User's guide of reflected BSDE with two barriers and other topic
Contents
1. dy H x H py PH 37 ldt z dB x 0 0 y T 0 input H m e m fos pa E 1 e E input p such that 2 pH Ha Ht lt 0 4 Ricatti Curve Stochastic Hamilton Curves depend on H i3 Curves Close of solution x y will be drawn by red and blue lines Figure 7 Interface for Stochastic Hamilton System On the right subfigure you can choose the parameters 1 0 5 1 e Hamilton matrix H with Ho H 2 whose default value is 0 5 2 1 1 1 1 e range is for choosing time interval which can be decided by slide bar e default eigenvalue p 4 and it satisfies pH H33 H lt 0 5 Click Ricatti Curve the a pair of solution of Ricatti equation is show in left upper portion by blue and green line 6 Click Stochastic Hamilton in left upper portion a pair of strategies 7 Click Curves user will see several pairs of strategies of solution xz yz basing on same Brownian motion strategy with parameter H12 changes 10 from 0 5 to 0 with each step 0 05 At last program will generate two new figures to draw these in 3 dimensional 7 For closing the figures there is two ways One is using the button Close on them The other is to click the little cross on the right up corner of the figure 2 1 Stochastic Laplace Transform Stochastic Laplace transform is a kind of BSDE For process f stochastic Laplace transform of f is L2. followed with a return Then the reflected BSDE window figure No 4 is prompted reflected BSDE X X b X ds o XdB Y T HG X Y Z ds A A TZdB te 0T T 1 calculate progress Y gt L OStST input X 1 f 1 input b P 1 input o x input H t x y z amp B 1 input B B exp 5 B L tX 0 input Y tx by Mingyu Xu in Institute of Mathematics Shandong University Figure 4 Interface reflected BSDE solution y solution z solution 7 2 solution 7 2 4 B M amp solution y Close An important feature of this program is its strong capacity of user interface Run command in the command window We get the main user interface shown in fig 2 At the left side of the user interface the reflected BSDE is shown on the upside to indicate you the meaning of input functions On the downside you can input three functions the generator g the terminal condition and the barrier S For example in fig 1 2 we will input X 1 b 1 o 1 H t x y 2 x exp 0 5 Br Le X So in the blank spaces we type g x exp 5 B W t x x 2 To see the expression in Matlab please read the tablet in Section 2 2 or use the help in Matlab For the coefficient H the function only depend on t 2 y z It can not support other variable expect t x y z And for the terminal co
3. f s u p 0 where p t q t is the solution of following complex valued BSDE on infinite interval 0 00 dp t sp t utjat f t dt q t dW t u t ax lon t 0 00 sEC ul 5 M The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present package is compressed as hamilton zip After download the compressed file you should 1 uncompress the file Xuamoption zip in the document C matlab work or D your Matlab is installed in the hard disk D 2 Run the Matlab command window 3 Then within this window single click File in the menu buttons and then among the prompted file buttons single click Set Path button Then within the prompted Path Browse window browse and add the direction C matlab work laplace in the Path 4 After these preparation you can run our program in Matlab s com mand window For Stochastic Laplace transform type slaplace followed with a return figure 8 is generated 11 Stochastic Laplace Transform ir stochastic process 0 97 f WW 0 87 input w x 0 7 exp x 0 6 L choose the parameter s X 0 57 0 44 from 2 i to 2 i 0 37 the defual parameter 0 27 0 17 Ps 6 0 i i i i i i i i i j a 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 lculat Stochastic Laplace Transform sie dp t s pO WO q fldt qodW 6 add line
4. main user interface shown in fig 2 At the left side of the user interface the reflected BSDE is shown on the upside to indicate you the meaning of input functions On the downside you can input three functions the generator g the terminal condition and the barrier S For example in fig 1 2 we will input g t y z 10 y z 1 E B B Le V t Bs 3 x B t 2 3 U Volt Bi B t 1 3 t 1 So in the blank spaces we type g 10 abs y z 1 abs x V1 t x 3 x 2 24 38 Vo t x x 1 2 3 t 1 To see the expression in Matlab please read the tablet in Section 2 2 or use the help in Matlab For the coefficient g the function only depend on t y z It can not support other variable expect t y z And for the terminal condition it s same only can have one variable x which take place of the B For the barrier W Ve it only has the variable t x t is for the time x is for the Brownian motion B After inputting the parameters you can use these programs to do the calculation 1 Clicking the button calculate on the left side once then the program of calculation will run For getting the result calculation will take certain seconds and then indicate the end of the calculation by jumping a dialogue box the calculation is complete 2 Click the next button progress the program will generate a new figure named calculating process of so
5. User s guide of reflected BSDE with two barriers and other topic Mingyu XU gt School of Mathematics and System Science Shandong University 250100 Jinan China Department of Financial Mathematics and Control science School of Mathematical Science Fudan University Shanghai 200433 China Institute of Applied Mathematics AMSS CAS 100190 Beijing China 1 Reflected Backward Stochastic Differential Equations 1 1 Introduction for reflected backward differential equa tions with one or two barriers The reflected backward stochastic differential equations RBSDE in short are firstly introduced by N El Karoui C Kapodjian E Pardoux S Peng and M C Quenez in 4 We need the following notation 10 4 es 0 lt t lt T is a 1 d F progressively processs t E sup lt o O lt t lt T S 0 T fyr 0 lt t lt T is a 1 d continuous increasing process s t E yr lt oo With assumptions i a terminal condition L7 ii a map g which satisfied g 0 T x Rx R gt R g 0 0 L and the Lipschitz condition V y1 21 Y2 22 R x R4 there exists a constant C gt 0 satisfying lg t y1 21 g t yo 22 lt C ly1 yo z1 22 Email xumy amss ac cn iii the lower barrier L upper barrier U which are F adapted continu ous processes s t E supy lt yer L7 E supgcrer U lt 00 And we assume there exists a semimartingale such that L l
6. ajectory of Bi yz resp Bi 21 On the above they are showed by a 3 d moving image the red resp blue lines show a trajectory of the solution y resp Brownian motion and the light red vertical lines indicate the relation between the two trajectories On the below this trajectory of solution y is showed in 2 d moving image by a red line with time being the x coordinate In the 3rd column we show the push A and K Clicking more button on this figure there will produce a different triple y zt A corresponding to a different Brownian motion path Like in the figure for the solution surface the button center right and up are for see the two 3 d image in different direction 5 Clicking the following button B M and solution y you will get a new figure solution y on the surface for the sample way of solution y on the solution surface On the above window of it a trajectory of Brownian Motion B w is showed on the ground while the solution y according to this Brownian motion is showed on the solution surface And the grey surface is for the barrier S On the below windows there are the trajectories of A on the left and A on the right The button more is still for a new group of lines produced in the different color and the button right up are for the same use like the ones in the figure surface for solution y 6 Click the next button distribution a n
7. ew figure distribution func tion is generated to show the distribution function of solution y at different time t On this figure from the time T 1 to t 0 the distribution func tions of solution y are showed by lines in different color by turn The button right up are for the same use 7 For closing the figures there is two ways One is using the button Close on them The other is to click the little cross on the right up corner of the figure In all the images the meanings of the coordinate axises are noted directly beside the corresponding axises So we omit the detailed explanation 1 3 User s guide Reflected BSDEs with one lower bar rier and a diffusion process The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present package is compressed as rebsdex zip After download the compressed file you should 1 uncompress the file rebsde zip in the document C matlab work or D your Matlab is installed in the hard disk D 2 Run the Matlab command window 3 Then within this window single click File in the menu buttons and then among the prompted file buttons single click Set Path button Then within the prompted Path Browse window browse and add the direction C matlab work rebsdex in the Path 4 After these preparation you can run our program in Matlab s com mand window type rebsdex2
8. ion is required The present package is compressed as rebsdetb zip After download the compressed file you should 1 uncompress the file rebsde zip in the document C matlab work or D your Matlab is installed in the hard disk D 2 Run the Matlab command window 3 Then within this window single click File in the menu buttons and then among the prompted file buttons single click Set Path button Then within the prompted Path Browse window browse and add the direction C matlab work rebsdetb in the Path 4 After these preparation you can run our program in Matlab s com mand window type rebsdetb followed with a return Then the reflected BSDE window figure No 3 is prompted reflected BSDE calculate Y 6 I g s Y Z ds FA Ar Kr K progress T z ji Z dB te 0 7 T 1 Brownian Motion L S SU OStST JTY L dA 0 Y U dK 0 solution LZA K solution y input g t y z 10 abs y z 1 solution z E amp B 1 input B x abs L P tB input Y tx 3 x 2 243 B M amp solution y distribution xt 1 243 t 1 U t B t input P t x Close realized by XU Mingyu in Universite de Maine Figure 3 Interface reflected BSDE An important feature of this program is its strong capacity of user interface Run command in the command window We get the
9. ions in Optimal Stochas tic Control J Math Anal Apl 44 pp 384 404 27 Cvitanic J and Karatzas I 1996 Backward Stochastic Differential Equations with Reflection and Dynkin Games Ann Probab 24 no 4 2024 2056 N El Karoui C Kapoudjian E Pardoux S Peng and M C Quenez 1997 Reflected Solutions of Backward SDE and Related Obstacle Prob lems for PDEs Ann Probab 25 no 2 702 737 Lepeltier J P and San Mart n J 2004 Backward SDE s with two bar riers and continuous coefficient An existence result Journal of Applied Probability vol 41 no 1 162 175 J M min S Peng M Xu 2002 Convergence of solutions of discret re flected backward SDE s preprint E Pardoux and S Peng Adapted solutions of Backward Stochastic Dif ferential Equations Systems Control Lett 14 1990 51 61 S Peng 2000 Problem of eigenvalues of stochastic Hamilton systems with boundary conditions Stochastic Processes and their application 88 259 290 13 9 Peng S and Xu M 2005 Smallest g Supermartingales and related Reflected BSDEs Annales of I H P Vol 41 3 605 630 10 Peng S and Xu M 2006 The numerical Algorithms and simulations for BSDEs arXiv math PR 0611864v1 14
10. lution y On this figure you will see the backward computation procedure of the function y t x dynamically and backwardly In the figure the red lines above resp blue lines below show the solution y resp Brownian motion The grid surface is the barrier At the end we can see a vertical red line which simplify the value of solution y at time 0 Then the above colorful surface of solution y is generated in a new figure which is named surface for solution y The green lines in this figure show the relation between the solution y and the Brownian motion while the blue line below indicate the range of descrete Brownian motion By clicking the button right up you can see the surface in different direction The next three buttons on the main user interface are for the simulations of the solved y z 3 Click the button Brownian motion a dynamically generated Brow nian path will appear on the new figure Sample way of Brownian Motion This path will terminated by a jump of a vertical line indicating the terminal value yr E w of this sample If you click the button more on this new figure then another Brownian path and the related w will be produced in a different color 4 Click the button solution y z A KY you will see a moving yr zt At Ki 4 on the generated new figure trajectories of solution y and z and A and K In the 1st resp the 2nd column we show the tr
11. ndition 9 it s same only can have one variable x which take place of the B For the barrier Y it only has the variable t x t is for the time x is for the Brownian motion Bz After inputting the parameters you can use these programs to do the calculation 1 Clicking the button calculate on the left side once then the program of calculation will run For getting the result calculation will take certain seconds and then indicate the end of the calculation by jumping a dialogue box the calculation is complete 2 Click the next button progress the program will generate a new figure named calculating process of solution y On this figure you will see the backward computation procedure of the function y t x dynamically and backwardly In the figure the red lines above resp blue lines below show the solution y resp Brownian motion The grid surface is the barrier At the end we can see a vertical red line which simplify the value of solution y at time 0 Then the above colorful surface of solution y is generated in a new figure which is named surface for solution y The green lines in this figure show the relation between the solution y and the Brownian motion while the blue line below indicate the range of descrete Brownian motion By clicking the button right up you can see the surface in different direction The next three buttons on the main user interface are for the simulati
12. ons of the solved y z 3 Click the button solution y you will see a moving y on the gen erated new figure in 2 dimensional Clicking more button on this figure there will produce a different triple y corresponding to a different Brownian motion path 4 Click the button solution z you will see a moving z on the gen erated new figure in 2 dimensional Clicking more button on this figure 7 there will produce a different triple 2 corresponding to a different Brownian motion path 5 Click the button solution y z you will see a moving y 24 on the generated new figure in 2 dimensional and 3 dimensional This left two subfigures are for y while the right two are for z Clicking more button on this figure there will produce a different triple y z1 corresponding to a different Brownian motion path 6 Click the button solution y z A you will see a moving yr 21 At on the generated new figure trajectories of solution y and z and A and K In the 1st resp the 2nd column we show the trajetory of B y resp Bi 2 On the above they are showed by a 3 d moving image the red resp blue lines show a trajectory of the solution y resp Brownian motion and the light red vertical lines indicate the relation between the two trajetories On the below this trajectory of solution y is showed in 2 d moving image by a red line with time being the x c
13. oordinate In the 3rd column we show the push A and y Li Clicking more button on this figure there will produce a different triple yr zt A corresponding to a different Brownian motion path Like in the figure for the solution surface the button center right and up are for see the two 3 d image in different direction 7 Clicking the following button B M and solution y you will get a new figure solution y on the surface for the sample way of solution y on the solution surface On the above window of it a trajectory of Brownian Motion B w is showed on the ground while the solution y according to this Brownian motion is showed on the solution surface And the grey surface is for the barrier L On the below windows there are the trajectories of A on the left and y L on the right The button more is still for a new group of lines produced in the different color and the button right up are for the same use like the ones in the figure surface for solution y 8 For closing the figures there is two ways One is using the button Close on them The other is to click the little cross on the right up corner of the figure In all the images the meanings of the coordinate axises are noted directly beside the corresponding axises So we omit the detailed explanation 2 Stochastic Hamilton system Stochastic Hamilton system is a linear forward backward
14. stochastic differen tial equation See more details in 8 dx Hax pHo2y pH23x dt pH31 pH32y H33 dB dy Hla Hioy pAisx dt zid B 2 0 0 y T 0 Obviously Yt 2 0 0 0 is its solution We study its nonzero solu tion p is eigenvalue of this system which permits this FBSDE has nonzero solution And p satisfies pH22 H33H lt 0 The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present package is compressed as hamilton zip After download the compressed file you should 1 uncompress the file Xuamoption zip in the document C matlab work or D your Matlab is installed in the hard disk D 2 Run the Matlab command window 3 Then within this window single click File in the menu buttons and then among the prompted file buttons single click Set Path button Then within the prompted Path Browse window browse and add the direction C matlab work hamilton in the Path 4 After these preparation you can run our program in Matlab s com mand window For Stochastic Hamilton system type shamilton followed with a return figure 7 is generated 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 Stochstic Hamiltonian systems 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Stochstic Hamiltonian systems dx H x PH y PH 32 ldt PH x PH3 y H3 2 ldB
15. t X lt Uj The solution of RBSDE w r t the lower barrier L is the triple Y Z A S 0 T x H 0 T x 0 T which satisfies T T i E olr Zr dr Ap Ar Z dB t t and L lt Y fr 0 lt t lt T JE Y Li dA 0 a s The solution of RBSDE w r t the lower barrier L and upper barrier U is the triple Y Z A K S 0 T x H 0 T x 0 T which satisfies T T i f lr Yn Zi dr Ap Ay Kr Ki f Z dB t t and I lt Y lt U for 0 lt t lt T JE Y 1 dA J Y U d K 0 a s And we also have program to simulation reflected BSDE with a diffusion process X i bX ds ig oX dB The reflect BSDE is as following T T Peca K HELY Zp dr Ap Ar Z dB t t and L lt Y for0 lt t lt T JI Y Li dA 0 as In the paper 4 of N El Karoui et al the existence and uniqueness for the solution of such RBSDE is proved Moreover in the Section 6 they consider a very clever method penalization method to prove the existence This method also stir out the study for the numerical solution for RBSDE How to calculate the numerical solution for RBSDE is written in the paper Convergence of solutions of discrete reflected Backward SDE s by J Memin and S Peng the simulation is done by M Xu 6 1 2 User s guide Reflected BSDEs with two barriers The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher vers
16. te 0 seC W JES WO ax Ty Close LIF ICs W p 0 Figure 8 Interface for Stochastic Laplace transform On the right subfigure you can choose the parameters e Stochastic process f t w W where W is Brownian motion De fault value is y x exp z e Choose parameter by menu there are three parameters s T a to choose The chosen parameter will change in a range This range is decided by following edit window default value is for s from 2i to 2i e Following are rest two parameter whose values are decided by following edit window The default value is T 6 a 1 5 Click calculate program will generate the curve of stochastic Laplace transform of f changing on chosen parameter in chosen range 12 6 Click add line program will generate the curve of stochastic Laplace transform of f in another color changing on same parameter in same range after user change any value of default parameters 7 For closing the figures there is two ways One is using the button Close on them The other is to click the little cross on the right up corner of the figure In all the images the meanings of the coordinate axises are noted directly beside the corresponding axises So we omit the detailed explanation References P Briand B Delyon J M min Donsker type theorem for BSDEs Elect Comm in Probab 6 2001 1 14 J M Bismut 1973 Conjugate Convex Funct
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