An efficient Monte Carlo scheme for Zakai equations Christian Beck13 Sebastian Becker1 Patrick Cheridito1 Arnulf Jentzen23and Ariel Neufeld4

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An eﬃcient Monte Carlo scheme for Zakai equations

Christian Beck1,3, Sebastian Becker1, Patrick Cheridito1,

Arnulf Jentzen2,3and Ariel Neufeld4

1Department of Mathematics, ETH Zurich, Switzerland

2School of Data Science and Shenzhen Research Institute of Big Data,

The Chinese University of Hong Kong, Shenzhen, China

3Applied Mathematics: Institute for Analysis and Numerics,

Faculty of Mathematics and Computer Science, University of M¨unster, Germany

4Division of Mathematical Sciences, School of Physical and Mathematical Sciences,

Nanyang Technological University, Singapore

Abstract

In this paper we develop a numerical method for eﬃciently approximating solutions of certain

Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE

with random coeﬃcients. We show that under suitable regularity assumptions on the coeﬃcients of

the Zakai equation, the corresponding random PDE admits a solution random ﬁeld which, for almost

all realizations of the random coeﬃcients, can be written as a classical solution of a linear parabolic

PDE. This makes it possible to apply the Feynman–Kac formula to obtain an eﬃcient Monte Carlo

scheme for computing approximate solutions of Zakai equations. The approach achieves good

results in up to 25 dimensions with fast run times.

Keywords: Zakai equation, nonlinear ﬁltering problems, stochastic partial diﬀerential equations,

Doss–Sussmann transformation, Feynman–Kac representation

1 Introduction

The goal of stochastic ﬁltering is to estimate the conditional distribution of a not directly observable

stochastic process blurred by measurement noise. The process of interest is usually called signal

process, while the observed process is referred to as observation process. Whereas the signal process

follows a hidden dynamic, probing the system only reveals the observation process, which, in general,

might depend nonlinearly on the signal process and, in addition, is blurred by measurement noise.

Stochastic ﬁltering problems were ﬁrst studied in connection with tracking and signal processing

(see the seminal works by Kalman [26] and Kalman & Bucy [27]) but soon turned out to also be

relevant in a variety of other applications in ﬁnance, the natural sciences and engineering. Among

others, nonlinear ﬁltering problems naturally arise in e.g., ﬁnancial engineering ([3, 10, 13, 18, 20, 21]),

weather forecasting ([8, 9, 11, 17, 19, 33]) or chemical engineering ([7, 12, 34, 35, 36, 38]). For further

applications of nonlinear ﬁltering, we refer to the survey paper [31]. Stochastic ﬁltering problems

are naturally related to stochastic partial diﬀerential equations (SPDEs) since in continuous time,

the (unnormalized) density of the unobserved signal process given the observations is described by a

suitable SPDE, such as the Zakai equation [39] or Kushner equation [30]. The SPDEs arising in this

context can typically not be solved explicitly but instead, have to be computed numerically. Moreover,

they often are high-dimensional as the number of dimensions corresponds to the state space dimension

of the ﬁltering problem.

In this paper, we focus on Zakai equations with coeﬃcients that satisfy certain regularity conditions.

Let us assume the signal follows the d-dimensional dynamics

Yt=Y0+Zt

µ(Ys)ds +σWt

arXiv:2210.13530v2 [math.NA] 21 Aug 2023

for a d-dimensional random vector Y0with density φ:Rd→[0,∞), a suﬃciently regular function

µ:Rd→Rd, a constant d×d-matrix σand a d-dimensional Brownian motion (Wt)t∈[0,T ]independent

of Y0, while we observe a k-dimensional process of the form

Zt=Zt

h(Ys)ds +Vt

for a suﬃciently regular function h:Rd→Rkand a k-dimensional Brownian motion (Vt)t∈[0,T ]inde-

pendent of Y0and (Wt)t∈[0,T ]. Then the solution of the corresponding Zakai equation

Xt(x) = φ(x) + Zt

01

2TraceRdσσTHess(Xs)(x)−div(µXs)(x)ds +Zt

Xs(x)⟨h(x), dZs⟩Rd(1)

describes the evolution of an unnormalized density of the conditional distribution of Ytgiven obser-

vations of Zs,s≤t; that is,

[Yt∈A|Zs, s ∈[0, t]] = RAXt(x)dx

RRdXt(x)dx for every Borel subset A⊆Rd.

Our numerical method is based on a transformation which transforms a Zakai SPDE of the form (1)

into a PDE with random coeﬃcients. We show that under suitable conditions on the coeﬃcients of

the Zakai SPDE, the solution of the resulting random PDE is ω-wise a classical solution of a linear

parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an eﬃcient Monte

Carlo scheme for the numerical approximation of solutions of high-dimensional Zakai equations. The

following is this paper’s main theoretical result.

Theorem 1. Let T∈(0,∞),d, k ∈N,σ∈Rd×d, and consider functions φ∈C2(Rd,[0,∞)),

µ∈C3(Rd,Rd)and h∈C4(Rd,Rk)such that φhas at most polynomially growing derivatives up to

the second order, µhas bounded derivatives up to the third order and hhas bounded derivatives up to

the fourth order. Let (Ω,F,(Ft)t∈[0,T ],)be a ﬁltered probability space satisfying the usual conditions1

which supports standard (Ft)t∈[0,T ]-Brownian motions W, U : [0, T ]×Ω→Rdand V: [0, T ]×Ω→Rk

with continuous sample paths such that Wand Vare independent. Let Y: [0, T ]×Ω→Rdand

Z: [0, T ]×Ω→Rkbe (Ft)t∈[0,T ]-adapted stochastic processes such that (Y0∈A) = RAφ(x)dx for

every Borel subset A⊆Rdand

Yt=Y0+Zt

µ(Ys)ds +σWt, Zt=Zt

h(Ys)ds +Vtfor t∈[0, T ].(2)

For all z∈C([0, T ],Rk),t∈[0, T ]and x∈Rd, let Rz,t,x : [0, t]×Ω→Rdbe an (Fs)s∈[0,t]-adapted

stochastic processes satisfying2

Rz,t,x

s=x+Zs

0σσT[Dh(Rv,t,x

r)]Tz(t−r)−µ(Rv,t,x

r)dr +σUsfor all s∈[0, t].(3)

Moreover, let for all z∈C([0, T ],Rk), the functions Bz, uz: [0, T ]×Rd→Rbe given by3

Bz(t, x) =1

2σT[Dh(x)]Tz(t), σT[Dh(x)]Tz(t)Rd−1

2⟨h(x), h(x)⟩Rk

2TraceRdσσTHessx(⟨h(x), z(t)⟩Rk)− ⟨µ(x),[Dh(x)]Tz(t)⟩Rd−div(µ)(x)(4)

and

uz(t, x) = hφ(Rz,t,x

t) expt

Bz(t−s, Rz,t,x

s)dsi, t ∈[0, T ], x ∈Rd.(5)

1A ﬁltered probability space (Ω,F,(Ft)t∈[0,T ],) is said to satisfy the usual conditions if for all t∈[0, T ), one has

SA∈F,(A)=0{B⊆Ω : B⊆A} ⊆ Ft=Ts∈(t,T ]Fs.

2By Dh(x) we denote the Jacobian matrix ∂

∂xjhi(x)1≤i≤k,1≤j≤d,x∈Rd.

3For d∈N, we denote by ⟨·,·⟩Rd:Rd×Rd→Rthe standard scalar product given by ⟨x, y⟩Rd=Pd

i=1 xiyiand by

∥.∥Rd:Rd→[0,∞) the corresponding norm ∥x∥Rd=p⟨x, x⟩.

Then

Xt(x, ω) = uZ(ω)(t, x) exp⟨h(x), Zt(ω)⟩Rk, t ∈[0, T ],x∈Rd,ω∈Ω,(6)

is, up to indistinguishability, the unique random ﬁeld X: [0, T ]×Rd×Ω→Rsatisfying the following

properties:

(i) for all t∈[0, T ]and x∈Rd, the mapping Xt(x): Ω →Ris Ft/B(R)-measurable,

(ii) for all ω∈Ω, the mapping (t, x)7→ Xt(x, ω)is in C0,2([0, T ]×Rd,R)and there exist constants

a(ω), c(ω)≥0such that

sup

t∈[0,T ]|Xt(x, ω)| ≤ a(ω)ec(ω)∥x∥Rdfor all x∈Rd,

(iii) Xt(x) = φ(x)+Zt

0h1

2TraceRdσσTHess(Xs)(x)−divµXs(x)ids+Zt

Xs(x)⟨h(x), dZs⟩Rk(7)

-a.s. for all t∈[0, T ]and x∈Rd.

Representation (6) makes it possible to approximate the solution Xt(x, ω) of the Zakai equation

(7) along a realization of the observation process (Zs(ω))s∈[0,t]by averaging over diﬀerent Monte Carlo

simulations of the process RZ(ω),t,x given in (3). We provide numerical results for a Zakai equation

of the form (7) for dimensions d∈ {1,2,5,10,20,25}in Section 2 below. The proof of Theorem 1 is

given in the Appendix.

The idea of transforming a stochastic diﬀerential equation into an ordinary diﬀerential equation

with random coeﬃcients goes back to Doss [16] and Sussmann [37]. An extension to SPDEs was

used by Buckdahn and Ma [4, 5] to introduce a notion of stochastic viscosity solution for SPDEs and

show existence and uniqueness results as well as connections to backward doubly stochastic diﬀerential

equations. The same approach was employed by Buckdahn and Ma [6] and Boufoussi et al. [2] to study

stochastic viscosity solutions of stochastic Hamilton–Jacobi–Bellman (HJB) equations. In this paper,

we analyze the regularity properties of such transformations and use them to develop a Monte Carlo

method for approximating solutions of Zakai equations. The numerical results in Section 2 below

show that it produces accurate results in high dimensions with fast run times. For diﬀerent numerical

approximation methods for Zakai equations, see e.g., [1, 14, 15, 22, 23].

2 Numerical experiments

Together with time-discretization, the trapezoidal rule and Monte Carlo sampling, Theorem 1 can be

used to approximate the solution of a given Zakai equation of the form (1) along a realization of the

observation process Z. We illustrate this in the following example: Choose T, α ∈(0,∞), β∈R,

d∈N, and let σ∈Rd×dbe given by σij =d−1

/2for all i, j ∈ {1, . . . , d}. Consider a d-dimensional

signal process with dynamics

Yt=Y0+Zt

βYs

1 + ∥Ys∥2

ds +σWt, t ∈[0, T ],(8)

for an F0-measurable random initial condition Y0: Ω →Rdwith density

φ(x) = α

2πd

/2exp−α

2∥x∥2

Rd, x ∈Rd,

deﬁned on a ﬁltered probability space (Ω,F,(Ft)t∈[0,T ],) satisfying the usual conditions and a stan-

dard (Ft)t∈[0,T ]-Brownian motion W: [0, T ]×Ω→Rdwith continuous sample paths. Assume the

observation process is of the form

Zt=Zt

γYsds +Vt, t ∈[0, T ],(9)

for a constant γ∈Rand a standard (Ft)-Brownian motion V: [0, T ]×Ω→Rdwith continuous

sample paths independent of W. Let U: [0, T ]×Ω→Rdbe another standard Brownian motion with

continuous sample paths and consider stochastic processes Rz,t,x : [0, t]×Ω→Rd,z∈C([0, T ],Rd),

t∈[0, T ], x∈Rdsatisfying

Rz,t,x

s=x+Zs

0"γσσTz(t−r)−βRz,t,x

1 + ∥Rz,t,x

r∥2

Rd#dr +σUs

for all z∈C([0, T ],Rd), t∈[0, T ], s∈[0, t] and x∈Rd. Let the mappings Bz, uz: [0, T ]×Rd→R,

z∈C([0, T ],Rd) be given by

Bz(t, x) =γ2

2⟨σTz(t) + x, σTz(t)−x⟩Rd−βγ(1 + ∥x∥2

Rd)−1⟨x, z(t)⟩Rd

−dβ(1 + ∥x∥2

Rd)−1+ 2β∥x∥2

Rd(1 + ∥x∥2

Rd)−2

and

uz(t, x) = φRz,t,x

texpt

Bz(t−s, Rz,t,x

s)ds,

z∈C([0, T ],Rd), t∈[0, T ], x∈Rd. By Theorem 1,

Xt(x, ω) = uZ(ω)(t, x) exp⟨γx, Zt(ω)⟩Rd, t ∈[0, T ], x ∈Rd, ω ∈Ω,(10)

is, up to indistinguishability, the unique random ﬁeld X: [0, T ]×Rd×Ω→Rsatisfying (i)–(ii) of

Theorem 1 and solving the Zakai equation

Xt(x) = φ(x) + Zt

Xs(x)⟨γx, dZs⟩Rd

+Zt

0"1

2Pd

i,j=1 ∂2

∂xi∂xjXs(x)−Pd

i=1 ∂

∂xi βxiXs(x)

1 + ∥x∥2

Rd!#ds -a.s.,

(11)

t∈[0, T ], x ∈Rd, corresponding to the dynamics (8)–(9).

We use representation (10) to approximate XT(x) for a given realization (z(t))t∈[0,T ]of (Zt)t∈[0,T ].

For numerical purposes, we generate a discrete realization of the observation process by choosing an

N∈Nand considering the following discretized versions of (8)–(9):

Y0∼ N 0,1

αId,Yn=Yn−1+βYn−1

1 + ∥Yn−1∥2

N+σ(WnT/N −W(n−1)T /N ),

Z0= 0,Zn=Zn−1+γYn−1+Yn

N+ (VnT/N −V(n−1)T /N ), n ∈ {1, . . . , N}.

Let U(i): [0, T ]×Ω→Rd,i∈N, be i.i.d. standard Brownian motions independent of Y0,W,V, and

consider R(x,i)

n: Ω →Rd,n∈ {0,1, . . . , N },x∈Rd,i∈N, given by R(i,x)

0=xand

R(x,i)

n=R(x,i)

n−1+ γσσTZN−n+1 −βR(x,i)

n−1

1 + ∥R(x,i)

n−1∥2

Rd!T

N+σU(i)

nT/N −U(i)

(n−1)T/N (12)

for x∈Rd,i∈Nand n∈ {1,2, . . . , N}. Deﬁne the mappings Bn,XM: Ω×Rd→R,n∈ {0,1, . . . , N},

M∈N, by

Bn(x) =γ2

2⟨σTZn+x, σTZn−x⟩Rd−βγ(1 + ∥x∥2

Rd)−1⟨x, Zn⟩Rd

−dβ(1 + ∥x∥2

Rd)−1+ 2β∥x∥2

Rd(1 + ∥x∥2

Rd)−2, n ∈ {1, . . . , N}, x ∈Rd,

and

XM(x) = 1

i=1

φR(x,i)

N(13)

×exp N

n=1

2NhBN−n(R(x,i)

n) + BN−n+1(R(x,i)

n−1)i+⟨γx, ZN⟩Rd)!, M ∈N, x ∈Rd.

It follows from the law of large numbers that, for M→ ∞,

XM(x)-a.s.

−→ [X1(x)| Z],

which approximates XT(x).

Table 1 below shows point estimates and 95% conﬁdence intervals for [X1(x)| Z] for diﬀerent

realizations of (Y,Z), α= 2π,β=1

/4,γ= 1, T=1

/2,N= 100 and x∈ {YN,2ZN}. For every

d∈ {1,2,5,10,20,25}we simulated ﬁve realizations of (Y,Z) and computed estimates of [X1(x)| Z]

for x∈ {YN,2ZN}by computing realizations of XM(x), x∈ {YN,2ZN}, for M= 4,096,000. Note that

in a typical application, the signal YNis not directly observable, while, in view of (9), 2ZN=ZN/(γT )

is a naive estimate of YNbased on the observation process Z. As expected, with a few exceptions,

the values XM(x) reported in Table 1 are higher for x=YNthan for x= 2ZN. The 95% conﬁdence

intervals were approximated, using the central limit theorem, with

XM(x)−sM(x)

√Mq0.975 ,XM(x) + sM(x)

√Mq0.975,

where q0.975 is the 97.5%-quantile of the standard normal distribution and s2

M(x) the sample variance

of (13) given by

M(x) = 1

M−1

i=1 φR(x,i)

N

×exp N

n=1

2NhBN−nR(x,i)

n+BN−n+1 R(x,i)

n−1i+⟨γx, ZN⟩Rd)!− XM(x)2

The reported runtimes are averages of the ten times needed to compute XM(x), x∈ {YN,2ZN}, for

ﬁve diﬀerent realizations of (Y,Z).

To approximate the whole function x7→ XT(x), our algorithm can be run simultaneously for

diﬀerent x∈Rd. To produce the plots in Figures 1–3, we divided the time interval into N= 20

subintervals and computed XM(x) for a given realization of (Y,Z) and diﬀerent x∈Rdbased on M=

102,400 independent copies of (12), which we generated simultaneously for diﬀerent x∈Rdusing the

same simulated Brownian increments. The ﬁrst plot in Figure 1 shows XM(x) for xon a regular grid

with 1024 grid points in the interval [YN−5,YN+ 5], while the second plot in Figure 1 shows XM(x)

for xon a regular grid with 1282grid points in the square [YN,1−5,YN,1+ 5] ×[YN,2−5,YN,2+ 5].

Figures 2–3 show XM(x, YN,2,...,YN,d) for xon a regular grid with 1024 grid points in the interval

[YN,1−5,YN,1+ 5] for d∈ {5,10,20,25}. The computation times for the results depicted in Figures

1–3 were 1.1s, 28.4s, 7.4s, 14.5s, 28.1s, 34.1s, respectively.

The numerical experiments presented in this section were implemented in Python using TensorFlow

on a NVIDIA GeForce RTX 2080 Ti GPU. The underlying system was an AMD Ryzen 9 3950X CPU

with 64 GB DDR4 memory running Tensorﬂow 2.1 on Ubuntu 19.10. The Python source codes can

be found in the GitHub repository https://github.com/seb-becker/zakai.

3 Conclusion

In this paper we have introduced a Monte Carlo method for approximating solutions of certain Zakai

equations in high dimensions. It is based on a Doss–Sussmann-type transformation which transforms

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AnefficientMonteCarloschemeforZakaiequationsChristianBeck1,3,SebastianBecker1,PatrickCheridito1,ArnulfJentzen2,3andArielNeufeld41DepartmentofMathematics,ETHZurich,Switzerland2SchoolofDataScienceandShenzhenResearchInstituteofBigData,TheChineseUniversityofHongKong,Shenzhen,China3AppliedMathematics:Ins...

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An efficient Monte Carlo scheme for Zakai equations Christian Beck13 Sebastian Becker1 Patrick Cheridito1 Arnulf Jentzen23and Ariel Neufeld4

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