Solving Stochastic Differential Equations with Maple

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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. 3
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