L2-regularity of solutions to linear backward stochastic heat equations, and a numerical application

Journal Pre-proof L2 -regularity of solutions to linear backward stochastic heat equations, and a numerical application

Yanqing Wang

PII:

S0022-247X(20)30032-9

DOI:

https://doi.org/10.1016/j.jmaa.2020.123870

Reference:

YJMAA 123870

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

13 October 2019

Please cite this article as: Y. Wang, L2 -regularity of solutions to linear backward stochastic heat equations, and a numerical application, J. Math. Anal. Appl. (2020), 123870, doi: https://doi.org/10.1016/j.jmaa.2020.123870.

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L2-regularity of Solutions to Linear Backward Stochastic Heat Equations, and a Numerical Application∗ Yanqing Wang†

Abstract 2

In this work, we mainly explore the L -regularity for the second component of the solutions to linear backward stochastic heat equations, which is crucial to obtain the convergence of the numerical solutions. As an application, we provide the convergence rate for time-discretized Galerkin approximation of these equations.

Keywords: Backward stochastic heat equations, L2 -regularity, time-discretized Galerkin approximation, convergence rate. AMS 2010 subject classification: 60H15, 65M60

1

Introduction

Let T ∈ (0, ∞), (Ω, F, F, P ) be a complete probability space and F = {Ft : t ∈ [0, T ]} be the natural filtration generalized by a 1-dimensional Wiener process {W (t) : t ∈ [0, T ]} satisfying the usual conditions (noting that W (·) may have high dimension). The purpose of this work is to present the L2 -regularity result on solution to the following linear backward stochastic heat equation: ⎧ ⎪ in [0, ×T ) × D, ⎨ dq(t, x) = (−Δq(t, x) + r(t, x) + f (t, x)) dt + r(t, x)dW (t), (1.1) q(t, x) = 0, on [0, T ) × ∂D, ⎪ ⎩ q(T, x) = p(T, x), in D, where D ⊂ Rd is a bounded domain with smooth boundary, ∂D is the boundary of the domain  D, Δ = dk=1 ∂ 2 /∂x2k is the Laplacian operator, and p(·, ·) solves the following forward stochastic heat equation: ⎧ ⎪ in [0, ×T ) × D, ⎨ dp(t, x) = (Δp(t, x) + b(t, x)) dt + (p(t, x) + σ(t, x)) dW (t), (1.2) p(t, x) = 0, on [0, T ) × ∂D, ⎪ ⎩ p(0, x) = p (x), in D. 0 As we know, when we study the convergence rate for Euler method of backward stochastic differential equations (BSDEs, for short), the main challenge is the regularity of the second component of the solution (martingale integral). Since the existence of this component comes from the ∗

This work is supported in part by the National Natural Science Foundation of China (11801467), the Chongqing Natural Science Foundation (cstc2018jcyjAX0148). † School of Mathematics and Statistics, Southwest University, Chongqing 400715, China. e-mail: [email protected].

1

martingale representation theorem, which is essentially dependent on terminal condition of BSDE. Hence, if the terminal condition turns out more information, then we can deduce more properties of the solution. Up to now, there exist two types of assumptions on terminal condition: (1) Malliavin derivability, (2) Markovian type. The reader can refer to [5, 8]. Under those assumptions, L2 -regularity can be obtained (see, [14, Theorem 3.1], [8, Theorem 2.6(b)]), and applying these regularity results, convergence rate for Euler method of BSDEs can be deduced. In the case of backward stochastic heat equation (1.1) with general terminal condition, applying Malliavin analysis such as that in [8], Wang in [13] also got the L2 -regularity result, and adopting the obtained result, he provided the convergence rate for time-discretized Galerkin approximation of that equation. However, the Malliavin derivability for a random variable is not easy to check, except for some special cases. In practical applications, BSDEs associated with some forward stochastic differential equations (SDEs, for short) are more frequently met. In the finite case, these equations are called Markovian ones. In the infinite case, we may also meet these equations. In what follows, we show an example to state that. Example 1.1. Let us consider a stochastic linear quadratic (SLQ, for short) problem: Problem (SLQ). Search a control u ¯(·) ∈ L2F (0, T ; L2 (D)), such that J(¯ u(·)) =

inf

u(·)∈L2F (0,T ;L2 (D))

J(u(·)),

associated with a stochastic heat system as follows: ⎧ ⎪ ⎨ dp(t, x) = (Δp(t, x) + u(t, x)) dt + p(t, x)dW (t), p(t, x) = 0, ⎪ ⎩ p(0, x) = p (x), 0 where 1 J(u(·)) = E 2



T 0

in [0, ×T ) × D, on [0, T ) × ∂D, in D,



α |p(t)|2L2 (D) + |u(t)|2L2 (D) dt + E|p(T )|2L2 (D) , 2

for some α ≥ 0. By Pontryagin maximum principle, Problem (SLQ) admits a unique optimal pair (¯ p(·), u ¯(·)), which is equivalent to the fact: the following backward stochastic heat equation ⎧ ⎪ in [0, ×T ) × D, ⎨ dq(t, x) = (−Δq(t, x) + p¯(t, x)) dt + r(t, x)dW (t), (1.3) q(t, x) = 0, on [0, T ) × ∂D, ⎪ ⎩ q(T, x) = −α¯ p(T, x), in D. has a unique solution, and the following maximum condition holds q(·) − u ¯(·) = 0, a.e.. Indeed, there exist some equations such as (1.1) coming from practical applications. Study on regularity of solutions, even the algorithm for numerical solutions, owns its potential value for solving specific problems. As an application of solution’s L2 -regularity, we would list a space-time discretization of (1.1), and prove its convergence rate in space and time variables. Up to now, the numerical methods for solving backward stochastic partial differential equations are few. Dunst and Prohl in [4] solved 2

linear backward stochastic heat equations by finite element method with respect to the space variables, and Euler method with respect to the time variable. Meanwhile, Wang in [13] presented a Galerkin approximation in space and Euler method in time for semi-linear backward stochastic heat equations. In [4], spacial discretization error is derived, while spacial and temporal discretization error is obtained in [13]. In [13], based on appropriate Malliavin derivability of terminal condition, 1 older continuity of martingale integral is deduced, and then the temporal discretization error 2 -H¨ is obtained. However, it seems that the regularity obtained by Malliavin analysis is too much in proving that error. The rest of the paper is organized as follows. In Section 2, we present some notations, assumptions and review the nonlinear Feymann-Kac Formula which is used to prove our main result. In Section 3, we mainly prove the L2 -regularity of martingale integral r(·). Applying deduced L2 regularity result, we provide the time-discretized Galerkin approximation of (1.1), and prove the convergence rate in Section 4.

2

Preleminaries

Let H be a Hilbert space with norm | · |H and inner product ·, ·H . The following spaces stochastic processes will be frequently used throughout this paper: • CF ([0, T ]; L2 (Ω; H)) is the space of all F-adapted, continuous, H-valued stochastic processes X satisfying |X|2CF ([0,T ];L2 (Ω;H)) = supt∈[0,T ] E|X(t)|2H < ∞; L2F (Ω; C([0, T ]; H)) is the subspace of CF ([0, T ]; L2 (Ω; H)) which satisfies E(supt∈[0,T ] |X(t)|2H ) < ∞; • L2F (0, T ; H) is the space of all F-adapted, H-valued processes Y satisfying |Y |2L2 (0,T ;H) = F

T E( 0 |Y (t)|2H dt) < ∞. Besides, denote by ·, · and | · | respectively the inner product and norm in different Euclidean spaces, which can be identified from the context; denote by In the unit matrix of size n. Define A : D(A) = H 2 (D) ∩ H01 (D) −→ L2 (D) by Af = Δf , for all f ∈ D(A). It is easy to show that A is the infinitesimal generator of a C0 -semigroup on L2 (D). Then (1.1), (1.2) can be rewritten as the following abstract form:  dq(t) = (−Aq(t) + r(t) + f (t))dt + r(t)dW (t), t ∈ [0, T ], (2.1) q(T ) = p(T ), and (2.2)



dp(t) = (Ap(t) + b(t))dt + (p(t) + σ(t))dW (t),

t ∈ [0, T ],

p(0) = p0 ,

respectively. The reader can refer to [3, 6, 10] for well-posedness of (2.2), and [9, 11] for well-posedness of (2.1). Throughout the paper, we shall make use of the following assumptions: (A). b, σ, f : R+ × Rd −→ Rd are deterministic functions which are 12 -H¨ older continuous with respect to t, i.e., there exists a positive constant L, such that |Φ(t1 , x) − Φ(t2 , x)| ≤ L |t1 − t2 |, ∀t1 , t2 ∈ R+ , x ∈ Rd , Φ = b, σ, f. 3

Moreover, p0 ∈ H01 (D), b(·, ·), σ(·, ·), f (·, ·) ∈ L2 (0, T ; H01 (D)) and sup |σ(t)|2L2 (D) ≤ L.

t∈[0,T ]

∞ Let {(−λi , φi )}∞ i=1 be the sequence of A’s eigenvalues and eigenfunctions, where {φi }i=1 forms 2 an orthonormal basis of L (D). Set Sn is the n-dimensional space generated by eigenfunctions {φ1 , φ2 , · · · , φn }, for n = 1, 2, · · · . Let π is a partition of [0, T ], where π : 0 = t0 < t1 < · · · < tN = T with the mesh size |π| = max0≤i≤N −1 (ti+1 − ti ). We will write

Δi = ti − ti−1 , Δi W = W (ti ) − W (ti−1 ), ∀i = 0, 1, · · · , N − 1. Throughout this work, we assume that the partition π is quasi-uniform, which means that, there exists a constant K such that, |π| ≤ K, ti+1 − ti

∀i = 0, 1, · · · , N − 1.

Definition 2.1. For a given partition π, if there exists a constant K, such that N −1  tk+1 

E |r(t) − r(tk )|2L2 (D) + |r(t) − r(tk+1 )|2L2 (D) dt ≤ K|π|, k=0

tk

we call r(·) owns the L2 -regularity. By virtue of eigenfunctions, let us decompose the solution (p(t, x), q(t, x), r(t, x)) to (1.1) and (1.2) as follows: for any (t, x) ∈ [0, T ] × D, (2.3)

p(t, x) =

∞

j=1

αj (t)φj (x), q(t, x) =

∞

βj (t)φj (x), r(t, x) =

j=1

∞

γj (t)φj (x).

j=1

For any n ∈ N, set αn (·) = (α1 (·), α2 (·), · · · , αn (·)) , β n (·) = (β1 (·), β2 (·), · · · , βn (·)) ,

(2.4)

γ n (·) = (γ1 (·), γ2 (·), · · · , γn (·)) . We know that (αn (·), β n (·), γ n (·)) is the solution to the following decoupled forward-backward stochastic differential equation: ⎧ n n n n n ⎪ ⎨dα (t) = (−Λn α (t) + b (t)) dt + (α (t) + σ (t)) dW (t), t ∈ [0, T ], (2.5) dβ n (t) = (Λn β n (t) + γ n (t) + f n (t)) dt + γ n (t)dW (t), t ∈ [0, T ], ⎪ ⎩ n α (0) = pn0 , β n (T ) = αn (T ), where Λn = diag {λ1 , λ2 , · · · , λn },   = b(·, x), φ1 (x)L2 (D) , b(·, x), φ2 (x)L2 (D) , · · · , b(·, x), φn (x)L2 (D) ,   σ n (·) = σ(·, x), φ1 (x)L2 (D) , σ(·, x), φ2 (x)L2 (D) , · · · , σ(·, x), φn (x)L2 (D) ,   f n (·) = f (·, x), φ1 (x)L2 (D) , f (·, x), φ2 (x)L2 (D) , · · · , f (·, x), φn (x)L2 (D) ,   pn0 (·) = p0 , φ1 L2 (D) , p0 , φ2 L2 (D) , · · · , p0 , φn L2 (D) . bn (·)

4

For simplicity, we will write b(·, x), φi (x)L2 (D) , σ(·, x), φi (x)L2 (D) , f (·, x), φi (x)L2 (D) , p0 , φi L2 (D) by bi (·), σi (·), fi (·), p0i , respectively. The following result is the nonlinear Feynman-Kac Formula, which will be used to represent the martingale integral r(·). Theorem 2.2. Suppose that u(·, ·) ∈ C 1,2 ([0, T ] × Rn ; Rn ) is a classical solution to the following PDEs: ⎧ 1 n  ⎪ i n i n ⎪ ⎪ ⎨∂t u (t, x) + 2 (α (t) + σ (t)) ∂xx u (t, x) (αn (t) + σ (t)) + ∂x ui (t, x)(−Λn αn (t) + bn (t)) + fi (t) = 0, (t, x) ∈ [0, T ] × Rn , i = 1, 2, · · · , n, ⎪ ⎪ ⎪ ⎩u(T, x) = x, x ∈ Rn . Then, it holds that (2.6)

β n (t) = u(t, αn (t)), γ n (t) = ∂x u(t, αn (t))(αn (t) + σ n (t)), t ∈ [0, T ].

Moreover, u(·, ·) is continuously differentiable in x with bounded derivatives and (2.7)

∂x u(t, αn (t)) = ∇β n (t)(∇αn (t))−1 , γ n (t) = ∇β n (t)(∇αn (t))−1 (αn (t) + σ n (t)), t ∈ [0, T ],

where ∇αn (·) ∈ L2F (0, T ; Rn×n ), and (∇β n (·), ∇γ n (·)) ∈ L2F (0, T ; Rn×n × Rn×n ) solve matrix-valued SDE:  t  t n n Λn ∇α (s)ds + ∇x(s)dW (s), t ∈ [0, T ], (2.8) ∇α (t) = In − 0

and matrix-valued BSDE: (2.9)

0



T

∇β n (t) = ∇αn (T ) − t

3

 Λn ∇β n (s)ds −

T

∇γ n (s)dW (s), t ∈ [0, T ].

t

L2 -regularity

In this section, we mainly prove the L2 -regularity of r(·). By the decomposition (2.3), we only need to get the L2 -regularity of γn (·). Since (βn (·), γn (·)) satisfies a Markovian BSDE, in what follows, we would study the regularity αn (·) firstly, and then βn (·), γn (·). Lemma 3.1. ∇αn (·) defined in (2.8) can be obtained     1 n (3.1) ∇α (t) = exp − Λn + In t + In W (t) , t ∈ [0, T ], 2 which is invertible; and (∇αn (·))−1 ∈ L2F (Ω; C([0, T ]; Rn×n )) is the solution to the following SDE:  t  t (3.2) (∇αn (s))−1 (Λn + In ) ds − (∇αn (s))−1 dW (s), t ∈ [0, T ]. (∇αn (t))−1 = In + 0

0

Furthermore, αn (·) can be represented by

(3.3)

 t (∇αn (s))−1 (bn (s) − σ n (s))ds αn (t) =∇αn (t)αn (0) + ∇αn (t) 0  t + ∇αn (t) (∇αn (s))−1 σ n (s)dW (s), t ∈ [0, T ]. 0

5

Moreover,

(3.4)

  t γ n (t) =∇β n (t) αn (0) + (∇αn (t))−1 σ n (t) + (∇αn (s))−1 (bn (s) − σ n (s))ds 0   t −1 n n (∇α (s)) σ (s)dW (s) , t ∈ [0, T ]. + 0

Proof. (3.1) can be checked directly, and (3.2) is deduced subsequently. By Itˆ o’s formula to (∇αn (t))−1 αn (·), we get (3.3). At last, (3.4) can be deduced by (3.3) and (2.7). Remark 3.2. Actually, ∇αn (·) ∈ L2F (Ω; C([0, T ]; Rn×n )) is the fundamental solution to the first equation in (2.5). Since ∇αn (·), ∇β n (·), ∇γ n (·) are all diagonal matrices, we only need to consider the diagonal elements reading ∇αi (·), ∇βi (·), ∇γi (·), for all i ∈ N. Lemma 3.3. Matrix-valued BSDE (2.9) can be unique solved by ∇β n (·) = diag {∇β1 , ∇β2 , · · · , ∇βn }, ∇γ n (·) = In , 

with ∇βi (t) = exp

λi −

1 2



 t + W (t) exp{−2λi T }, ∀t ∈ [0, T ].

Furthermore, for any p ≥ 2, it holds that (3.5)

E sup |∇βi (t)|p ≤ C exp{−λi pT }, t∈[0,T ]

E|∇βi (t) − ∇βi (s)|p ≤ C|t − s|p/2 , ∀t, s ∈ [0, T ], i ∈ N,

where C is a positive constant independently on i. Proof. By (2.9) and (3.1), we can get     1 t + W (t) , t ∈ [0, T ], (3.6) ∇αi (t) = exp − λi + 2    and it is easy to check that (∇βi (·), ∇γi (·)) = (exp λi − 12 · +W (·) exp{−2λi T }, 1) solves the following 1-dimensional BSDE:  (3.7)

∇βi (t) = ∇αi (T ) −

T t

 λi ∇βi (s)ds −

T t

∇γi (s)dW (s), t ∈ [0, T ].

That is the first desired result. In what follows, we would adopt martingale to estimate ∇βi (·). Indeed, for any  the exponential  p2 p ≥ 2, noting that Mp (t) ≡ exp pW (t) − 2 t is an exponential martingale, one gets   1 pt + pW (t) exp{−2λi pT } |∇βi (t)| = exp λi − 2    1 p = exp λi − + pt − 2λi pT Mp (t). 2 2 

p

(3.8)

6



Then E|∇βi (t)| ≤ exp p

   p(p − 1) p(p − 1) t EMp (t) = exp t , 2 2

and by Doob’s maximal inequality, E sup |∇βi (t)|p ≤ exp{−λi pT }E sup |M1 (t)|p t∈[0,T ]

t∈[0,T ]

p p E|M1 (T )|p p−1   p  p p(p − 1) = exp{−λi pT } exp T EMp (T ) p−1 2   p  p p(p − 1) exp T . = exp{−λi pT } p−1 2 

≤ exp{−λi pT } (3.9)

Now, we tend to estimate E|∇βi (t) − ∇βi (s)|p . Without loss of generality, taking s ≤ t, for p ≥ 2, we have |∇βi (t) − ∇βi (s)|p =| exp{−2λi T } [exp {λi t} M1 (t) − exp {λi s} M1 (s)] |p ≤2p−1 exp{−2pλi T } [exp {λi t} − exp {λi s}]p |M1 (t)|p

(3.10)

+ 2p−1 exp{−2pλi T } exp {pλi s} |M1 (t) − M1 (s)|p . Since

[exp {λi t} − exp {λi s}]2 = exp {2λi t} − 2 exp {λi (t + s)} + exp {2λi s}  t   =2λi exp{2λi τ } − exp{λi (τ + s)} dτ s

≤4λi exp{2λi T }|t − s|, by (3.10), we can see that (3.11)

E|∇βi (t) − ∇βi (s)|p  p/2 ≤C λi exp{−pλi T }|t − s|p/2 E|M1 (t)|p + exp{−pλi T }E|M1 (t) − M1 (s)|p .

By (3.9), we find that E|M1 (t)|p ≤ C. Now, we deal with E|M1 (t) − M1 (s)|p . Since M1 (t) solves dM1 (t) = M1 (t)dW (t), t ∈ [0, T ], by martingale moment inequality, we can see that  t p    E|M1 (t) − M1 (s)| = E  M1 (τ )dW (τ ) s  t p/2 ≤CE |M1 (τ )|2 dτ p

s

≤CE sup |M1 (t)|p |t − s|p/2 t∈[0,T ]

 ≤C

p p−1

p

 exp

p(p − 1) T 2 7

 |t − s|p/2 ,

which, together with (3.11), leads to (3.12)

p/2

E|∇βi (t) − ∇βi (s)|p ≤ Cλi

exp{−pλi T }|t − s|p/2 ,

where C is a constant only dependently on p. Finally, the second inequality in (3.5) can be showed by the fact: for a given positive constant K, there exists a constant C, such that x exp{−Kx} ≤ C, ∀x ∈ R. That completes the proof. Using the similar procedure, we can prove: Lemma 3.4. The following statements hold, for any p ≥ 2,  p E sup (∇αi (t))−1  ≤ C exp{λi pT }, t∈[0,T ]  (3.13) −1 −1 p E| (∇αi (t)) − (∇αi (s)) | ≤ C λpi |t − s|p/2 + 1 exp{pλi T }|t − s|p/2 , ∀t, s ∈ [0, T ], where C only depends on p. Now, we are in the position to prove the L2 -regularity of r(·), the second component of the solution to backward stochastic heat equation (1.1). Theorem 3.5. Suppose that Assumption (A) holds. Then we have the following estimate:  N

E (3.14)

k=1



tk tk−1

≤C 1 +



|r(t) − r(tk−1 )|2L2 (D) + |r(t) − r(tk )|2L2 (D) dt

|p0 |2L2 (D)





T

+

|b(s)|2H 1 (D) 0

0

+

|σ(s)|2H 1 (D) 0



 ds |π|.

older continuous with respect to the time variable, specifically, Moreover, if σ(·) ≡ 0, then r(·) is 12 -H¨ (3.15)

E|r(t) −

r(s)|2L2 (D)

 ≤C

|p0 |2L2 (D)



T

+ 0

|b(s)|2H 1 (D) ds 0

 |t − s|.

Here C only depends on T, L. Proof. We divide our proof into two steps. In the first step, we prove (3.15), and then prove (3.14) in the second step. Step 1. In this step, we consider the case of σ(·) ≡ 0. By (3.4) in Lemma 3.1, we know that, for any t ∈ [0, T ],    t −1 (3.16) (∇αi (s)) bi (s)ds . γi (t) =∇βi (t) αi (0) + 0

8

Thus, for 0 ≤ s < t ≤ T , it follows that E|γi (t) − γi (s)|2   =E(∇βi (t) − ∇βi (s))αi (0) 



+ ∇βi (t)

(3.17)

t



−1

s

−1

 2  bi (τ )dτ 

bi (τ )dτ − ∇βi (s) (∇αi (τ )) 0  2  s   −1 2  ≤3E|(∇βi (t) − ∇βi (s))αi (0)| + 3E (∇βi (t) − ∇βi (s)) (∇αi (τ )) bi (τ )dτ  (∇αi (τ ))

0

0

 2  t   + 3E ∇βi (t) (∇αi (τ ))−1 bi (t)dτ  s

≡3I1 + 3I2 + 3I3 . By Lemma 3.3 and 3.4, we find that I1 ≤ C|t − s||αi (0)|2 ,

(3.18) and

 I3 ≤ E sup |∇βi (t)|4 

(3.19)

t∈[0,T ] t

≤C  ≤C



s T 0

2  4  t    E sup (∇αi (t))−1   bi (s)ds t∈[0,T ]

s

|bi (s)|2 ds|t − s| |bi (s)|2 ds|t − s|.

For I2 , due to (3.12) and Lemma 3.4, we have   I2 ≤

(3.20)

E |∇βi (t) − ∇βi (s)|

 ≤C

T

0

 4   −1   E sup (∇αi (τ ))  

4

τ ∈[0,s]

s 0

2  bi (τ )dτ 

λi |bi (τ )|2 dτ |t − s|.

Combining with (3.17)-(3.20), we have E|r(t) − r(s)|2L2 (D) = ≤



∞

C |αi (0)|2 +

i=1



≤C

|p0 |2L2 (D)

 + 0

E|γi (t) − γi (s)|2

i=1



T

∞

T 0

(1 + λi )|bi (τ )|2 dτ

|b(τ )|2H 1 (D) dτ 0

That is (3.15).

9

 |t − s|

 |t − s|.

Step 2. Now, we prove the result (3.14) for general σ(·). Also, (3.4) in Lemma 3.1, leads to

(3.21)

E|γi (t) − γi (tk−1 )|2   ≤4E(∇βi (t) − ∇βi (tk−1 ))αi (0)  2   t  tk−1  (∇αi (τ ))−1 bi (τ )dτ − ∇βi (tk−1 ) (∇αi (τ ))−1 bi (τ )dτ  + ∇βi (t) 0 0  2   −1 + 4E ∇βi (t) (∇αi (t)) σi (t) − ∇βi (tk−1 ) (∇αi (tk−1 ))−1 σi (tk−1 )  2  t  tk−1   −1 −1  + 4E ∇βi (t) (∇αi (s)) σi (s)ds − ∇βi (tk−1 ) (∇αi (s)) σi (s)dτ  0

0

  t   −1 n n  + 4E ∇βi (t) (∇α (s)) σ (s)dW (s) − ∇βi (tk−1 ) 0

tk−1

n

(∇α (s))

−1

0

2  σ (s)dW (s) n

≡4J1 + 4J2 + 4J3 + 4J4 . In Step 1, we have showed that   J1 ≤ C |αi (0)|2 +

(3.22)

T 0

(1 + λi )|bi (τ )|2 dτ

 |π|,

and applying the similar procedure, we also can show that  T λi |σi (t)|2 dt|π|. (3.23) J3 ≤ CE 0

Thus, we only need to estimate J2 , J4 . For J2 , noting that ∇βi (t) (∇αi (t))−1 = exp{2λi (t − T )}, we find that  2 !   J2 ≤2E  ∇βi (t) (∇αi (t))−1 − ∇βi (tk−1 ) (∇αi (tk−1 ))−1 σi (t)  2   −1 + 2E ∇β (t ) (∇α (t )) (σ (t) − σ (t ))  i k−1 i k−1 i i k−1  (3.24) ≤Cλi |t − tk−1 ||σi (t)|2 + 2|σi (t) − σi (tk−1 )|2   ≤C λi |σi (t)|2 |π| + |σi (t) − σi (tk−1 )|2 . For J4 , such as I2 , I3 , we can get (3.25)  2  tk−1   −1 n n  (∇α (s)) σ (s)dW (s) J4 ≤2E (∇βi (t) − ∇βi (tk−1 )) 0  2  t     −1 n n + 2E ∇βi (tk−1 ) (∇α (s)) σ (s)dW (s)   tk−1     4  tk−1  4 −1  2 ≤C E |∇βi (t) − ∇βi (tk−1 )| E sup (∇αi (s))  |σi (s)| ds + C  ≤C

T 0

λi |σi (s)|2 ds|π| + C



0

s∈[0,T ]

tk tk−1

|σi (s)|2 ds.

Here, martingale moment inequality is utilized. 10

t tk−1

|σi (s)|2 ds

At last, (3.23), (3.24), (3.25), together with (3.21) and (3.22), yield to N

k=1



tk

E

tk−1

|γi (t) − γi (tk−1 )|2 dt

  ≤C |αi (0)|2 +

T 0

 N

E +C

tk

k=1

tk−1



  ≤C |αi (0)|2 +

 (1 + λi )|bi (s)|2 ds |π|

N

 λi |σi (s)|2 + |σi (s) − σi (tk−1 )|2 ds|π| + C

T 0



  (1 + λi ) |bi (s)|2 + |σi (s)|2 ds |π| + C





tk

k=1 tk−1 N

 tk k=1

tk−1

tk tk−1

|σi (s)|2 dsdt

|σi (s) − σi (tk−1 )|2 ds|π|.

Furthermore, we conclude that N



tk

E

tk−1

k=1

|r(t) − r(tk−1 )|2L2 (D) dt =

  2 ≤C |p0 |L2 (D) + L +

0

T

 N

E k=1



|b(s)|2H 1 (D) 0

+

tk

∞

tk−1 i=1

|σ(s)|2H 1 (D) 0



|γi (t) − γi (tk−1 )|2 dt  ds |π|.

The another part of (3.14) can be deduced by the similar procedure. That completes the proof. Remark 3.6. By the above result, we can find that the regularity of r(·) is mainly influenced by σ(·) in the diffusion term of the forward stochastic heat equation (1.2), and is totally not influenced older continuity. by nonhomogeneous f (·) in the drift term of (1.1). If σ(·) ≡ 0, we can get r(·)’s 12 -H¨ But if σ(·) = 0, we only get the L2 -regularity in the mean sense. The following result presents the estimate for martingale integral in the point-wise sense. Theorem 3.7. Suppose that Assumption (A) holds. Then it follows that: " #  T |b(t)|2L2 (D) dt , (3.26) sup E|r(t)|2L2 (D) ≤ C |p0 |2L2 (D) + sup |σ(t)|2L2 (D) + t∈[0,T ]

0

t∈[0,T ]

where C is a constant. Proof. We only need to estimate E|γi (t)|2 , for any t ∈ [0, T ]. By (3.4), we only need to estimate the following terms. (3.27)

E|∇βi (t)αi (0)|2 ≤ C|αi (0)|2 ,

(3.28)

E|∇βi (t) (∇αi (t))−1 σi (t)|2 = | exp{2λi (t − T )}σi (t)|2 ,

(3.29)

 2  t   −1 E ∇βi (t) (∇αi (s)) (bi (s) − σi (s))ds 0  2  4  t    −1   4 ≤ E sup |∇βi (t)| E sup (∇αi (t))   (bi (s) − σi (s))ds  ≤C

t∈[0,T ] T

0



0

t∈[0,T ]

 |bi (s)|2 + |σi (s)|2 ds, 11

and  2  t   −1  E ∇βi (t) (∇αi (s)) σi (s)dW (s) 0   t  2 2    −1 4 ≤C E sup |∇βi (t)| E (∇αi (t)) σi (s) ds t∈[0,T ]

(3.30) ≤C

 E sup |∇βi (t)|4 t∈[0,T ]

 ≤C



T 0

0

 4  t   E sup (∇αi (t))−1  |σi (s)|2 ds 0

t∈[0,T ]

|σi (s)|2 ds.

Combining with (3.27)-(3.30), we can get (3.26). Theorem 3.8. If Assumption (A) is true, then for any s, t ∈ [0, T ], the following regularity for q(·) holds E|q(t) − q(s)|2L2 (D) $ ≤C

|p0 |2H 1 (D) 0

+ sup t∈[0,T ]

|σ(t)|2L2 (D)



T

+ 0



|b(t)|2L2 (D)

+

|f (t)|2L2 (D)

+

|σ(t)|2H 1 (D) 0



% dt |t − s|,

where C is a constant. Proof. To get the result, it is sufficient to estimate E|βi (t) − βi (s)|2 . By BSDE (2.5), we can get  t  t   E|βi (t) − βi (s)|2 ≤6E |γi (τ )|2 dτ |λi βi (τ )|2 + |γi (τ )|2 + |fi (τ )|2 dτ |t − s| + 2E s s (3.31)  t  T   |γi (τ )|2 dτ. |λi βi (τ )|2 + |γi (τ )|2 + |fi (τ )|2 dτ |t − s| + 2E ≤6E 0

Now, we tend to estimate E that  2 λi E|βi (t)| + E

s

T 0

T

|λi βi (t)|2 dt appeared in (3.31). Applying Itˆo’s formula, we see 

T

|λi βi (τ )|2 dτ  T  T 2 λi βi (τ ), γi (τ )dτ − 2E λi βi (τ ), fi (τ )dτ =λi E|βi (T )| − 2E t t  T  T  T |λi βi (τ )|2 dτ + 2E |γi (τ )|2 dτ + 2 |fi (τ )|2 dτ, ≤λi E|βi (T )|2 + E t

2

λi |γi (τ )| dτ + 2E

t

which deduces that  (3.32) E

T 0

2

t

t

2

|λi βi (τ )| dτ ≤ λi E|βi (T )| + 2E

t



T 0



 |γi (τ )|2 + |fi (τ )|2 dτ.

For the SDE part, we still can apply above trick, and obtain  t |λi αi (τ )|2 dτ λi E|αi (t)|2 + 2E 0  t  t   1/2 1/2 |λi αi (τ )|2 + |λi σi (τ )|2 dτ. ≤λi E|αi (0)|2 + E |λi αi (τ )|2 + |bi (τ )|2 dτ + E 0

0

12

Then λi E|αi (t)|2 + E

(3.33)



t 0

  |λi αi (τ )|2 dτ ≤ C λi |αi (0)|2 +

T

0

 1/2 |bi (τ )|2 + |λi σi (τ )|2 dτ .



Finally, combining (3.31)-(3.33) with the fact βi (T ) = αi (T ), and Theorem 3.7, we have E|βi (t) − βi (s)|2 $ ≤C

  1/2 sup |σi (t)|2 + |αi (0)|2 + |λi αi (0)|2 +

t∈[0,T ]



T

%



+ 0

T 0



 |bi (t)|2 + |fi (t)|2 dt

1/2 |σi (t)|2 + |λi σi (t)|2 dt |t − s|.

Therefore, we deduce that E|q(t) − q(s)|2L2 (D) $ ≤C

sup t∈[0,T ]

|σ(t)|2L2 (D)

+

|p0 |2H 1 (D) 0



T

+ 0



|b(t)|2L2 (D)

+

|f (t)|2L2 (D)

+

|σ(t)|2H 1 (D) 0



% dt |t − s|.

That completes the proof.

4

An application on numerical scheme

In this section, based on the results in Section 3, we list the time-discretized Galerkin approximation for (1.1), and prove its convergence rate. Out totally discrete method is stated as follows: (Step 1.) Spacial Galerkin approximation: approximating (1.1) by n 1-dimensional BSDEs ⎧ ⎪ ⎨dαi (t) = (−λi αi (t) + bi (t)) dt + (αi (t) + σi (t)) dW (t), t ∈ [0, T ], (4.1) dβi (t) = (λi βi (t) + γi (t) + fi (t)) dt + γi (t)dW (t), t ∈ [0, T ], ⎪ ⎩ αi (0) = p0i , βi (T ) = αi (T ), i = 1, 2, · · · , n. (Step 2.) Temporal discretization: solving (4.1) by implicit Euler method, ⎧ ! −1 π π π ⎪ (t ) = (1 + λ |π|) (t ) + b (t )Δ + (α (t ) + σ (t )) Δ W α α ⎪ i i i k+1 k k k+1 k k k+1 i i i ⎪ ⎪ ⎪ ! ⎪ ⎪ ⎪ ⎨βiπ (tk ) = (1 + λi |π|)−1 E (βiπ (tk+1 )|Ftk ) − (γiπ (tk ) + fi (tk )) Δk ,   (4.2)  Δk+1 W π ⎪ π ⎪  ⎪ γ k = 0, 1, · · · , N − 1, (t ) = E β (t ) F tk , k+1 i k ⎪ ⎪ Δk+1 i ⎪ ⎪ ⎪ ⎩ π αi (0) = p0i , βiπ (tN ) = αiπ (T ). In what follows, we take (qnπ (·), rnπ (·)) as (q(·), r(·))’s time-discretized Galerkin approximation, where qnπ (t) =

n

i=1

βiπ (tk )φi , rnπ (t) =

n

γiπ (tk )φi , t ∈ [tk , tk+1 ), k = 0, 1, · · · , N − 1.

i=1

The following is the convergence rate for our numerical scheme. Since the proof is standard, we only list the result. For more details, the reader can refer to [13] for Galerkin approximation, and [13, 14] for Euler approximation. 13

Theorem 4.1. Suppose that Assumption (A) holds. Then E|q(tk ) − qnπ (tk )|2L2 (D) +

(4.3)

 N

E

tk tk−1

k=1

|r(t) − rnπ (tk−1 )|2L2 (D) ≤ C



 1 + λ2n |π| . λn

Furthermore, if martingale integral r(·) does not appear in the drift term in (1.1), then it holds that   1 π 2 (4.4) E|q(tk ) − qn (tk )|L2 (D) ≤ C + λn |π| . λn Proof. For the convergence rate (4.3), the reader can refer to [13, Theorem 4.1]. Note that when we deal with the error E|q(T )−qnπ (T )|2L2 (D) , we use the fact E|q(T )−qnπ (T )|2L2 (D) = E|p(T )−pπn (T )|2L2 (D) ,  and the error estimate E|p(T ) − pπn (T )|2L2 (D) ≤ C λ1n + λn |π| (see [7, Theorem 1]). We only prove (4.4). For this special case, we borrow the idea from error estimate of finite element method for forward stochastic partial differential equations. Set ek = β n (tk ) − β0 (tk ), where (β0 (·), γ0 (·)) solves the following auxiliary BSDE: ⎧  tk+1  tk+1   ⎪ ⎨β0 (t ) − β0 (t) = γ0 (s)dW (s), t ∈ [t , t ], Λn β0 (t ) + f n (t ) ds + k+1

k

t

⎪ ⎩β (T ) = β n (T ). 0

k

k

t

k+1

It is easy to see that β0 (tk ) = β n,π (tk ) ≡ (β1π (tk ), β2π (tk ), · · · , βnπ (tk )) , k = 0, 1, · · · , N.

(4.5)

Subsequently, we have  tk+1    ek+1 − ek = Λn (β n (t) − β0 (tk )) + (f n (t) − f n (tk )) dt + tk

tk+1 tk

(γ n (t) − γ0 (t))dW (t).

Multiplying ek on both side and taking expectation, we find that  tk+1 E(|ek |2L2 − ek , ek+1 ) + E Λn ek , ek ds tk



tk+1

=−E =−E

tk  tk+1 tk

&

Λn (β (t) − β (tk )) + (f (t) − f (tk )), ek dt − E n

n

n

n

tk+1 tk

' (γ (t) − γ0 (t))dW (t), ek n

Λn (β n (t) − β n (tk )) + (f n (t) − f n (tk )), ek dt.

Since

 1  E(|ek |2L2 − ek , ek+1 ) = E |ek |2 − |ek+1 |2 + |ek − ek+1 |2 , 2

and  E

tk+1 tk

Λn (β n (t) − β n (tk )) + (f n (t) − f n (tk )), ek dt

1 |π| ≤ E|ek |2 + E 2 2



tk+1 tk

 | Λn ek |2 + | Λn (β n (t) − β n (tk ))|2 + |(f n (t) − f n (tk ))|2 dt,

14

for |π| ≤ 12 , we can get E|ek |2 + E



tk+1

|



tk

Λn ek |2 ds 

2

≤ (1 + 2|π|) E|ek+1 | + 2E

tk+1 tk

 | Λn (β n (t) − β n (tk ))|2 + |(f n (t) − f n (tk ))|2 dt

≤··· (4.6)

≤ (1 + 2|π|)N −k E|eN |2 +

N −1

2 (1 + 2|π|)

j−k

 E

tj

j=k

≤ (1 + 2|π|)N −k E|eN |2 +

N −1

tj+1

N

2 (1 + 2|π|) E



tj+1 tj

j=k

 | Λn (β n (t) − β n (tj ))|2 + |(f n (t) − f n (tj ))|2 dt

 | Λn (β n (t) − β n (tj ))|2 + |(f n (t) − f n (tj ))|2 dt.

By the definition of ek and β0 (·), we know that eN ≡ 0. By the proof of Theorem 3.8 we can get (4.7)

E|

Λn (β n (t) − β n (s))|2 ≤ Cλn |t − s|.

older’s continuity of f , we obtain By the 12 -H¨ (4.8)

|f n (t) − f n (s)|2 ≤ C|t − s|.

(4.7), (4.8), together with (4.5), (4.6), the fact (1 + 2|π|)N ≤ C, yield (4.9)

E|β n (tk ) − β n,π (tk )|2 ≤ Cλn |π|, ∀k = 0, 1, · · · , N,

which, together with [13, Theorem 3.1], yields the desired result. That completes the proof. Remark 4.2. If r(·) does not appear in the drift term, then the L2 -regularity of r(·) can not be used to get the error estimate of E |q(tk ) − qnπ (tk )|2L2 (D) . But, if we still tend to obtain the error

tk  π 2 2 estimate for N k=1 E tk−1 |r(t) − rn (tk−1 )|L2 (D) , L -regularity of r(·) is essential and necessary. Remark 4.3. Besides Assumption (A), if the data have more high regularity, saying p0 ∈ H01 (D) ∩ H 2 (D), sup |σ|2H 1 (D) ≤ L, σ(·) ∈ L2 (0, T ; H01 (D) ∩ H 2 (D)), t∈[0,T ]

0

then, in the case r(·) is not in the drift term, using the similar procedure as that in Theorem 4.1, we can prove that   1 E|q(tk ) − qnπ (tk )|2L2 (D) ≤ C + |π| . λn Remark 4.4. Since the original equation is stochastic, in Euler method (4.2), conditional expectations appear, whose computation is complicated. This is the main difference between deterministic equations and stochastic ones. In the stochastic PDEs case, compared with the finite element method in spacial discretization, conditional expectations appeared in our method are much easier to compute. There are some method to approximate conditional expectations subject to BSDEs, such as Malliavin calculus technique ([2]), regression based least squares Monto Carlo method ([1]), finite transposition method ([12]), and so on. 15

References [1] C. Bender and R. Denk, A forward scheme for backward SDEs, Stochastic Process. Appl., 117 (2007), pp. 1793–1812. [2] B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), pp. 175–206. [3] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992. [4] T. Dunst and A. Prohl, The forward-backward stochastic heat equation: numerical analysis and simulation, SIAM J. Sci. Comput., 38 (2016), pp. A2725–A2755. [5] N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), pp. 1–71. [6] F. Flandoli, Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions, Stochastics Stochastics Rep., 29 (1990), pp. 331–357. [7] W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc., 54 (1996), pp. 79–85. [8] Y. Hu, D. Nualart, and X. Song, Malliavin calculus for backward stochastic differential equations and application to numerical solutions, Ann. Appl. Probab., 21 (2011), pp. 2379– 2423. [9] Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), pp. 445–459. [10] N. V. Krylov, A W2n -theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98 (1994), pp. 389–421. [11] G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Anal. Appl., 14 (1996), pp. 461–486. [12] P. Wang and X. Zhang, Numerical solutions of backward stochastic differential equations: a finite transposition method, C. R. Math. Acad. Sci. Paris, 349 (2011), pp. 901–903. [13] Y. Wang, A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations, Math. Control Relat. Fields, 6 (2016), pp. 489–515. [14] J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), pp. 459–488.

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