View source for Stochastic differential equation

''for a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900801.png" /> with respect to a [[Wiener process|Wiener process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900802.png" />''

An equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900805.png" /> are non-anticipative functionals, and the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900806.png" /> plays the part of the initial value. There are two separate concepts for a solution of a stochastic differential equation — strong and weak.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900807.png" /> be a probability space with an increasing family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900808.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s0900809.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008010.png" /> be a Wiener process. One says that a continuous stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008011.png" /> is a strong solution of the stochastic differential equation (1) with drift coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008012.png" />, diffusion coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008013.png" /> and initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008014.png" />, if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008015.png" /> with probability one:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where it is supposed that the integrals in (2) are defined.

The first general result on the existence and uniqueness of a strong solution of a stochastic differential equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

was obtained by K. Itô. He demonstrated that if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008018.png" /> the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008020.png" /> satisfy a Lipschitz condition with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008021.png" /> and increase not faster than linearly, then a continuous solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008022.png" /> of the equation (3) exists, and this solution is unique in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008023.png" /> is another continuous solution, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008024.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008025.png" />, the measurability and boundedness of the drift coefficient (vector) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008026.png" /> guarantees the existence and uniqueness of a strong solution of (3). The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008027.png" />, generally speaking, does not have a strong solution for any bounded non-anticipative functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008028.png" />.

When studying the concept of a weak solution of the stochastic differential equation (1), the probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008029.png" /> with the family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008030.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008031.png" />, the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008032.png" /> and the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008033.png" /> are not fixed in advance, but the non-anticipative functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008035.png" />, defined for continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008036.png" />, and the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008037.png" /> (so to speak, the initial value) are fixed. Then by a weak solution of the equation (1) with given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008040.png" /> one understands a set of objects

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008041.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008042.png" /> is a Wiener process relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008045.png" /> are related by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008046.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008047.png" />. The term  "weak solution"  sometimes applies only to the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008048.png" /> that appears in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008049.png" />. A weak solution of equation (3) exists under weaker hypotheses. It is sufficient, for example, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008050.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008051.png" /> be continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008052.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008053.png" /> be measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008054.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008055.png" />.

The development of the theory of stochastic integration (see [[Stochastic integral|Stochastic integral]]) using semi-martingales (cf. [[Semi-martingale|Semi-martingale]]) and random measures has led to the study of more general stochastic differential equations, where semi-martingales and random measures are used as generators (along with a Wiener process). The following result is typical. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008056.png" /> be a probability space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008057.png" /> be an increasing family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008058.png" />-algebras, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008059.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008060.png" />-dimensional semi-martingale, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008061.png" /> be a matrix consisting of non-anticipative functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008062.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008063.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008064.png" /> do not increase too rapidly (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008065.png" />). Then the stochastic differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008067.png" />, has a unique strong solution.

If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008071.png" />, satisfy a Lipschitz condition (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008072.png" />) and do not increase faster than linearly, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008073.png" /> of equation (3) (unique up to [[Stochastic equivalence|stochastic equivalence]]) will be a Markov process. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008075.png" /> are continuous in all variables, then this will be a diffusion process. Using stochastic differential equations, starting only from a Wiener process, it is thus possible to construct Markov and diffusion processes.

Given certain extra conditions of smoothness on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008077.png" />, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008078.png" /> of equation (3) with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008079.png" /> is such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008080.png" />, given a sufficiently smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008081.png" />, satisfies the backward Kolmogorov equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008082.png" /></td> </tr></table>

in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008084.png" />, with the boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090080/s09008085.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Gikhman,   A.V. Skorokhod,   "Stochastic differential equations and their applications" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Liptser,   A.N. Shiryaev,   "Statistics of random processes" , '''1–2''' , Springer  (1977–1978)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Arnold,   "Stochastic differential equations" , Wiley  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bunke,   "Gewöhnliche Differentialgleichungen mit zufällige Parametern" , Akademie Verlag  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Freedman,   "Stochastic differential equations and applications" , '''1''' , Acad. Press  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.Z. [R.Z. Khasmins'kii] Hasminski,   "Stochastic stability of differential equations" , Sijthoff &amp; Noordhoff  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Ikeda,   S. Watanabe,   "Stochastic differential equations and diffusion processes" , North-Holland &amp; Kodansha  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T.T. Soong,   "Random differential equations in science and engineering" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.K. Srinivasan,   R. Vasudevan,   "Introduction to random differential equations and their applications" , Amer. Elsevier  (1971)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.L. Stratonovich,   "Topics in the theory of random noise" , '''1–2''' , Gordon &amp; Breach  (1963–1967)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.W. Stroock,   S.R.S. Varadhan,   "Multidimensional diffusion processes" , Springer  (1979)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Th. Gard,   "Introduction to stochastic differential equations" , M. Dekker  (1988)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  B. Øksendahl,   "Stochastic differential equations" , Springer  (1987)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P. Protter,   "Stochastic integration and differential equations" , Springer  (1990)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S. Albeverio,   M. Röckner,   "Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms"  ''Probab. Th. Rel. Fields'' , '''89'''  (1991)  pp. 347–386</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  K.D. Elworthy,   "Stochastic differential equations on manifolds" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  M. Emery,   "Stochastic calculus in manifolds" , Springer  (1989)  ((Appendix by P.A. Meyer.))</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  K. Sobczyk,   "Stochastic differential equations. With applications to physics and engineering" , Kluwer  (1991)</TD></TR></table>