A branching particle system approximation for a class of FBSDEs

A branching particle system approximation for a class of FBSDEs Probability, Uncertainty and Quantitative Risk Probability, Uncertainty and Quantitative Risk (2016) 1 9 DOI 10 1186/s41546 016 0007 y R[.]

Probability, Uncertainty and Quantitative Risk

Probability, Uncertainty and Quantitative Risk (2016) 1:9

DOI 10.1186/s41546-016-0007-y

A branching particle system approximation for a

class of FBSDEs

Dejian Chang · Huili Liu · Jie Xiong

Received: 6 April 2016 / Accepted: 10 August 2016 /

© The Author(s) 2016 Open Access This article is distributed under the terms of the Creative Commons

Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Abstract In this paper, a new numerical scheme for a class of coupled backward stochastic differential equations (FBSDEs) is proposed by using branchingparticle systems in a random environment First, by the four step scheme, we intro-duce a partial differential Eq (PDE) used to represent the solution of the FBSDEsystem Then, infinite and finite particle systems are constructed to obtain the approx-imate solution of the PDE The location and weight of each particle are governed bystochastic differential equations derived from the FBSDE system Finally, a branch-ing particle system is established to define the approximate solution of the FBSDEsystem The branching mechanism of each particle depends on the path of the parti-cle itself during its short lifetime = n −2α , where n is the number of initial particles

forward-andα < 1

convergence are obtained

Keywords Forward-backward stochastic differential equation· Partial differentialequations· Branching particle system · Numerical solution

Since the work of Pardoux and Peng (1990), forward-backward stochastic differentialequations (FBSDEs) have been extensively studied and have found important appli-cations in many fields, including finance, risk measure, stochastic control and so on(cf Cvitani´c and Ma (1996); El Karoui et al (1997); Ma and Yong (1999); Xiongand Zhou (2007), and Yong and Zhou (1999)) For instance, we consider a risk min-

can be interpreted as cash-balance, wealth and an intrinsic value in different fields

Suppose that x (·) is governed by



dx v (t) = (A(t)x v (t) + B(t)v(t))dt + (C(t)v(t) + D(t))dW(t),

x v (0) = x0,

wherev(·) is the control strategy of a policymaker and A(·), B(·), C(·), D(·) are

quan-tity x v (1), where the risk measure is convex in the sense of F¨ollmer and Schied

(1999) Recently, Rosazza Gianin (2006) established the relationship between therisk measureρ(·) and the g-expectation E v

g(see Peng (2010)):

ρ(x v (1)) = E g v [−x v (1)]

the generator of the following BSDE:

In previous work, Ma et al (1994) studied the solvability of the adapted tion to the FBSDEs, in particular, they designed a direct scheme, called the four stepscheme to solve the FBSDEs explicitly However, in most cases, it is often diffi-cult to get the solution in closed form so it is important to study numerical methodsfor solving FBSDEs Following the earlier works of Bally (1997) and Douglas et

solu-al (1996), various efforts have been made to find efficient numerical schemes forFBSDEs In the decoupled forward-backward case, these include the PDE method

in the Markovian case (e.g., Chevance (1997)), random walk approximations (e.g.,Briand et al (2001) and Ma et al (2002)), Malliavin calculus and Monte-Carlomethod (e.g., Zhang (2004), Ma and Zhang (2005), and Bouchard and Touzi (2004))and so on However, in the case of coupled FBSDEs, to our knowledge, there arevery few works in the literature, such as Milstein and Tretyakov (2006), Delarue andMenozzi (2006), Cvitani´c and Zhang (2005), and Ma et al (2008)

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 3 of 34

In this paper, we are interested in investigating a new numerical scheme for aclass of coupled FBSDEs by a branching particle system approximation There arevarious studies about particle system representations for stochastic partial differ-ential equations with application to filtering since the pioneering work of Crisanand Lyons (1997) and Del Moral (1996) Here we list a few which are closelyrelated to the present work: Kurtz and Xiong (1999); Kurtz and Xiong (2001),Crisan (2002), Xiong (2008), Liu and Xiong (2013), Crisan and Xiong (2014).Particle system representations for FBSDEs are studied in Henry-Labord`ere

et al (2014) when the forward part is independent of the backward one, namely, thedecoupled case In this case, the approximation of the solution of a PDE and that

of the forward SDE can be constructed separately However, for the coupled case,the construction of the branching particle system must consider both the PDE andthe SDE in a delicate manner This paper can be regarded as a first attempt in thisdirection One of the main advantages of this method is the circumventing of thecomputation of conditional expectations via regression methods

, , F , { F t}0≤t≤T , P be a filtered complete probability space, where{F t}0≤t≤T denotes the natural filtration generated by a standard Brownian motion

{W t}0≤t≤T , F =F T and T > 0 is a fixed time horizon We consider the following

FBSDE in the fixed duration[0, T ]:

In what follows, we make the following assumption:

(A1) The generator g has the following form: for z = (z1 , · · · , z l ),

and b (x, y), σ(x), g(x, y, z), f (x), C(x, y) and D(x, y) are all bounded and

Lip-schitz continuous maps with bounded partial derivatives up to order 2 Furthermore,the matrixσσ∗is uniformly positive definite, and the function f is integrable Here

σ∗denote the transpose of the matrixσ.

Remark 1.1For the generators associated with g-expectation, the condition

g (y, 0) = 0 (we omit the variable t) together with an extra differentiability condition,

con-Relying on the idea of the four step scheme, we know that the solution to the above

FBSDE has the relation Y (t) = u(t, X(t)), Z(t) = ∂ x u (t, X (t)) σ (X(t)), where

u (t, x) is a solution to the PDE

i j,σ = (σ1, · · · , σ l ) and b i being the ith coordinate of b.

For 0≤ t ≤ T , assume v (t, x) = u (T − t, x) Note that

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 5 of 34

b (R d ) denotes the collection of all bounded functions with bounded

continu-ous derivatives up to order 2 In Theorem 2.2, we will show that the density function

of V (t) determined by the above infinite particle system is v(t, x), which is exactly

the solution to PDE (1.2)

The rest of this paper is organized as follows In “Particle system approximation”Section, we construct infinite and finite particle systems to respectively get theapproximate solution of the PDE and prove the convergent results “Branching part-icle system approximation” Section is devoted to the formulation of a branching parti-cle system to represent the approximate solution of the PDE In “Numerical solution”Section, we present the numerical solution of the FBSDE system and its error bound.Finally, “Conclusion” Section concludes the paper

Particle system approximation

For two integrable functionsv1, v2, we define their distance

In this paper we regard K with or without subscript as a constant which assumes

different values at different places By the boundedness of the coefficient assumed in(A1), we can verify the following condition:

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 7 of 34

ProofBy the law of large numbers, we have

where∇∗denotes the transpose of the gradient operator∇

By the boundedness of˜c, it is easy to show that there is a constant K such that

Integrating and averaging both sides of (2.3), we see that (2.2) holds

Theorem 2.2The solution to particle system (1.4) is unique and its density function is the solution to partial differential equation (1.3).

Proof Firstly, we know for any fixed i = 1, 2, · · · , the SDE

d X i (t) = ˜b(X i (t), v(t, X i (t)))dt + σ(X i (t))d B i (t)

has a unique solution because of the Lipschitz condition on the coefficients Since

we know the partial differential Eq (1.3) has a unique solution, then

d A i (t) = A i (t)˜c (X i (t), v (t, X i (t))) dt

is solvable The i.i.d property of {(A i (0), X i (0))} and independence of {B i }, i =

1, 2, · · · ensures that V (t) is well-defined.

Following similar steps as in Theorem 2.1, for anyφ ∈ C2

Since (2.5) is a parabolic PDE satisfying the uniform elliptic condition, by

stan-dard PDE theory, it is well-known that V (t) is absolutely continuous with respect to

the Lebesgue measure We denote the density function byv(t, x) Then,

12

i.e.v(t, x) is the solution to Eq (1.3).

Remark 2.1For any φ ∈ C2

b (R d ), it is obvious that

v(t), φ = E (A i (t)φ (X i (t))) , where i = 1, 2, · · ·

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 9 of 34

Next we introduce a finite particle system to get the approximation solution: forfixedδ > 0, t ∈ (0, T ],

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 11 of 34

On the other hand,

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 13 of 34

where the last inequality follows from (2.8), (2.15) and the fact thatρ ≤ 2.

Lemma 2.2For0≤ t ≤ T , we have Eρ˜v n,δ (t), v δ (t)≤ √K δ n

adapt the argument of Crisan and Xiong (Crisan and Xiong 2014) to the current setup

Lemma 2.3There exists a constant C3(T ), such that

sup

x∈Rd v δ (t, x) ≤ C3(T ) ProofBy the convolution form of (2.2), we have

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 15 of 34



Rd ∂ j v δ (s, y)L i j p t −s (y, x)dyds

+

 t0

As a consequence of Lemmas 2.3 and 2.4, we have the following lemma

Lemma 2.5There exists a constant K T , such that Eρv δ (t), v(t)≤ K T√

δ ProofWe define ¯v δ = v δ − v Then

here we used, in the first inequality, the integrability condition onv(0, ·) = f

Applying Gronwall’s inequality, we get

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 17 of 34

Remark 2.2Set c∞t = sup

y∈Rd c t (y), similarly, we have

c t∞≤ K T√

δ.

Theorem 2.3The distance of v n,δ (t) to v(t) is bounded by K√δ,T n + K T√

δ ProofCombining the conclusions from Lemmas 2.1, 2.2 and 2.5, we get

Eρv n ,δ (t) , v (t)≤ K√δ,T n + K T√

δ.

Branching particle system approximation

hence, the error of the approximation grows exponentially fast To avoid this back of the numerical scheme, we introduce a branching particle system to modifythe weights of the particles at the time-discretization steps

draw-Firstly, we rewrite our infinite particle systems governed by the following tic differential equations: for any fixedδ > 0, t ∈ (0, T ], i = 1, 2, · · ·

Now, we are ready to construct the branching particle system For fixedδ > 0,  =

n −2α , 0 < α < 1, there are n particles initially, each with weight 1 at locations

X n i ,δ, (0), i = 1, 2, · · · , n which are i.i.d random variables in R d Assume the timeinterval is[0, T ] and N∗ = T

which is the largest integer not greater than T



Define(t) = j for j ≤ t < ( j + 1) In the time interval [ j, ( j + 1)), j ≤ N∗,

A n,δ, i ( j, t) = exp{˜c δ

X n,δ, i ( j) , V n ,δ, ( j) (t − j)}, where the initial values are defined as: X i n,δ, (0) = x, A i n,δ, (0, 0) = 1, m n

0 = n.

At the end of the interval, the ith particle branches into ξ i

j+1offsprings such thatthe conditional expectation and the conditional variance given the information prior

to the branching satisfy

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 19 of 34

Proof For simplicity of notation, we only consider the case when j = 0 and denote

A n i ,δ, (0, t) and X n i ,δ, (t) by A(t) and X(t).

By the independent increments of B (t) and note that θ B

By Itˆo’s formula, we have

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 21 of 34

By Itˆo’s formula, we know

Convergence ofV n ,δ, (t) to V δ (t) at any point t ∈ [0, T]

Proposition 3.1For j = 1, 2, · · · , N∗, there exists a constant K, such that

⎞

⎠2

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 23 of 34

l=1

φ X l n ,δ, (t) −n1

m n [t/]

Letψ s , 0 ≤ s ≤ N be the solution to the PDE (3.2) with t replaced by N Note

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 25 of 34

Lemma 3.3There exist a constant K such that E (I1 )2≤ K n −(1−2α)

ProofNote that

= n12

N

j=1E

= n12

N

j=1E

2 in the following paragraph.

Theorem 3.2For any t ∈ [0, T ], δ > 0,  = n −2α and0< α < 1

2, there exists a constant K δ,T , such that Eρ2

1



V n ,δ, (t), V δ (t)≤ K T n −(1−2α) + K δ,T n −2α Proof From the Eq (3.3), it is obvious that I2 can be separated into the sum ofthree parts:

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 27 of 34

X l n ,δ, (r) , V n ,δ, (( j − 1)) − ˜c δ

X l n ,δ, (r) , V n ,δ, (r)  + ˜c δ

X l n,δ, (r) , V n,δ, (r) − ˜c δ

X l n,δ, (r), V δ (r) 

≡ Q1+ Q2+ Q3.

Then, we have

E (Q1 )2 ≤ EK2X n,δ,

l (( j − 1)) − X n,δ, l (r)2

= EK2 ˜b δ

X l n,δ, (( j − 1)) , V n,δ, (( j − 1)) (r − ( j − 1)) + σ X l n,δ, (( j − 1)) (B l (r) − B l (( j − 1)))2

≤ K δ n −2α + K δ ρ2

1

V n,δ, (r), V δ (r).

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 29 of 34



V n ,δ, (r), V δ (r)dr

By triangle inequality, we have

l=1p δ

x − X l n,δ, (t) , i.e the smooth density of

V n,δ, (t), as the numerical approximation of v(t, x) and u n,δ, (t, x) = v n,δ, (T −

t , x) as the numerical approximation of u(t, x) Then, we have the following

corollary:

Corollary 3.1For any t ∈ [0, T ], 0 < α < 1

2, there exists a constant K δ , such that

E | u n,δ, (t, x) − u(t, x) |≤ K δ,T

n−1−2α2 ∨ n −α + K T√

δ Proof We set u δ (t, x) = v δ (T − t, x) Then

obtained by the same numerical scheme with u n,δ, (t, x) Firstly, we apply the Euler

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 31 of 34

Then, by the result of the four step scheme, we define Y n,δ, (t) = u n,δ, (t, ˜X n t ,δ, )

as the numerical solution of Y (t) in FBSDE (1.1) We have the following theorem:

Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 33 of 34

Conclusion

In this paper we investigated a new numerical scheme for a class of coupled backward stochastic differential equations Combining the four step scheme and theEuler Scheme, we defined a new numerical solution of the FBSDE system by branch-ing particle systems in a random environment and proved related convergent results.Prior to our work, there was no literature about particle system representations forthe numerical approximations of FBSDE systems

forward-Acknowledgements We would like to thank an anonymous referee for his/her constructive suggestions which lead to the improvement of this article.

Liu acknowledges research support by National Science Foundation of China NSFC 11501164 Xiong acknowledges research support by Macao Science and Technology Fund FDCT 076/2012/A3 and Multi- Year Research Grants of the University of Macau No MYRG2014-00015-FST and MYRG2014-00034- FST.

Bally, V: Approximation scheme for solutions of BSDE, Backward stochastic differential equations, (Paris,

1995–1996), Pitman Res Notes Math Ser., vol 364, pp 177–191 Longman, Harlow (1997) Bouchard, B, Touzi, N: Discrete-time approximation and Monte-Carlo simulation of backward stochastic

differential equations Stoch Process Appl 111, 175–206 (2004)

Briand, P, Delyon, B, M´emin, J: Donsker-type theorem for BSDEs Electron Comm Probab 6, 1–14

(2001)

Chevance, D: Numerical methods for backward stochastic differential equations, Numerical methods in

finance, Publ Newton Inst., pp 232–244 Cambridge Univ Press, Cambridge (1997)

Crisan, D: Numerical methods for solving the stochastic filtering problem, Numerical methods and

stochastics (Toronto, ON, 1999), Fields Inst Commun., 34, Amer Math Soc., pp 1–20, Providence,

to the branching satisfy

Probability,... that θ B

By Itˆo’s formula, we have