User's guide of g-expectation and reflected backward stochastic
Contents
1. 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 2 3 User s guide the penalization BSDEs for RBSDEs For an RBSDE with the parameters g L satisfying the assumptions i iii in section 2 1 denote Y Z S 0 T x H 0 T for each p N be the unique pair of F progressively processes satisfying T T T YP E olr YP Z dr p f YP S dr Z dB t t t and let A fe VP S dr Such equation is called the penalization BSDEs w r t 9 L In 3 N El Karoui et al Section 6 we know that the solution Y ZP A of the penalization BSDEs converges to the solution Y Z A of RBSDE in the space S 0 T x H 0 T x 0 T Then we use the numerical method for BSDEs to calculate the penalization BSDEs 10 The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present pagage is co2. The power of n x xD Absolute value abs x Squrare root 2 sqrt x Exponential e exp x Natural logarithm log log x Sine sin z sin x The plus and minus are just and respectively And for the trigonometric functions the expressions are also like usual so we did not list them out in the table 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 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 4 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 At the end we can see a vertical red line which simify the value of solution y at time 0 Then the colorful surface of solution y is generated in a new figure which is named surface for solution y The green lines in this fig
3. f Ilt Yn Zi dr f 7AB t t 1 where g satisfied i g 0 T x Rx R R and g 0 0 L ii the Lipschitz condition Y y1 21 Y2 22 Rx R there exists a constant C gt 0 satisfying l9 t y1 21 g t y2 22 lt C ly yo 21 22 and Li Although the history of the theory of backward stochastic differential equation is short but it is developing very fast For itself has many inter esting properties as well as many useful applications It can solve problems in partial differential equation theory differential geometry and mathemat ics finance Using backward stochastic differential equations Professor Peng gave a definition g expectation which extend the definition of classical ex pectation And it can be used widely in Mathematical finance For linear BSDEs we can solve them explicitly by using duality method But for nonlinear case unfortunately the equations have no explicit solution Thus the method of numerical solutions are in order A numerical algorithms had been proposed and calculated in 1994 by the research group of BSDE under the direction of S Peng in Shangdong University And we obtain numerical results of BSDEs and the convergence of this method has been proved by Philippe Briand Bernard Delyon and Jean M min 2001 in 1 Using this method we have developed a user interface of programs for calculations and simulations of BSDEs With this user interface people
4. solution y on the surface expect that on the 3 d window the price of the stock X take the place of the Brownian motion B 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 Note If you have any suggestion for this user s guide or for the programs please contact me xvmyQ math sdu edu cn References P Briand B Delyon J M min Donsker type theorem for BSDEs Elect Comm in Probab 6 2001 1 14 2 J M Bismut 1973 Conjugate Convex Functions in Optimal Stochas tic Control J Math Anal Apl 44 pp 384 404 27 3 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 4 J M min S Peng simulatiob by M Xu 2002 Convergence of solutions of discret reflected backward SDE s preprint 5 E Pardoux and S Peng Adapted solutions of Backward Stochastic Dif ferential Equations Systems Control Lett 14 1990 51 61 18
5. Brownian motion and the light red vertical lines indicate the relation between the two trajetories On the below this trajetory of solution y is showed in 2 d moving image by a red line with time being the x coordinate In the above of the 3rd column the push A corresponding to the solution y is shown And in the below of the 3rd column it s the difference of the y and S i e ye S Clicking more button on this figure there will produce a different triple y1 zt Az 9 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 y S 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 new figure
6. 0 0 and y satisfies yt gt Se for0O lt t lt 7 where S x k where 7 is a stopping time and T inf t y S lt 0 In finance y is wealth process x is price of stock 7 stand for exit time from market for the investor At this time he will do the action buying or selling the stock In this problem we are interested in the y z and 7 To solve it we consider it as a reflected BSDE with the barrier S then T inf t A gt 0 Reflected BSDE applied to American put option corresponds to the case yr E k zr S k x The investor s wealth y satisfies the RBSDE T T k 27 f rye p r zi ds Ar A z dBs 0 lt t lt 1 IV T Yt St dA gt 0 f y S dAy 0 where x is the price of the underlying stock which is a geometric Brownian motion t t Lt Xo f px ds f ox dBy 0 0 At the stopping time 7 inf t A gt 0 The investor will execute the contract and his return is k x 13 3 1 User s guide for American options The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present pagage is compressed as amoption zip To run the progam the following steps are needed 1 uncompress the file amoption 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
7. 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 0 B B S V t B 3 x B t 1 1 So in the blank spaces we type g 10 abs y z 1 abs x V t x 3 x 1 2 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 Y it only has the variable t x t is for the time x is for the Brownian motion By 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
8. solution y on the solution surface The differences with the button B M and solution y is the window below this window is for A JE Y S dr 6 The next button difference is to show the difference between the numerical solution of penelization BSDE and reflected BSDE After load the data of penalization BSDE and reflected BSDE first the program will check if the parametres g S of the two equations are same if they are not same a dialogue window will be generated CANNOT COMPARE The function or terminal condition or obstacle is defferent If they are same a new figure named the difference of PBSDE and RBSDE will be shown with the surface of the difference The button right up are for the same use 7 For closing the figures there is two ways One is using the button 12 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 3 Applications to American options Now we will discuss some applications of the reflected BSDE In the math ematical finance we know that American option can be described by the linear reflected BSDE For the American call option the wealth y satisfies the following BSDE T T y zr k rys u r zs ds j ozsdB 0 lt t lt 1 t t ie S fi f px ds f ox dB
9. who have no experience in stochastic calculations and numerical simulations could quickly learn how to use our programs to calculate to simulate and to study the BSDEs Briefly speaking after inputting his parameters i e the function g gly z the terminal condition in our BSDE one can type them into our user interface then execute and simulate the BSDE by clicking the related buttons on the user interface shown in fig 1 1 We think that the visibility of the numerical results of our programs will be useful for specialists and non specialists in BSDE Such knid of user interface is firstly developped by Zhou Haibin from 1996 These program that we put on the website are made by Xu Mingyu from 2000 Non linear g expectation calculate y E J g s yy z ds T z dB poper te 0 7 T 1 brownian motion E AS 1 y solution z z 2 input g t y z EM solution y amp amp B 1 input P x sin abs x distribution Close by Institute of Mathematics Shandong University Figure 1 Interface for g expectation BSDE In fact this group of programs can be used to calculate the BSDE not only the g expectation since the g expectation is a specialized BSDE with a condition of g s t g y 0 0 for Vy E R 1 1 User s guide for BSDE g expectations The programs are realized by Matlab s p files To run these programs Matlab 5 3 or highe
10. 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 amoption in the Path 4 After these preparation you can run our program in Matlab s com mand window For American call option type amoption followed with a return Then the American call option window figure 4 is prompted For American pull option type aoptionpu following with a enter the American pull option window figure 5 is generated 14 American Option calculate ay r y U r z dt z dB Yr xp k T 1 enS dx u x dt 0 x dB x 0 xo Seic brownian motion and Y 2 S O lt t lt ST solution Z A input r oa input 0 5 BM8 solution y input i Price amp solutiony input Xp 100 input k 100 Close by Institute of Mathematics Shandong University Figure 4 Interface for American call option For the American call option figure 4 on the left side of the user interface the reflected BSDE for American call option is shown on the upside to indicate you the meaning of input parametres On the downside you can input the parametres r 4 0 o k The default values are r 0 1 u 06 0 1 zo 100 k 100 For the American pull option figure 5 the arranement is same with the window for Ame
11. 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 simify 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 descret 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 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 you will see a moving Yt 2 At on the generated new figure trajetories of solution y and z and A In the lst resp the 2nd column we show the trajetory of B y resp Bz Zt On the above they are showed by a 3 d moving image the red resp blue lines show a trajetory of the solution y resp
12. User s guide of g expectation and reflected backward stochastic differential equation Xu Mingyu Mathematics Department of Shandong University September 3 2003 1 Backward Stochastic Differential Equation g Expectation Let Q F P be a probability space and Bs s gt 0 be a 1 dimensional Brow nian motion defined on this space Set F0 lt t lt T be the naturel filtration generated by the Brownian motion B F 0 B 0 lt s8s lt t where Fo contained all P null sets of F All processes metioned in this paper are supposed to be F adapted and we are interested in the behav ior of processes on a given interval 0 T The space L 0 7 R is the space of F adapted square integrable process on the interval 0 T the space H 0 T R is the space of F predictable square integrable process on the interval 0 7 and L is the space of F7 measurable square integrable ran dom variable The linear backward stochastic differential equations BSDE in short is firstly introduced by J M Bismut in 1973 2 and systematically studied by A Bensousan in order to obtain the stochastic maximum principle of Pontrya gin s type of optimal control systems E Pardoux and S Peng in 1990 firstly proved the uniqueness and existence of the solution to backward stochastic differential equation 5 that is there exists a unique pair of F processes Y Za L 0 T R x H 0 T R satisfied the following equation T T i
13. d SDE s by J Memin and S Peng the simulation is done by M Xu 4 2 2 User s guide RBSDEs The programs are realized by Matlab s p files To run these programs Matlab 5 3 or higher version is required The present pagage is compressed as rebsde zip To run the progam the following steps are needed 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 rebsde in the Path 4 After these preparation you can run our program in Matlab s com mand window type rebsde followed with a return Then the reflected BSDE window figure No 2 is prompted reflected BSDE calculate Y i gY Z ds Ar A fr ZdB progress Brownian Motion solution 2A input g y z 10 abs y z 1 B M amp solution y amp B B 1 input B x abst probability S t B O input Y t x 3 x 1 2 1 Close Figure 2 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 main user interface shown in fig 2 At the
14. e 0 Then the above colorful surface of solution y is generated in a new 16 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 descret 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 you will see a moving Yt zt At on the generated new figure trajetories of solution y and z and A In the lst resp the 2nd column we show the trajetory of Bi y resp Bi 21 On the above they are showed by a 3 d moving image the red resp blue lines show a trajetory of the solution y resp Brownian motion and the light red vertical lines indicate the relation between the two trajetories On the below this trajetory of solution y is showed in 2 d moving image by a red line with time being the
15. e 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 The button more is still for a new group of lines produced in the same 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 new 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 2 Reflected Backward Stochastic Differential Equations 2 1 Introduction for reflected backward differential
16. e will input g t y z 10 y z 1 B B p 10 S Vit B 3 x B t 1 1 So in the blank spaces we type g 10 abs y z 1 abs x p 10 U t x 3 x 1 2 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 Y 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 On this figure the functions of the buttons on the right side are almost same with the ones on the figure for RBSDE You can find the detail explanation in the Section 2 2 1 The button calculate is for calculation 2 The button progress is for show the procedure of the function y t x dynamically and backwardly The next three buttons on the main user interface are for the simulations of the solved y 2 3 The button Brownian motion is for a dynamically generated Brow nian with the terminal value yf w of this sample 4 The button solution y z is for trajetories of solution y and 2 Same as the one on the figure g expectation 5 The button B M and solution y is for the sample way of
17. equa tions 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 3 We need the following notation S 0 T fta 0 lt t lt T is a 1 d F progressively processs t E sup p lt o O lt t lt T S10 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 L4 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 lot y1 21 g t ye 22 lt C ly yel 21 221 iii the lower barrier L a F adapted continuous process s t E supg lt p lt r Lt lt oo 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 S 0 T which satisfies T T Ve f CO AA E E E f Z dB t t and L lt Y for0 lt t lt T EY dA 0 a s In the paper 3 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 Backwar
18. mpressed as pebsde zip To run the progam the following steps are needed 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 pebsde in the Path 4 After these preparation you can run our program in Matlab s com mand window type pebsde followed with a return Then the penaliza tion BSDE window figure No 3 is prompted BSDE under Penalization calculate yP 6 T lgay cP pvp Slds I lt PaB progress te 0 T T 1 brownian motion e a solution y amp z input g t y z B M on surface amp B 1 input p 10 difference input p Poon EAE sm 20 t S tB O input Lx mom SES by Institute of Mathematics Shandong University Figure 3 Interface penalization BSDE 11 At the left side of the user interface the penalization 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 penalization s parameter p the terminal condition and the barrier S For example in fig 1 2 w
19. r version is required The present pagage is compressed as gexp zip To run the progam the following steps are needed 1 uncompress the file gexp 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 gexp in the Path 4 After these preparation you can run our program in Matlab s com mand window type gexpectation followed with a return Then the g expectation window figure No 1 is prompted 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 1 1 At the left side of the user interface the g expectation is shown on the upside to indicate you the meaning of input functions On the downside you can input two functions the generator g and the terminal condition For example in fig 1 1 we will input g t y z 2 B sin B So in the blank spaces we type g z 2 sin abs x Here we offer a table of the transform between the formulas and the expressions in Matlab the left are the fomula in mathematics the right are the expression in Matlab Multiply x A Divide yi
20. rican call option The default values are r 0 1 u 04 0 1 zo 100 k 100 15 American Option calculate dy ry U r z dt 0 zdB ype k Xp yt T 1 progress dx x dt o6 x dB x 0 xo S Ck x yt brownian motion and Y 2S O lt tST solution Y amp a28A input r 0 1 input WL 0 5 B M on surface input E Price on surface input xg 100 input k 100 Close Figure 5 Interface for American pull option After inputting the parameters you can use these programs to do the calculation The functions of the buttons on the windows for American call or pull option are same So we write their fnctions together 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 simify the value of solution y at tim
21. ure show the relation between the solution y and the Brownian motion while the blue line below indicate the range of descret 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 you will see a moving y 24 on the generated new figure trajetories of solution y In the 1st resp the 2nd column we show the trajetory of Bi y resp Bi z On the above they are showed by a 3 d moving image the red resp blue lines show a trajetory of the solution y resp Brownian motion and the light red vertical lines indicate the relation between the two trajetories On the below this trajetory of solution y is showed in 2 d moving image by a red line with time being the x coordinate Clicking more button on this figure there will produce a different couple y z1 corresponding to a different Brownian motion path Like in the figure for th
22. x coordinate In the above of the 3rd column the push A corresponding to the solution y is shown And in the below of the 3rd column it s the difference of the y and S i e ye S Clicking more button on this figure there will produce a different triple yz zt At 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 On this figure and the following two the big blue or grey point is for the exit time 7 inf t A gt O 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 y S 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 Price and solution y a new figure Price on 17 the result surface is generated This figure is almost same as the figure
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