array}{l}\hfill d{X}_t=\mu X_{t}dt+\sigma X_{t}d{W}_t,\end{array}$

and its solution is

$\begin{array}{l}\hfill X_t=X_{0}exp\left(\left(\mu -\frac{1}{2}{\sigma }^2\right)t+\sigma W_t\right).\end{array}$

Hence, the process only takes positive values, and follows a log-normal distribution with parameters $\left(\mu -\frac{1}{2}{\sigma }^2\right)$ and σ ² t.

Remark 9

The parameter μ represents the mean growing rate per capita and σ the intensity of the random effects on the growing rate.

•

An Ornstein–Uhlenbeck process is a process that is attracted to some constant level but is also continuously perturbed by noise. It satisfies the following equation:

$\begin{array}{l}\hfill d{X}_t=-\left(\frac{X_t-\alpha }{\tau }\right)dt+\sigma d{W}_t\end{array}$

where τ has units of time, and the autocorrelation of the process is corr(X _t , X _t+s ) = e ^−s/τ (autocorrelation decreases with a factor of 1/e after τ units of time). Its solution is

$\begin{array}{l}\hfill X_t=X_0{e}^{-t/\tau }+\alpha (1-e^{-t/\tau })+e^{-t/\tau }{\int }_0^t{e}^{s/\tau }\sigma d{W}_s.\end{array}$

Remark 10

The Ornstein–Uhlenbeck process is a stochastic process that describes the velocity of a Brownian particle under the influence of friction. The process is a stationary Gauss–Markov process, that is, it is a Gaussian process, a Markov process, and is homogeneous, which tends to drift toward its long-term mean (i.e., mean-reverting).

Remark 11

Note, the amplitude of the noise does not change over time for the Wiener and the Ornstein–Uhlenbeck process, whereas for Geometric Brownian motion the amplitude of the noise is proportional to the state variable. When the diffusion term does not depend on the state variable X _t as in the Wiener process with drift and the Ornstein–Uhlenbeck process, we say that it has additive noise and in the case of Geometric Brownian motion, we say that it has multiplicative noise.

Remark 12

Since a Wiener process is nondifferentiable, Itô's formula is often used to find the explicit solution of simple stochastic differential equations with a Wiener process. When no explicit solution is available, we can approximate different characteristics of the process by simulation, such as sample paths, moments, qualitative behavior, etc. Usually such simulation methods are based on discrete approximations of the continuous solution to a stochastic differential equation. Some of the principal discrete schemes are Euler–Maruyama and Milstein.

In this chapter, we provide examples of various types of analysis that can be carried out for understanding dynamics of some stochastic models (Fig. 3).

Fig. 3. Different types of analysis of a stochastic process collected in this chapter are shown here in schematic flow diagram.

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