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 (2016) 1:9 DOI 10.1186/s41546-016-0007-y Probability, Uncertainty and Quantitative Risk RESEARCH Open Access A branching particle system approximation for a class of FBSDEs Dejian Chang · Huili Liu · Jie Xiong Received: 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 forwardbackward stochastic differential equations (FBSDEs) is proposed by using branching particle systems in a random environment First, by the four step scheme, we introduce a partial differential Eq (PDE) used to represent the solution of the FBSDE system Then, infinite and finite particle systems are constructed to obtain the approximate solution of the PDE The location and weight of each particle are governed by stochastic differential equations derived from the FBSDE system Finally, a branching particle system is established to define the approximate solution of the FBSDE system The branching mechanism of each particle depends on the path of the particle itself during its short lifetime  = n −2α , where n is the number of initial particles and α < 12 is a fixed parameter The convergence of the scheme and its rate of convergence are obtained Keywords Forward-backward stochastic differential equation · Partial differential equations · Branching particle system · Numerical solution MSC (2010) Classification 60H35 · 60H15 · 62J99 D Chang School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China H Liu Department of Mathematics, Hebei Normal University, Shijiazhuang 050024, People’s Republic of China J Xiong () Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macao, Special Administrative Region of China e-mail: jiexiong@umac.mo Page of 34 D Chang et al Introduction Since the work of Pardoux and Peng (1990), forward-backward stochastic differential equations (FBSDEs) have been extensively studied and have found important applications 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); Xiong and Zhou (2007), and Yong and Zhou (1999)) For instance, we consider a risk minimizing economic management problem x(·) denotes an economic quantity, which can be interpreted as cash-balance, wealth and an intrinsic value in different fields Suppose that x(·) is governed by  v d x (t) = (A(t)x v (t) + B(t)v(t))dt + (C(t)v(t) + D(t))d W (t), x v (0) = x0 , where v(·) is the control strategy of a policymaker and A(·), B(·), C(·), D(·) are bounded and deterministic Let ρ(x v (1)) denote the risk of the economic quantity x v (1), where the risk measure is convex in the sense of Făollmer and Schied (1999) Recently, Rosazza Gianin (2006) established the relationship between the risk measure ρ(·) and the g-expectation Egv (see Peng (2010)): ρ(x v (1)) = Egv [−x v (1)] where the functional g : [0, 1] × R × R × R → ×R satisfies g(t, y, 0) = and is the generator of the following BSDE:  −dy v (t) = g(t, y v (t), z v (t))dt − z v (t)d W (t), y v (1) = −x v (1) Thus, the objective of the policymaker is equivalent to minimizing J (v(·)) = E [y (0) + v v  (v(t) − M(t))2 dt] subject to the FBSDE In previous work, Ma et al (1994) studied the solvability of the adapted solution to the FBSDEs, in particular, they designed a direct scheme, called the four step scheme to solve the FBSDEs explicitly However, in most cases, it is often difficult to get the solution in closed form so it is important to study numerical methods for solving FBSDEs Following the earlier works of Bally (1997) and Douglas et al (1996), various efforts have been made to find efficient numerical schemes for FBSDEs 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-Carlo method (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 are very few works in the literature, such as Milstein and Tretyakov (2006), Delarue and Menozzi (2006), Cvitani´c and Zhang (2005), and Ma et al (2008) Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page of 34 In this paper, we are interested in investigating a new numerical scheme for a class of coupled FBSDEs by a branching particle system approximation There are various studies about particle system representations for stochastic partial differential equations with application to filtering since the pioneering work of Crisan and Lyons (1997) and Del Moral (1996) Here we list a few which are closely related 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, the decoupled 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 and the SDE in a delicate manner This paper can be regarded as a first attempt in this direction One of the main advantages of this method is the circumventing of the computation of conditional expectations via regression methods   Let , , F , {Ft }0≤t≤T , P be a filtered complete probability space, where {Ft }0≤t≤T denotes the natural filtration generated by a standard Brownian motion {Wt }0≤t≤T , F = FT and T > is a fixed time horizon We consider the following FBSDE in the fixed duration [0, T ]: ⎧ ⎨ d X (t) = b (X (t) , Y (t)) dt + σ (X (t)) d W (t), −dY (t) = g(X (t), Y (t), Z (t))dt − Z (t)d W (t), (1.1) ⎩ X (0) = x, Y (T ) = f (X (T )), where b : Rd × Rk → Rd , σ : Rd → Rd×l , g : Rd × Rk × Rk×l → Rk and f : Rd → Rk In what follows, we make the following assumption: (A1) The generator g has the following form: for z = (z , · · · , zl ), g(x, y, z) = C(x, y)y + l D j (x, y)z j , j=1 and b(x, y), σ (x), g(x, y, z), f (x), C(x, y) and D(x, y) are all bounded and Lipschitz continuous maps with bounded partial derivatives up to order Furthermore, the matrix σ σ ∗ is uniformly positive definite, and the function f is integrable Here σ ∗ denote the transpose of the matrix σ Remark 1.1 For the generators associated with g-expectation, the condition g(y, 0) = (we omit the variable t) together with an extra differentiability condition, we have  g(y, z) = D j (y, z)z j j=1 The case of D j (y, z) depending on z is more technically demanding in the construction of the branching particle systems We hope to return to this case in a future work Page of 34 D Chang et al 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  = Lu(t, x) + C (x, u (t, x)) u (t, x) + − ∂u(t,x) ∂t u(T, x) = f (x), l j=1 D j (x, u (t, x))∂x u (t, x) σ j (x) and L=  σσT d d j ∂xi ,x j + bi ∂xi , i, j=1  i=1 , σ = (σ1 , · · · , σl ) and bi being the ith coordinate of b with j = ij For ≤ t ≤ T , assume v (t, x) = u (T − t, x) Note that  ∂v(t,x) ∂t = Lv(t, x) + C (x, v (t, x)) v (t, x) + v (0, x) = f (x) l j=1 D j (x, v (t, x))∂x v (t, x) σ j (x) (1.2) Remark 1.2 According to Proposition 4.2 in Ma et al (1994), the above nonlinear parabolic partial differential equation has a unique solution The nonlinear parabolic partial differential Eq (1.2) can be written as: ⎧ d d ⎪ ∂v(t, x) ⎪ ⎪ a (x)∂ v x) + bi (x, v) ∂xi v(t, x) + C (x, v) v (t, x) = (t, i j x ,x ⎪ ∂t i j ⎪ ⎪ ⎪ i, j=1 i=1 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ + l D j (x, v)∂x v (t, x) σ j (x) j=1 v (0, x) = f (x) (1.3) By rules of derivative, we have ∂v(t, x) ∂t d d   = ∂xi ,x j j (x)v(t, x) − ∂xi ,x j j (x)v(t, x) 2 i, j=1 − + d d d ∂xi j (x)∂x j v (t, x) − i=1 j=1 d i=1 + i, j=1 d i=1 ∂xi (bi (x, v)v (t, x)) − d ∂x j j (x)∂xi v (t, x) j=1 i=1 d ∂xi bi (x, v) v (t, x) i=1 d ∂xi D˜ i (x, v)v (t, x) − ∂xi D˜ i (x, v) v (t, x) + C (x, v (t, x)) v (t, x) i=1 Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page of 34 ⎛⎛ ⎞ ⎞ d d d   = ∂xi ,x j j (x)v(t, x) − ∂xi ⎝⎝ ∂x j j (x) − bi (x, v) − D˜ i (x, v)⎠v (t, x)⎠ i, j=1 i=1 j=1 ⎛ ⎞ d d d + ⎝C (x, v(t, x)) − ∂xi bi (x, v) − ∂xi D˜ i (x, v) + ∂xi ,x j j (x)⎠ v (t, x) i=1 ≡ d i=1   ∂xi ,x j j (x)v(t, x) − i, j=1 i, j=1 ∂xi b˜i (x, v) v (t, x) + c˜ (x, v (t, x)) v (t, x) d i=1 where c˜ (x, v (t, x)) = C (x, v(t, x)) − d ∂xi bi (x, v) − i=1 b˜i (x, v(t, x)) = d ∂xi D˜ i (x, v) + i=1 d d ∂xi ,x j j (x), i, j=1 ∂x j j (x) − bi (x, v) − D˜ i (x, v) , j=1 and D˜ i (x, v) = l D j (x, v(t, x)) σi j (x) j=1 Comparing this equation with (1.1) in Kurtz and Xiong (1999) formally, we now construct an infinite particle system {X i (t) : i ∈ N} with locations in Rd and time varying weights {Ai (t) : i ∈ N} governed by the following equations: for < t ≤ T, i = 1, 2, · · · ⎧ ˜ i (t), v(t, X i (t)))dt + σ (X i (t))d B i (t), ⎨ d X i (t) = b(X d Ai (t) = Ai (t)c˜ (X i (t), v (t, X i (t))) dt (1.4) ⎩ V (t) = limn→∞ n1 nj=1 A j (t)δ X j (t) with i.i.d initial random sequence {(X i (0), Ai (0)), i ∈ N} taking values in Rd × R, where {Bi (t), i ∈ N} are independent standard Brownian motions and  V (0), φ = φ(x) f (x)d x, for any φ ∈ Cb2 (Rd ), Rd where Cb2 (Rd ) denotes the collection of all bounded functions with bounded continuous derivatives up to order 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 the approximate solution of the PDE and prove the convergent results “Branching particle system approximation” Section is devoted to the formulation of a branching particle 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 Page of 34 D Chang et al Particle system approximation For two integrable functions v1 , v2 , we define their distance  ρ (v1 , v2 ) = |v1 (x) − v2 (x)|d x Rd Now we construct infinite particle systems governed by the following stochastic differential equations: for any fixed δ > 0, t ∈ (0, T ], i = 1, 2, · · ·   ⎧ v δ (t, X iδ (t)) dt + σ (X iδ (t))d B i (t), ⎨ d X iδ (t) = b˜ X iδ (t),  d Aδ (t) = Aiδ (t)c˜ X iδ (t), v δ (t, X iδ (t)) dt (2.1) ⎩ δ i v (t, x) = limn→∞ n1 nj=1 Aδj (t) pδ (x − X δj (t)) where pδ is the heat kernel given by − d2 pδ (x) = (2π δ)  |x|2 ex p − 2δ  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: (I)  2 ˜  ˜ v)|2 ≤ K b(x, v) + |σ (x)|2 + |c(x, We also condition  make the following on the initial data: δ δ (II) (Ai (0), X i (0)) , A j (0), X j (0) is an i.i.d sequence and  2  2     E |Ai (0)|2 + E |X i (0)|2 + E Aδj (0) + E X δj (0) < ∞ Theorem 2.1 Assume that {Aiδ (0), X iδ (0)} is i.i.d and independent of {Bi } Under (A1), for every φ ∈ Cb2 (Rd ), we have     δ v (t, ·), φ = E Aiδ (t)Tδ φ(X iδ (t)) and  t  t   δ     δ  δ  δ v (s, ·), L v δ φ ds + v (s, ·), Lφ ds, (2.2) v (t, ·), φ = v (0, ·), φ + 0 where  Tδ φ(x) = Lφ(x) = and L v φ(x) = d pδ (x − y)φ(y)dy, Rd d j (x)∂xi ,x j φ(x), i, j=1 ˜ v)φ(x), b˜i (x, v)∂xi φ(x) + c(x, i=1 while b˜i is the ith coordinator of b˜ and j = (σ σ ∗ )i j Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page of 34 Proof By the law of large numbers, we have  n    δ Ai (t)Tδ φ(X iδ (t)) = E Aiδ (t)Tδ φ(X iδ (t)) v δ (t, ·), φ = lim n→∞ n i=1 Applying Itˆo’s formula to (2.1),        d Aiδ (t)Tδ φ X iδ (t) = Aiδ (t)c˜ X iδ (t), v δ (t, X iδ (t)) Tδ φ X iδ (t) dt     ˜ iδ (t), v δ (t, X iδ (t)))dt +Aiδ (t)∇ ∗ Tδ φ X iδ (t) σ (X iδ (t))d B i (t)+ b(X n     + Aiδ (t) ∂xi ,x j Tδ φ X iδ (t) j X iδ (t) dt i, j=1     δ = Ai (t)L vδ Tδ φ X iδ (t) dt + Aiδ (t)L T δ φ X iδ (t) dt   +Aiδ (t)∇ ∗ Tδ φ X iδ (t) σ (X iδ (t))d B i (t), (2.3) 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 E sup Aiδ (t)2 = sup E sup Aiδ (t)2 ≤ K < ∞ 0≤t≤T i (2.4) 0≤t≤T Hence, the martingale term on the right hand side of (2.3) can be estimated as follows: 2  n   1 t     E sup  Aiδ (s)∇ ∗ Tδ φ X iδ (s) σ (X iδ (s))d B i (s)   n 0≤t≤T i=1  T n  2   ≤ E Aiδ (s)2 ∇ ∗ Tδ φ X iδ (s) σ (X iδ (s)) ds n i=1 ≤ ∇Tδ φ 2∞ K T E sup Aiδ (t)2 → as n → ∞ n 0≤t≤T Integrating and averaging both sides of (2.3), we see that (2.2) holds Theorem 2.2 The 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 ˜ i (t), v(t, X i (t)))dt + σ (X i (t))d B i (t) d X i (t) = b(X 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 Ai (t) = Ai (t)c˜ (X i (t), v (t, X i (t))) dt is solvable The i.i.d property of {(Ai (0), X i (0))} and independence of {Bi }, i = 1, 2, · · · ensures that V (t) is well-defined 2    +6E c˜ X iδ (s), v˜ n,δ (s, X iδ (s)) − c˜ X iδ (s), v˜in,δ (s, X iδ (s))    2  +6E c˜ X iδ (s), v˜in,δ (s, X iδ (s)) − c˜ X iδ (s), v δ (s, X iδ (s))   2   ≤ 2K E X in,δ (s) − X iδ (s) ⎛ ⎛ ⎞⎞2 n  n    1     n,δ n,δ δ δ δ +6E ⎝ K ⎝ d A j (s) X j (s) − X j (s)⎠⎠ A j (s) − A j (s) + d nδ j=1 nδ + j=1 ⎛ ⎞⎞ ⎛ n K 1 Aiδ (s) + +6E ⎝ d ⎝ Aδj (s)⎠⎠ n(n − 1) δ2 n −1 j=1   2  +6E c˜ X iδ (s), v˜in,δ (s, X iδ (s)) − c˜ X iδ (s), v δ (s, X iδ (s))   2   ≤ 2K E X in,δ (s) − X iδ (s) n n 2 2 6K 12K  n,δ 12K  n,δ   E M j (s) − M δj (s) + d+1 E X j (s) − X δj (s) + d d nδ nδ δ n j=1 j=1    2  +6E c˜ X iδ (s), v˜in,δ (s, X iδ (s)) − c˜ X iδ (s), v δ (s, X iδ (s))  , + (2.9) where the last inequality follows from Cauchy-Schwarz inequality and the fact that for τn,δ ≥ T , n δ A j (s) ≤ δ −1/2 n j=1 Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page 11 of 34 On the other hand,  2   E sup X in,δ (t) − X iδ (t) r ≤t  t 2   ≤12E σ (X in,δ (s)) − σ (X iδ (s)) ds  t 2  ˜ n,δ n,δ n,δ ˜ iδ (s), v δ (s, X iδ (s))) ds + 3tE b(X (s), v (s, X (s))) − b(X i i  t  2   K X in,δ (s) − X iδ (s) ≤6(3t + 2)E (2.10) n n  2 2 2K  n,δ K4  n,δ   δ δ + (s) − M (s) + (s) − X (s) M X   + d j j j j d d+1 nδ nδ δ n j=1 j=1   2   + b˜ X iδ (s), v˜in,δ (s, X iδ (s)) − b˜ X iδ (s), v δ (s, X iδ (s))  ds 2K Let  2   gn (t) = E sup Min,δ (r ) − Miδ (r ) r ≤t and  2   f n (t) = E sup X in,δ (r ) − X iδ (r ) r ≤t   2  E b˜ X iδ (s), v˜in,δ (s, X iδ (s)) − b˜ X iδ (s), v δ (s, X iδ (s))   2   n n   δ δ δ δ δ δ  ≤ KE A j (s) pδ (X i (s) − X j (s)) − lim A j (s) pδ (X i (s) − X j (s)) n→∞ n − n   j=1, j=i j=1  2  n   Aδj (s) pδ (X iδ (s) − X δj (s)) − E Aδj (s) pδ (X iδ (s) − X δj (s))  = K E  n −  j=1, j=i = K (n − 1)2 n  2 E Aδ1 (s) pδ (X iδ (s) − X 1δ (s)) − E Aδ1 (s) pδ (X iδ (s) − X 1δ (s)) j=1, j=i  2 K E Aδ1 (s) pδ (X iδ (s) − X 1δ (s)) n−1 K ≤ d E Aδ1 (s)2 δ (n − 1) = (2.11) Similarly,    2   E c˜ X iδ (s), v˜in,δ s, X iδ (s) − c˜ X iδ (s), v δ (s, X iδ (s))  ≤ Then,  gn (t)≤ t  {K gn (s) f n (s) f n (s) + d + d+1 δ δ  + K2 K EAδ1 (s)2 δ d (n − 1) }ds δd n (2.12) Page 12 of 34 and D Chang et al  gn (s) f n (s) f n (t)≤ {K f n (s) + d + d+1 + K d }ds δ δ n δ Adding (2.12) and (2.13), for t ≤ T , we have   t K6 gn (t) + f n (t) ≤ K + d+1 (gn (s) + f n (s)) ds + d δ n δ  t  By Gronwall’s inequality, we have K6 K gn (t) + f n (t) ≤ d e δ n Then, we have and 1+ δ d+1 t = (2.13) K δ,T n  2 K δ,T   E sup Min,δ (t) − Miδ (t) ≤ n t≤T  2 K δ,T   E sup X in,δ (t) − X iδ (t) ≤ n t≤T Lemma 2.1 For ≤ t ≤ T , we have √   K δ,T Eρ v n,δ (t), v˜ n,δ (t) ≤ √ + K T δ n Proof   ρ v n,δ (t), v˜ n,δ (t)      n   n n,δ n,δ δ δ  dx  A (t) p (x − X (t)) − A (t) p (x − X (t)) = δ δ j j j j   n Rd  n j=1  j=1  n      n,δ    n,δ n,δ ≤ A j (t)− Aδj (t) pδ x − X j (t) + Aδj (t) pδ x − X j (t) − pδ x − X δj (t)  d x Rd n j=1 n   ... random walk approximations (e.g., Briand et al (2001) and Ma et al (2002)), Malliavin calculus and Monte-Carlo method (e.g., Zhang (2004), Ma and Zhang (2005), and Bouchard and Touzi (2004)) and... Probability, Uncertainty and Quantitative Risk (2016) 1:9 Page of 34 In this paper, we are interested in investigating a new numerical scheme for a class of coupled FBSDEs by a branching particle. .. 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