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Effect of Additive Perturbations on the Solution of Reflected Backward Stochastic Differential Equations

By Jasmina Ðorđević

Submitted: October 4th 2020Reviewed: January 7th 2021Published: February 4th 2021

DOI: 10.5772/intechopen.95872

Downloaded: 34


This chapter has as a topic large class of general, nonlinear reflected backward stochastic differential equations with a lower barrier, whose generator, final condition as well as barrier process arbitrarily depend on a small parameter. The solutions of these equations which are obtained by additive perturbations, named the perturbed equations, are compared in the Lp-sense, p∈]1,2[, with the solutions of the appropriate equations of the equal type, independent of a small parameter and named the unperturbed equations. Conditions under which the solution of the unperturbed equation is Lp-stable are given. It is shown that for an arbitrary a>0 there exists ta≤T, such that the Lp-difference between the solutions of both the perturbed and unperturbed equations is less than a for every t∈taT.


  • reflected
  • backward
  • stochastic
  • perturbation
  • estimate

1. Introduction

This chapter is dedicated to the problem of additive perturbations of reflected backward stochastic differential equations (shorter RBSDEs) with one lower barrier. Motivation for the topic comes from a large application of perturbation problems in real life problems from one side, and reflected backward stochastic differential equations in finance from another. Perturbed stochastic differential equations are widely applied in theory and in applications. Randomness from the environment can be introduced via stochastic models with perturbations. In such manner, complex phenomena under perturbations in analytical mechanics, control theory, population dynamics or financial models, can be compared and approximated by appropriate unperturbed models of a simpler structure, i.e. the problems are translated on more simple and familiar cases which are easier to solve and investigate (see [1, 2, 3] for example). Problem of additively perturbed backward stochastic differential equations is analysed by Janković, M. Jovanović, J. Ðorđević in [4], while generally perturbed reflected backward stochastic differential equations are already observed by Ðorđević and Janković in [5]. Topic of this chapter is additive type of perturbations for reflected backward stochastic differential equations as a special type of mention general problem for reflected backward stochastic differential equations, and a more general one than the additive perturbation problem for simple backward stochastic differential equations. Finer and more precise estimates are deduced and generalizations emphasised.

Backward stochastic differential equations (BSDEs for short) was introduced and developed by Pardoux and Peng [6, 7, 8] in the 90s. Notation of nonlinear BSDE and proof of the existence and uniqueness of adapted solutions is given in their fundamental paper [6]. After that, many applications incited to introduction various types of BSDEs, in mathematical problems in finance (see [9]), stochastic control and stochastic games (see [10, 11]), stochastic partial differential equations, semi-linear parabolic partial differential equations (PDEs) (see [8, 12]) etc. (for further reading see also [13, 14, 15, 16, 17]).

Type of RBSDEs which is observed in this chapter have been first introduced in literature by El-Karoui et al. in [18]. Introduced RBSDEs with one lower barrier has following form,


where one of the components of the solution is forced to stay above a given barrier/obstacle process L=Ltt0T. The solution is a triple of adapted processes YtZtKtt0Twhich satisfies Eq. (1). The process K=Ktt0Tis nondecreasing and its purpose is to push upward the state process Y=Ytt0Tin order to keep it above the obstacle L.

As it was already mentioned, RBSDEs are connected with a wide range of applications within which, the pricing of American options (constrained or not) in markets is most famous one. Further, the important applications of RBSDEs are in mixed control problems, partial differential variational inequalities, real options (see [9, 18, 19, 20, 21] and the references therein) etc. El-Karoui et al. proved in [18] the existence and uniqueness of the solution to Eq. (1) under conditions of square integrability of the data and Lipschitz property for the coefficient (also called driver) f. Field of RBSDEs is developing in two directions, some authors deal with the issue of the existence and uniqueness results for RBSDEs under weaker assumptions (than the ones in [6] which are for the general BSDEs), while others are introducing some new types of those equations by adding jumps, introducing second barrier etc.

Systematization of the papers which are done in the framework of RBSDEs can be found in paper [5] by Ðorđević and Janković.

Recently, Hamèdene and Popier in [22] proved that if ξ,supt0TLt+and 0Tft,0,0dtbelong to Lpfor some p]1,2[, then RBSDE (1) with one reflecting barrier associated with fξLhas a unique solution. Aman gave [23] a similar result for a class of generalized RBSDEs with Lipschitz condition on the coefficients, and he extended these results under non-Lipschitz condition in his paper [24]. There are several papers by Hamadène [25] and Hamadène and Ouknine [26], Matoussi [27], Lepeltier and Xu [28] and Ren et al. [29, 30] in which authors emphasise the significance of the case when the data are from Lpfor some p]1,2[.

The aim of this chapter is a study Eq. (1) if the terminal condition ξand generator fare p-integrable, p]1,2[. Regarding that in several applications such as in finance, control, games, PDEs, etc., data are not square integrable, and the influence of some random external factors on the system can be seen as perturbations of the solution of Eq.(1), it is natural to introduced additive perturbation in the parameters of equation ξ, fand barrier process L, in order to better describe the change of the system and find some measurement for the change.

This chapter is organized in following way; In Section 2 elementary notations, definitions and preliminary results regarding RBSDEs are introduced. Next section is dedicated to the formulation of the main problem, i.e. problem of additively perturbed RBSDEs with one lower obstacle is stated. Together with the set up for the problem, some auxiliary estimates are proved in this section. In Section 4, conditions under which the solutions are stable are given, and estimates for the stability are derived. Section 5 contains the most interesting result, i.e. the estimation of a time interval for a given closeness of the solutions. The chapter is finished with the Section 6, Conclusions remarks, where the highlights of the chapter are emphasised and ideas and open problems for the future research are stated.

2. Preliminaries

All random variables and processes are defined on a filtered probability space ΩFFtP, where Ftt0Tis a natural filtration of a standard d-dimensional Brownian motion B=Btt0T, that is, it is right continuous and complete. Also, all stochastic processes are defined for t0T, where Tis a positive, fixed, real constant, and they take values in Rnfor some positive integer n. For any kNand xRk, xdenotes the Euclidean norm of x.

Further, for any real constant p]1,2[, we recall on standard notations which will be used:

  1. SpRis the set of R-valued, adapted and continuous processes Xtt0Tsuch that


The space SpRendowed with the norm Spis a Banach type.

  • Mpis the set of predictable processes Ztt0Twith values in Rdsuch that


    Likewise, MpRnendowed with the norm Mpis a Banach space.

  • The space Sp×Mpwill be denoted by Bp.

  • Let ξbe an R-valued and FT-measurable random variable and let a random function f:0T×Ω×R×RdRbe measurable with respect to P×BR×BRd, where Pdenotes the σ-field of progressive subsets of 0T×Ω, while LLtt0Tis a continuous progressively measurable R-valued process.

    The following hypothesis are introduced for ξ,fand L:



    1. The process ft,0,0t0Tsatisfies E0Tft,0,0dtp<;

    2. (ii) (Lipschitz condition) there exists a constant k>0such that for all t0T,yz,yzR×Rd,


    H3The barrier process Lsatisfies:

    1. LTξ;

    2. L+L0SpR.

    The definition of the unique solution to Eq. (1), associated with the triple ξfL, and the existence and uniqueness theorem under Lipschitz condition are given in [22].

    Definition 1

    1. (Existence of the solution.) The triple YtZtKtt0Tis an Lp-solution to RBSDE (1) with a continuous lower reflecting barrier L, terminal condition ξand drift/generator/driver fif:

      1. YtZtt0Tbelongs to Bp;

      2. K=Ktt0Tis an adapted continuous nondecreasing process such that K0=0andKTLpΩ;

      3. Yt=ξ+tTfsYsZsds+KTKttTZsdBsa.s.,t0T;

      4. YtLt,t0T;

      5. 0TYsLsdKs=0Pa.s.

    2. (Uniqueness of the solution.) The triple YtZtKtt0Tis a unique Lp-solution to RBSDE (1) if for any other solution Y¯tZ¯tK¯tt0T, the following holds,


    Proposition 1 [Hamèdene, Popier [22]] Let H1H3hold for ξ,fand L. Then, RBSDE (1) with one continuous lower reflecting barrier Lassociated with ξfLhas a unique Lp-solution, p]1,2[, i.e. there exists a triple of processes YtZtKtt0Tsatisfying Definition 1.

    The following lemma is well known result and it is widely used in stability estimates.

    Lemma 1 [Hamèdene, Popier [22]] Assume that YZBpis a solution of the equation



    1. fis a function satisfying the previous assumptions;

    2. The process Att0Tis Pa.s.of bounded variation.

    Then, for any 0tuTit follows that


    where cp=pp12and ŷ=yy1y0.

    When the model of some phenomenon is described by RBSDE, than, some change of the system can be treated as additive perturbation of the initial equation. The size of the change could be estimated as the difference between the solutions of the initial equation and the perturbed one. In view of this direction, together with Eq. (1), we study the following perturbed RBSDE,


    where ξε,fεand the barrier Lεare defined as ξ,fand L, respectively, they depend on a small parameter ε01, and they are of a special additive form


    For a given fεξεLε, a triple of adapted processes YtεZtεKtεt0Tis a solution to Eq. (3). In the sequel Eq. (1) will be named the unperturbedequation, while Eq. (3) a additivelyperturbedone. It is usually expected that the additively perturbed Eq. (3) is more general and more complexed than the unperturbed one. Furthermore, it is obvious that in case when βεαtyzεltε0, additively perturbed equation reduces to unperturbed equation. This fact is a basic motivation for us to introduce conditions guaranteeing the closeness of the solutions of the additively perturbed and unperturbed equations in the Lp-sense, and to estimate the conditions for the additive parameters in order for the solutions of these equations to stay close in the Lp-sense in some way.

    After basic notations, definitions and results are present, the formulation of the main problem is given in following section.

    3. Formulation of the problem of additively perturbed RBSDEs with one lower obstacle &and auxiliary results

    In order to deduce estimates for the closeness of the solutions of additively perturbed and unperturbed equations, following assumptions are introduced;

    A0For the additional part in final condition of perturbed equation βTε, such that ξε=ξ+βTε, while ξε,ξLpΩ, there exists a non-random function β1ε,ε01, such that


    A1For the additional part in the generator/driver integral, αtyzε, there exists a non-random function α1ε,ε01, such that


    A2For the additional part in barrier processes ltε, there exists a non-random function l1ε,ε01, such that


    We give first an auxiliary result for the stability of the solutions which we will use to prove main result.

    Proposition 2 Let p]1,2[and let YtZtKtt0Tand YtεZtεKtεt0Tbe the solutions to additively unperturbed and perturbed Eqs. (1) and (3), respectively. Let also assumptions A0A2and conditions H1H3be satisfied. Then,


    where c1=p1+pk+pk2p1and C1=β1ε+α1pεT+l1p1pεEK̂Tp1p.

    Proof: Let us denote for t0Tthe differences of the processes of the solutions,


    If we subtract Eqs. (1) and (3), we obtain


    Applying Lemma 1 on Yt̂p, we have


    where Iit,i=1,2,3,4are the appropriate integrals. In order to estimate I1t, we apply the elementary inequality ap1bp1pap+1pbp,a,b0and assumption A1. Then,


    In order to estimate I2t, we use the elementary inequality 2aba22+2b2,


    where cp=pp1/2.

    For estimation of member I3t, we will use mapping xaθ˜xa=xap21xaxa, xaR×R. Function xθ˜xais non-decreasing, while the function aθ˜xais non-increasing. As it is known, lsε=LsεLsand since YsεLsε,YsLs, then


    Substituting estimates for Iit,i=1,2,3in in (6), we obtain


    Taking expectation on last inequality, and taking into account that expectation of I4is 0, we have


    As KT,KTεLpΩ, it follows that EK̂Tp<. So (4) holds straightforwardly by applying the Gronwall-Bellman inequality ([31], Theorem 1.5):

    Let utbe a continuous function in ab, ftbe Riemann integrable function in aband c=const>0. If ut=ft+ctbusds,tab, then utft+ctbfsecstds,tab.

    The theorem is proved.

    For the introduced problem, conditions for the stability of the solutions and estimates for the stability of the solutions are derived In next section.

    4. Stability estimates for additive perturbations

    For the estimate of Lp-difference between the solutions to Eqs. (1) and (3), the Lp-stability of the solution to Eq. (1) is necessary.

    Following theorem provides the result, that in case of small additive perturbations case, we can expect that the difference of the solutions of perturbed and unperturbed equations tends to zero, when the perturbations are sufficiently small.

    Theorem 1 Let all the conditions of Proposition 2 be satisfied and let the functions β1ε,α1ε,l1εtend to zero as εtends to zero, uniformly in t0T. Then it follows that


    Proof: Let us define


    From Proposition 2, we have that C1ϕεC˜, where C˜=1+T+EK̂Tp1pand, therefore,


    Since ϕε0as ε0, then for every t00T,


    In order to estimate the Lp-closeness between the processes Ztand Zεt, as well as Ktand KεT, we need estimate Esupt0TŶtp, that is to estimate I4t. By applying the Burkholder-Davis-Guandy inequality [32] and Young inequality, uαv1ααu+1αv,v0,α01, we have


    We can conclude that


    It follows that




    where A1t0is a generic positive constant. By the assumption of the theorem, ϕεas ε0, then Esuptt0TŶtp0,asε0. The desired estimate holds if we take t0=0.

    Now we can estimate the other two parts.

    For every i0,1,2and arbitrary t00T, let us define stopping times


    Clearly, τiTa.s.when i. If we apply the Ito formula to ektŶt2,tt0τi, we find that


    where estimates J1and J2are the appropriate integrals. For λ1>0that


    Similarly, for λ2>0,




    Substituting (14), (15) and (16) in (13) yields


    It follows that


    The last inequality can be written as




    By applying the inequality i=1maikmk11i=1maik,ai0,k0on (18) and by taking expectation, we obtain


    It is left to estimate two integrals with respect to Brownian motion, which will be done by applying the Burkholder–Davis–Guandy inequality,


    where λ3>0, c3=1λ3Cp222p2epkT, and Cp=32/pp/2and Cp2=64/pp/4are the universal constants. Substituting previous estimates in (19), it follows that




    The constants λ2,λ3can be chosen such that 14p2λ2p2Cpλ3>0, then, from (12) and (20) it follows that


    where A2t0is a positive generic constant. By the Fatou’s Lemma,


    Then, second estimate holds if we take t0=0.

    It is left to estimate the difference between the processes Kand Kε. From (5), we have


    In view of (12) and (21), we derive that




    in accordance with the last estimate, we have that


    Hence, it follows from that there exists a generic constant A3t0>0such that


    Then, the last estimate of the theorem holds if we take t0=0, which completes the proof.

    In this section complete proof for the stability of the solutions is given, which as a strong result and it enable us to estimate the time interval for a given closeness of the solutions. This result is proved in next section.

    5. Time interval for a given closeness of the solutions

    Theorem 1 provides that the state processes Ytεand Yt, the control processes Ztεand Zt, as well as Kεand Kcould be arbitrarily close for εsufficiently small. I.e., if perturbations are small enough, closeness of the solutions can be provided. But, from the perspective of applications and modelling, it is usually important to study the closeness between Ytεand Ytnear to the terminal values ξεand ξ. Per example, for the application in pricing American options, an agent would be interested how will the price behave near the exercise time. It is interesting and useful to find the time interval on which we could preserve the wanted closeness, i.e. that for some permissible a>0and εsufficiently small, find t¯a=t¯0Tso that the rate of the closeness between Ytεand Ytdoes not exceed aon t¯T. Even-more, estimate of the closeness between the control processes Ztεand Zton t¯Tcan be estimated.

    Theorem 2 Let all the conditions of Theorem 1 hold. Also, let the function ϕε,ε01defined with (9) be continuous and monotone increasing. Then, for an arbitrary constant a>0and ε0Φ1a, there exists t¯0T, where


    such that


    and A2t¯and A3t¯are constants defined in (21) and (23), respectively.

    Proof: Let us introduce function SεTt,t0T, such that


    where c1is given in Proposition 1 and C˜in Theorem 1. For an arbitrary a>0,it must be


    that is,


    Since ϕεdecreases if εdecreases, it follows that


    where ϕ1is the inverse function of ϕ. For every εε1ε2, it is now easy to determine t̂from the relation SεTt̂=a, that is,


    If ε0ε1, then a>SεT. If ε0ε2, let us take


    Hence, for every ε0ε2, it is easy to see that


    Clearly, t¯Tas εε2and t¯0as εε1, that is, t¯0as ε0.

    This section illustrates the most important result of the chapter. Indeed, estimate of a time interval, for the given, precise closeness of the solutions is very important in the applications. Per example, if some random observation is modelled by RBSDE, and its behaviour (value) on fixed time Tis familiar, as well as its change up to some other value in capital moment, and if the driver of the model is supposed to linearly change, it is interesting to estimate the time interval on which we could. “control” the observations, i.e. under which our change under linearisation of final value and the drift will remain within the boundaries we impose.

    6. Conclusions and remarks

    It should be noted that this is a special case of generally perturbed problem observed by Ðorđević and Janković in [5], but we have provided and explicit, concrete estimates for the additive type of perturbations. Interesting in this case also is, that even-though we introduce the hypothesis (H2), i.e. Lipschitz condition for the drift/driver/generator function, this hypothesis is not explicitly used in the estimates for perturbations. It is necessary to have it in order to have the existence of the solutions for perturbed and unperturbed equations, but it is not necessary for the perturbation estimates with the given assumptions A0A2. It follows that results from this chapter can be generalized in several ways:

    1. assumption A1can be weaken in the sense that it can be per example of the form:

      1. Lipschitz condition


      for some Lipschitz constant Land nonrandom function α1tε.

    2. Non-Lipschitz condition.

      There exist constants C>0such that for any ωtΩ×0Tand y1z1,y2z2Rk×Rk×d,


    where ρ:0T×R+R+satisfies: For fixed t0T, ρtis: a concave and non-decreasing function with ρt00;

    • for fixed u, 0Tρtudt<;

    • for any M>0, the ODE


    has a unique solution ut0, t0T.

  • Linear growth condition


    for some constant Kand nonrandom function α1tε.

    In all alternatives, further assumption is that there exist nonrandom function α¯εsuch that


  • Conditions of existence and uniqueness of the solutions of perturbed and unperturbed equations can be generalized in a sense for the driver f,fεof Eqs. (1) and (3) to satisfy some of mentioned conditions: non-Lipschitz or linear growth one. In this manner, these assumptions would hold for the additional function αin the perturbed driver also.

  • In the case when we change the initial conditions and assumptions, the steps will be similar, while the main inequality at the end of the estimates will be established by applying Bihari inequality and not Gronwall-Bellman one.


    Jasmina Ðorđvić is supported by STORM-Stochastics for Time-Space Risk Models, granted by Research Council of Norway - Independent projects: ToppForsk. Project nr. 274410 and by “Functional Analysis and Applications”, 2011-2020, Faculty of Natural Sciences and Mathematics, University of Niš, Serbia, Project 174007, MNTRS.


    We would like to thank Prof. Svetlana Jankovć for the comments and suggestions.

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    Jasmina Ðorđević (February 4th 2021). Effect of Additive Perturbations on the Solution of Reflected Backward Stochastic Differential Equations [Online First], IntechOpen, DOI: 10.5772/intechopen.95872. Available from:

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