Stochastic Differential Equations - Science topic

Fixed Point theorem.

Julia differential equation solvers have high order and adaptive methods (https://diffeq.sciml.ai/stable/tutorials/sde_example/). But more importantly, these implementations are compatible with automatic differentiation, making it easy to do things like gradient descent for doing parameter estimation and model calibration. Tutorials of this can be found in the Julia SciML libraries, such as https://sensitivity.sciml.ai/dev/sde_fitting/optimization_sde/. The other advantage of course is performance, where the Julia performance over Python and MATLAB solvers is a few orders of magnitude (https://benchmarks.sciml.ai/html/MultiLanguage/wrapper_packages.html)

For those who were following this question, after a long search I couldn't find any package that implements the CIR model under a Bayesian framework. So I wrote up a Python script to do that. Interested readers can find the code in my GitHub repository https://github.com/Kwabena16108/CIR-Bayesian-Estimation.

Hope this helps.

Write the equation in a form more useful for calculating anything:

dX/dt=(a(t)+b(X-X0))X+cXη(t)

where η(t) is the noise.

Now write a(t)X+b(X-X0)X=-dU(X)/dX, which shows that this equation describes the motion of a particle in a potential U(X)=-(1/2)(a(t)-bX0)X^2-(b/3)X^3, in the presence of multiplicative noise.

So, for a(t)=0, the term that’s quadratic in X, does describe oscillations with frequency squared bX0, with the cubic term describing escape from the well.

However the multiplicative noise complicates things.

If we divide out by X and set Y(t)=ln X, the noise for Y is additive.

you should have a qualitative motivation to propose a specific model to fit a dataset, you do not just try models by chance. The models you propose possess a stationary distribution, while your dataset seems to suggest an illuminated growth. Have you considered this fact?

Some of your models could have problems if zero is reached, you should specify parameters range in which this does not happen and verify that your estimates are compatible

If you have a reason to model the data with one of the side you propose you'd better observe that for all your models the transition density is known in an explicit form. That means that you can do maximum likelihood (without PSEUDO!), and that you can also apply goodness of fit tests (if you really like). However, things are a bit long to discuss here and it seems that you need some guidance.

Thanks @Ette Etuk

In my opinion, one cannot avoid to learn the theory of SDEs without any theorems/proofs. I recommend Bernt Oksendal's book "Stochastic Differential Equations", which has some examples about the applications of SDEs in finance.

The short answer is No. And there are many examples where the probability distribution at long times is, indeed, independent of the initial conditions. The dependence on the properties of the noise is a bit more subtle, since Brownian motion is determined by the 1- and 2-point functions and, unless the probability distribution at long times is a Gaussian, its 2-point function isn't related in a simple way with the 1- and 2-point function of the noise.

The Fokker-Plank equation can be treated as a so-called forward Kolmogorov equation for a certain diffusion process.

To derive a stochastic equation for this diffusion process it is very useful if you know a generator of this process. Finally, to find out a form of the generator you have to consider a PDE, dual to the Fokker-Plank equation which is called the backward Kolmogorov equation. The elliptic operator in the backward Kolmogorov equation coincides with the generator of the required disffusion process. Let me give you an example.

Assume that you consider the Cauchy problem for the Fokker-Plank type equation

u_t=Lu, u(0,x)=u_0(x),

where Lu(t,x)=[A^2(x)u(t,x)]_{xx}-[a(x)u(t,x)]_x.

The dual equation is h_t+L^*h=0, where L^*h= A^2(x)h_{xx}+a(x)h_x.

As a result the required diffusion process x(t) satisfies the SDE

dx(t)=a(x(t))dt+A(x(t))dw(t), x(0)= \xi,

where w(t) is a Wiener process and \xi is a random variable independent on w(t) with the distribution density u_0(x).

You may see the book Bogachev V.I., Krylov N.V., Röckner M., Shaposhnikov S.V. "Fokker-Planck-Kolmogorov equations"

From my practical experience if f(0)=0, then g(0) doesn't equal 0, because of the sensors noises.

With Markov chains having a detailed balance distribution is a stronger condition then having a stationary distribution (that you can check by Lyapunov theory).

Every detailed balance distribution is stationary, but not all stationary distributions are detailed balanced. The proof is very easy. Detail balance means there exist a distribution pi(k) such that

pi(k)q_kj =pi(j)q_jk,

where q_jk is any non-diagonal entry of the generator matrix Q.

Now if on both sides you take the sum over all the k that are different from j you get that stationarity follows (pi Q=0).

You can check it on Durret's book Essentials of stochastic processes, page 130

I'm not sure what happens with diffusions, but it shuold be the same.

This paper might help

The question about ``equilibrium solution" and probably also many other questions concerning Fokker-Planck equation are answered in the book of H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications. For this one search the key-word detailed balance, as suggested above. The question abut stability of SDE is discussed for example here http://www6.cityu.edu.hk/ma/ws2010/doc/mao_notes.pdf .

Vortex flows are ubiquitous at all scales of matter organization, from quantum systems to large structures of the universe. In the most general mathematical sense, it is useful to look at these structures in a unified way. When trying to organize my ideas in this field, I have encountered a book on the general theory of vortices that I recommend as a valuable source of information placing the subject in a multidisciplinary context; for the synopsis please see:

Regarding the research suggestions, I agree with the previous comments, but I can add some specific answers:

>>Do you think this topic is linked with dynamic systems analysis? if yes: how this analysis should be done?<<

The answer is definitely yes. You can consider the following paper as an illustration of the methods derived from Dynamical Systems Theory.

>>What kind of theoretical analysis (based on differential equations formulation) could be added to my research (especially to the vortex's stability and/or stochastic factors)?<<

The theory of stable and unstable manifolds discussed in the reference above. It is also useful to consult a book by Ottino: The kinematics of mixing: stretching, chaos and transport

>>What's your best suggestion for making sure that the results obtained (from experiments) are dependable? (Validation by CFD?)<<

The best way to obtain reliable results is to set an experiment as carefully as possible. CFD calculations are generally validated by experiment. However, the use of CFD to validate the experimental results is very useful (I always look at numerical simulations as a parallel experiment).

Dear Dr. Ali Namadchian

Greating!

The following links may be useful to answer

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With best wishes.

Thanks

Dear Arif, see attached pdf, Chap.4, point 4.5 . Gianluca

Dear Amita,

It is possible to use stochastic Lyapunov's second method (stochastic Lyapunov functions). See, for example, links below.

Something like least-squares minimisation in the programming language of your choice would do it. There seems to be Matlab code to do exactly what you want here - http://jmahaffy.sdsu.edu/courses/f09/math636/lectures/lotka/qualde2.html#fitparameter

Dear Shaker,

A nice review of bifurcation theory for physicist is: Crawford, J. D. (1991). Introduction to bifurcation theory. Reviews of Modern Physics, 63(4), 991.

and if you wish to learn finer details, I would suggest this book:  Manneville, P. (1995). Dissipative structures and weak turbulence. In Chaos—The Interplay Between Stochastic and Deterministic Behaviour (pp. 257-272). Springer Berlin Heidelberg.

Cheers,

Carles

check out Gardiner's handbook. it helped me a lot with understanding this link

Crispin W. Gardiner, Handbook of stochastic methods: For Physics, Chemistry and the Natural Sciences

I would use code [2} above with two parameters theta[1] and theta[2]. 

Also,  it is very easy to code this directly without using any packages by applying the Euler Maruyama approximation method (which is described in E. Allen's book).

Also, see the book by Linda J.S. Allen which has ALL THE CODES for such examples problems given in the book . So may copy into R directly and implement:

Linda J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second Edition

Also, the following papers contain more examples of somewhat more complicated stochastic differential equations which have been solve in MATLAB (similar to R) using Euler Maruyama approximation: 

A.S. Ackleh and S. Hu, Comparison between Stochastic and Deterministic Selection-Mutation Models. Mathematical Biosciences and Engineering, 4(2007), 133-157.

A.S. Ackleh, K. Deng and Q. Huang, Stochastic Juvenile-Adult Models with Application to a Green Tree Frog Population. Journal of Biological Dynamics, 5(2011), 64-83.

Yes, see: Sensitivity to the Service-Time Distribution in the Nonstationary Erlang Loss Model

Jimmie L. Davis, William A. Massey and Ward Whitt

Management Science

Vol. 41, No. 6 (Jun., 1995), pp. 1107-1116.

I don't have page number, but this reference is a good start.

Some related papers are given below.

I am not sure of the formulation of your problem. So, I propose 3 interpretations in the following. It will be easier to explain my point with some notations.

Let X be your state vector, Y your observation, theta your vector of parameters if there are any.

Your stochastic differential equation:

d X_t = F(t, X_t, theta) dt + G(t, X_t, theta) dBt  (Eq. 1) .

Your observation equation:

Y_t = H(t, X_t, epsilon, theta),  (Eq. 2)

with epsilon being random.

1) If you want to maximize the likelihood for a "state estimation" i.e. to estimate the state X_t at several times t knowing Y_t' at several times t'

However, this problem is trivial and I do not think this is what you want to do.

Using the Markov property of your system, in a discrete-time point of view, it is easy to show that the function to be maximized reads:

p( Y_1 , ... , Y_n | X_1, ... , X_n) = p( Y_1 | X_1) ... p( Y_n | X_n) .

It only relies on Eq. 2 (your model of observation) and not on Eq.1 (your hidden dynamics).

2) If you want to maximize the likelihood for a "parameter estimation" i.e. to estimate the vector of parameter, theta, knowing Y_t' at several times t'. 

I think Sergiy Prykhodko understood your problem that way. By the way, the papers he suggested seem interesting.

For this problem, a very powerful tools exist in stochastic calculus, even if X_t is not Gaussian. Based on Girsanov theorem, an explicit expression of the log-likelihood can be derived in a general case. You can find it for instance in:

P. Rao. Statistical inference for diffusion type processes. Arnold, 1999

3) If you want to focus on "state estimation" i.e. to estimate the state X_t at several times t knowing Y_t' at several times t' using the likelihood p(Y_t | X_t) (without dealing with parameters).

This problem, so-called filtering problem or smoothing problem, is well documented and widely used. I advice the following papers:

 A. Doucet, N. De Freitas, and N. Gordon. Sequential Monte Carlo methods in practice . Springer, 2001.

A. Doucet and A. Johansen. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of Nonlinear Filtering , 12:656–704, 2009.

Dear Mahmood, you can see the book:

Shaikhet L. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Dordrecht, Heidelberg, New York, London, 2013. http://www.springer.com/engineering/control/book/978-3-319-00100-5

Thank you for answers. Let me explain more precisely my question: How can one simulate the following Ornstein Uhlenbeck stationary Gaussian process,

dX_t=X_0-\alpha X_tdt +dB_t, where X_0=\int_0^\infty e^{\alpha s} dB_s and B is a fractional brownian motion.

Dear Igor; I think if you try enough you can transform Bessel differential equation, which is known has oscillatory solutions (i.e., Hankel Fct)  into the Volterra linear integral format.  First try to find the Green's function as such that it satisfies the boundary condition  G(r-r', t-t') = 0    r' at  S(BC), and the initial condition where G should go every where zero in the domain of  interest, exception at point   r where it should go to infinity represented by a Generalized function, which would be the point source solution of the differential equation. In the case of Parabolic DE, one has:

G(r-r', t-t')= 8^-1[Pi Kappa(t-t')]^-3/2 exp{[- (r-r')²/kappa(t-t')]} 

Note: The Greens functions of many DE which are popular in science and engineering are are known in the literature. Since in Finite or Boundary Element methods of numerical solution of DE,  we are employing Integral Equation format to convert them into the matrix form as a final step for the machine computation practices.

Best Luck , New Year Greetings.  

Tarık  

Note: Elements of Partial Differential Equations by Ian N  SNEDDON.

Dear Mantas Gabrielaitis,

Thank you very much for your answer to my question, I investigated your answer and this answer is enough for now, but associated with other versions of the FP equation will give you information, see you as soon as possible...

thank you very much,actually i have achieved to your answer. There is not any other non-parametric method (without knowing any information about functional form of SDE).

May be this answer comes to late, but I got the question only some days ago. Since I am not familiar with GARCH I focus on SDEs.

For me, DEs are describing the change of the mean of a quantity over time, and SDEs are a generalization to account for variability. As a result, a system that is stable or reverting when described by a DE may be less stable or unstable when the varibility increases. SDEs of the Ito type have diffusion processes, i.e., Markov processes with continuous paths as a solution. These processes are well understood and many results are known. However, the Markov property can be a restriction if the future of a system does not only depend on the present, but also on the past (an issue which can sometimes be remedied by increasing the state space). SDEs have been much used in the financial sector, e.g., to describe the course of share prices, and to some extent in the life sciences.

I do not see a stochastic differential here so I do not see an SDE. In fact, here is an ODE (which might dependent on a stochastic parameter, though this is not detailed here) of the form D_t f_0 = <B, D_t f >, where B = (β_1, β_2 , β_3) is a constant vector

and f = (f_1, f_2, f_3) is a vector-valued function. The equation implies

f_0 (t) - <B, f(t) > = f(0) - <B, f(0) >.

Certainly, there is a huge lot of functions satisfying the above.

However, if the question has been misformulated and in fact f is a 3-dim semi-martingale (say, a Wiener process W_t) and the equation actually means df_0 = B df_t then the answer is still the same: f_0 (t) - <B, f(t) > = f(0) - <B, f(0) >.

Okay, so the answer is that this is a single-factor Ornstein-Uhlenbeck process with additive noise and where the mean-reversion level is a deterministic exponential, which is a function that gets along well with the structure of the O-U process.

You may find necessary and sufficient conditions (not only for Gaussian measure) in

1) Kunita H. Stochastic flows and stochastic differential equations. Theorem 4.3.2.

or

2) T.E. Harris: Brownian motions on the diffeomorphisms of the

plane, Ann. Prob. 9 (1981), 232–254.

Appendices in "Expectation Puzzles, Time-varying Risk Premia, and Dynamic Models of the Term Structure", Dai and Singleton