Solving Stochastic Differential Equations with Maple

Solving Stochastic Differential Equations with
Maple
Sasha Cyganowski
Introduction
The effects of intrinsic noise within physical phenomena is
ignored when mathematical models of their behavior are constructed
using deterministic differential equations. Such equations
describe what is occurring on average to the system.
However, more information is often required to obtain a full
picture of the behavior of the model, especially when nonlinearities
are involved. In these circumstances a stochastic
model is used. Stochastic Differential Equations (SDEs) include
a random term which describes the intrinsic noise, or
randomness, within dynamical systems. SDEs have found
many applications in areas such as Chemistry, Physics, and
Mathematical Biology. A general, one-dimensional SDE takes
the form
dX t = a(t; X t)dt + b(t; X t)dW t; (1)
where a(t; X t) is the drift coefficient and b(t; X t) is the diffusion
coefficient. W t is a Wiener process which represents
intrinsic noise in the dynamical system, and is often referred
to as white noise. The subscript t denotes time-dependence.
The drift coefficient models the dominant action of the system
whilst the diffusion coefficient represents randomness
along the dominant curve.
The Wiener process is the simplest random or stochastic
process which provides an adequate model for Brownian motion.
Details of the Wiener process can be found in Kloeden
and Platen [3]. This random “function” is of significance as
it is the starting point of stochastic calculus. Now, the Wiener
process is not of bounded variation, thus if we write equation
(1) as an integral equation of the form
Xt(w) = Xt0 (w) +
+
Z t
t0
Z t
t0
a[s; X s(w)] ds
b[s; X s(w)] dW s(w); (2)
the second integral is not defined by the rules of classical calculus.
This problem was overcome in the early 1950s when
Ito formulated his definition of the Ito integral, for which the
second integral is defined. Ito stochastic calculus exhibits
many peculiarities and does not conform to the rules of classical
calculus, so care must be taken when constructing methods
to solve SDEs. Efficient methods are now available and
can be found in Kloeden and Platen [3].
School of Computing and Mathematics, Deakin University, Geelong,
Vic 3217, Australia. sash@deakin.edu.au
The author has developed a Maple package, called “stochastic”,
containing routines which return explicit solutions to
those SDEs known to have such solutions. This paper will
concentrate on these particular routines. The package also
contains routines which construct efficient, high order stochastic
numerical schemes. Readers wishing to approximate solutions
to SDEs via numerical methods will benefit from reading
Cyganowski [1].
Explicit Solutions to SDEs through Maple
In the subsections below, specific forms of SDEs that are
known to have explicit solutions will be introduced and procedures
from the “stochastic” package will be provided which
automate the solving of such SDEs. General solutions are
simply given, as found from Kloeden and Platen [3]. Firstly,
linear SDEs will be considered followed by non-linear SDEs
that are reducible to linear form.
LINEAR SDES
Linear SDEs are explicitly solvable, exhibiting the general
form
dX t =(a(t)X t + c(t))dt +(b(t)X t + d(t))dW t; (3)
with general solution
where
X t = t;t0 (X t0 +
t;t0
+
= exp(
Z t
t0
Z t
t0
Z t
t0
,1
s;t0
(c(s) , b(s)d(s)) ds
,1
s;t0 d(s) dW s); (4)
(a(s) , 1
2 b2 (s)) ds +
Z t
t0
b(s) dW s) (5)
is the fundamental solution. A suitable Maple procedure for
solving such SDEs is given below.
1

Stochastic DEs with Maple
> linear:=proc(a:algebraic,b:algebraic)
> local temp1,alpha,beta,gamma,delta,fundsoln,
> fundsoln2,soln,default1,default2,default3;
> if diff(a,x,x) 0
> or diff(b,x,x) 0 then
> ERROR(`SDE not linear, try a reducible
> procedure`)
> else
> alpha := diff(a,x);
> alpha := subs(t = s,alpha);
> beta := diff(b,x);
> beta := subs(t = s,beta);
> if diff(beta,s) = 0 then
> temp1 := beta*W;
> else temp1 := Int(beta,W = 0 .. t);
> fi;
> gamma := coeff(a,x,0);
> gamma := subs(t = s,gamma);
> delta := coeff(b,x,0);
> delta := subs(t = s,delta);
> fundsoln := exp(int(alpha-1/2*(beta**2),
> s = 0 .. t)+temp1);
> fundsoln2 := subs(t = s,fundsoln);
> if beta = 0 then
> soln := fundsoln*(X[0]+
> int(1/fundsoln2*
> (gamma-beta*delta),s = 0 .. t)
> +Int(1/fundsoln2*delta,W = 0 .. t))
> else soln := fundsoln*(X[0]+
> Int(1/fundsoln2*
> (gamma-beta*delta),s = 0 .. t)
> +Int(1/fundsoln2*delta,W = 0 .. t))
> fi;
> default1 := Int(0,W = 0 .. t) = 0;
> default2 := Int(0,W = 0 .. s) = 0;
> default3 := Int(0,s = 0 .. t) = 0;
> soln := X[t] = subs(default1,default2,
> default3,soln)
> fi
> end:
To invoke such a procedure we use the following command.
linear(a(t,x),b(t,x));
Figure 1 Syntax for procedure “linear”.
In Figure 1, a(t; x) and b(t; x) denote the drift and diffusion
coefficients of the SDE, respectively. Note that both arguments
must be given in the variables x and t. This syntax
will be required for all the procedures in this paper.
REDUCIBLE SDES
SDEs are in this category if they have the general form
dX t =( b(X t)h + 1
with general solution
2
X t = h ,1 (exp t h(X0) + exp t
2 b(X t)b 0 (X t))dt + b(X t)dW t; (6)
Z t
0
exp , s dW s); (7)
where
h =
Z x ds
b(s)
: (8)
A suitable Maple procedure for solving such SDEs is given
below.
> reducible:=proc(a:algebraic,b:algebraic)
> local beta,temp1,h,alpha,soln,soln1;
> h := int(1/b,x);
> temp1 := alpha*b*h+1/2*b*
> simplify(diff(b,x));
> temp1 = a;
> alpha := simplify(solve(",alpha));
> beta := alpha*h;
> if diff(alpha,x) = 0 then
> if alpha=0 then
> soln:=h=subs(x=X[0],h)+W;
> X[t]=simplify(solve(soln,x));
> else
> soln1 := h = exp(alpha*t)*
> subs(x = X[0],h)
> +exp(alpha*t)*Int(exp(-alpha*s),
> W = 0 .. t);
> X[t] = solve(soln1,x);
> fi
> elif diff(beta,x) = 0 then
> X[t] = simplify(solve(h=beta*t+W+
> subs(x=X[0],h),x));
> else ERROR(`non-linear SDE not reducible`)
> fi
> end:
GENERAL MAPLE PROCEDURE FOR FINDING
EXPLICIT SOLUTIONS.
The Maple procedure “explicit” is given below, which is a
combination of procedures “linear” and “reducible”. This
procedure determines whether an SDE is reducible or linear,
and solves it accordingly. If the SDE does not fall into one
of the above categories, the user is returned with an error
message.
> explicit:=proc(a:algebraic,b:algebraic)
> if diff(a,x,x) = 0 and diff(b,x,x) = 0 then
> linear(a,b)
> else
> reducible(a,b)
> fi
> end:
The procedure “explicit” is more user-friendly than “linear”
or “reducible” as it is not necessary for the user to predetermine
which category the SDE belongs to.
Applications
(I) Consider the SDE
dX t = dt +2
p XtdW t: (9)
Here we see how the procedure “explicit” can be used to find
the explicit solution of equation (9).

explicit(1,2*sqrt(x));
X t =2
(II) Consider the SDE
p X0W + W 2 + X0
dX t =lnX tdt + X tdW t: (10)
In this example the SDE is reported to have no known
explicit solution.
> explicit(ln(x),x);
Error, (in reducible) non-linear SDE not reducible
Conclusion
Stochastic calculus has been introduced in this paper through
the computer algebra system Maple. Procedures were presented
and used to find solutions to SDEs. It should be noted
that the methods provided here for solving SDEs are suitable
for real-valued, one-dimensional SDEs with real-valued,
one-dimensional Wiener processes.
Symbolic computation in stochastic analysis has been investigated
by Kendall [2] and Valkeila [6], while Maple code
suitable for use in stochastic numerics has been developed
by Cyganowski [1] and Kloeden and Scott [5]. Previously
Maple had not been used in stochastic calculus for finding
explicit solutions to SDEs.
The procedures in this paper are part of the Maple package
“stochastic” which has been developed by the author
and recently accepted into Maple's share library. It will become
publicly available when the next electronic update of
the share library occurs. Those who are interested in acquiring
the package now are welcome to contact the author for
a free copy, complete with help files and installation instructions.
Alternatively, the package can be obtained via the Internet
at http://www.cm.deakin.edu.au/~sash/maple.html.
Other functions contained in the “stochastic” package include
procedures which construct strong stochastic numerical
schemes up to order 2, weak stochastic numerical schemes
up to order 3, as well as procedures which check for commutative
noise of the first and second kind, and a procedure
which converts SDEs with white noise into colored noise
form. Additional procedures are included within the package
so that users can easily construct numerical schemes other
than those already available.
Future extensions to this package will include a set of
procedures which plot approximate solutions to SDEs. The
input required for these “graphical” procedures will be easily
obtainable as it will be the output from the “algebraic”
procedures discussed in this paper.
References
Stochastic DEs with Maple
[1] Cyganowski, S.O., A MAPLE Package for Stochastic
Differential Equations, Computational Techniques
and Applications: CTAC95, editors A. Easton, R. May,
World Scientific, (to appear).
[2] Kendall, W.S., Computer algebra and stochastic calculus,
Notices Amer. Math. Soc., 37, 1990, 1254–1256.
[3] Kloeden, P.E. and Platen, E., Numerical Solution
of Stochastic Differential Equations, Springer-Verlag,
1992.
[4] Kloeden, P.E., Platen, E. and Schurz, H., Numerical Solution
of Stochastic Differential Equations through Computer
Experiments, Springer-Verlag, 1993.
[5] Kloeden, P.E. and Scott, W.D., Construction of Stochastic
Numerical Schemes through Maple, Maple Technical
Newsletter, 10, 1993, 60–65.
[6] Valkeila, E., Computer algebra and stochastic analysis,
some possibilities, CWI Quarterly, 4, 1991, 229–238.
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