We obtain weak rates for approximation of an integral functional of a Markov process by integral sums. An assumption on the process is formulated only in terms of its transition probability density, and, therefore, our approach is not strongly dependent on the structure of the process. Applications to the estimates of the rates of approximation of the Feynman–Kac semigroup and of the price of “occupation-time options” are provided.
Markov processesintegral functionalweak approximation ratesFeynman-Kac formulaoccupation-time option60J5560F17Introduction and main results
Let Xt, t≥0, be a Markov process with values in Rd. We consider the following objects:
1) the integral functional
IT(h)=∫0Th(Xt)dt
of this process;
2) the sequence of integral sums
IT,n(h)=Tn∑k=0n−1h(X(kT)/n),n≥1.
The problem we are focused on is obtaining upper bounds on the accuracy of approximation of the integral functional IT(h) by the integral sums IT,n(h) without any regularity assumption on the function h. The function h is only assumed to be measurable and bounded. Therefore, the class of functionals IT(h) contains, for example, the occupation time of the process X in a set A⊂B(Rd) (in this case, h=IA).
The problem of estimating the expectation of expressions that contain both the value of a process and the value of an integral functional of this process arises naturally in a wide range of probabilistic problems. Two of them related with the Feynman–Kac semigroup and the price of an occupation-time option are discussed in Section 3. An exact calculation of such expressions, if possible, can be performed only under substantial assumptions on the structure of functionals and processes; see, for example, [3], where the price of an occupation-time option is precisely calculated for a Lévy process with only negative jumps. For more complicated models, it is natural to use approximative methods, which naturally require estimates of approximation errors. This motivates the main aim of the paper to evaluate the error bounds for discrete approximations of the integral functional IT(h).
In what follows, Px denotes the law of the Markov process X conditioned by X0=x, and Ex denotes the expectation with respect to this law. Both the absolute value of a real number and the Euclidean norm in Rd are denoted by |·|; ‖·‖ denotes the sup-norm in L∞.
The following result was obtained in [2] as a part of the proof of a more general statement (see Theorem 2.5 in [2]).
Suppose that X is a multidimensional diffusion process with bounded Hölder continuous coefficients and that its diffusion coefficient satisfies the uniform ellipticity condition(a(x)θ,θ)Rd≥c|θ|2,x,θ∈Rd,c>0.0.\]]]>
Then there exists a positive constant C such that|ExIT(h)−ExIT,n(h)|≤C‖h‖lognn.
The scheme of the proof of this result can be extended straightforwardly to the case of arbitrary Markov process that satisfies the following assumption (see Proposition 2.1 [1]):
The process X possesses a transition probability density pt(x,y) that is differentiable with respect to t and satisfies
|∂tpt(x,y)|≤CTt−1qt,x(y),t≤T,CT≥1,
for some measurable function q such that for any fixed t and x, the function qt,x is a distribution density.
A diffusion process satisfies condition X with
qt,x(y)=c1t−d/2exp(−c2t−1|x−y|2)
and properly chosen c1,c2. The other examples of the processes satisfying condition (2) are provided in [1]. Among them, we should mention an α-stable process.
Under assumption X, Proposition 1 and Proposition 2.1 in [1] give bounds for the rate of approximation of expectations of the integral functionals of the process X. Such approximation rates are called weak. StrongLp-rates, that is, the bounds for
Ex|IT(h)−IT,n(h)|p,
have been recently obtained in [4] for diffusion processes and in [1] without restrictions on the structure of the processes. In this paper, we provide another generalization of the weak rate (1), namely, the rates of approximation for expectations of more complicated functionals. Let us formulate the main result of this paper.
Suppose thatXholds. Then for eachk∈Nand any bounded function f,|Ex(IT(h))kf(XT)−Ex(IT,n(h))kf(XT)|≤6k2CTTk‖h‖k(lognn)‖f‖.
Clearly, Proposition 2.1 in [1] is a particular case of Theorem 1. The latter statement is a substantial extension of the former one: it contains both the moments of any order of the integral functional and the value of the process in the final time moment. Using the Taylor expansion, we obtain the following corollary of Theorem 1.
Consider any analytic function g defined in a neighborhood of 0 and constants Dg,Rg>00$]]> such that |g(m)(0)m!|≤Dg(1Rg)m for any natural m.
Suppose thatXholds. Then for any bounded function f and a function h such thatT‖h‖<Rg, we have:|Exg(IT(h))f(XT)−Exg(IT,n(h))f(XT)|≤CT,h,Dg,Rg(lognn)‖f‖,whereCT,h,Dg,Rg=6DgCTT‖h‖Rg(1+T‖h‖Rg)1(1−T‖h‖Rg)3.
We provide the proof of Theorem 1 in Section 2. In Section 3, we give an application of Theorem 1 to estimates of the rates of approximation of the Feynman–Kac semigroup and of the price of an occupation-time option.
Proof of Theorem <xref rid="j_vmsta37cnf_stat_004">1</xref>
Denote
Sk,a,b:={(s1,s2,…,sk)∈Rk|a≤s1<s2<⋯<sk≤b},k∈N,a,b∈R,
and for each t∈[kT/n,(k+1)T/n), put ηn(t):=kTn.
We have:
1k!(Ex[(IT(h))k−(IT,n(h))k]f(XT))=Ex∫Sk,0,T[∏i=1kh(Xsi)−∏i=1kh(Xηn(si))]f(XT)∏i=1kdsi=∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)(∏i=1kpsi−si−1(yi−1,yi))pT−sk(yk,z)×dz∏j=1kdyj∏i=1kdsi−∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))pT−ηn(sk)(yk,z)dz∏j=1kdyj∏i=1kdsi,
where s0=0, y0=x.
Rewrite the expression under the integral
(∏i=1kpsi−si−1(yi−1,yi))pT−sk(yk,z)−(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))pT−ηn(sk)(yk,z)
in the form
(∏i=1kpsi−si−1(yi−1,yi))pT−sk(yk,z)∓pηn(s1)(x,y1)(∏i=2kpsi−si−1(yi−1,yi))pT−sk(yk,z)−(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))pT−ηn(sk)(yk,z)=(∏i=1kpsi−si−1(yi−1,yi))pT−sk(yk,z)∓pηn(s1)(x,y1)(∏i=2kpsi−si−1(yi−1,yi))pT−sk(yk,z)∓pηn(s1)(x,y1)ps2−ηn(s1)(y1,y2)(∏i=3kpsi−si−1(yi−1,yi))pT−sk(yk,z)−(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))pT−ηn(sk)(yk,z)⋮=J1+J2+⋯+J2k−1+J2k,
where
J1=(ps1(x,y1)−pηn(s1)(x,y1))(∏i=2kpsi−si−1(yi−1,yi))pT−sk(yk,z),J2=pηn(s1)(x,y1)(ps2−s1(y1,y2)−ps2−ηn(s1)(y1,y2))×(∏i=3kpsi−si−1(yi−1,yi))pT−sk(yk,z),J3=pηn(s1)(x,y1)(ps2−ηn(s1)(y1,y2)−pηn(s2)−ηn(s1)(y1,y2))×(∏i=3kpsi−si−1(yi−1,yi))pT−sk(yk,z),J4=pηn(s1)(x,y1)pηn(s2)−ηn(s1)(y1,y2)(ps3−s2(y2,y3)−ps3−ηn(s2)(y2,y3))×(∏i=4kpsi−si−1(yi−1,yi))pT−sk(yk,z),J5=pηn(s1)(x,y1)pηn(s2)−ηn(s1)(y1,y2)(ps3−ηn(s2)(y2,y3)−pηn(s3)−ηn(s2)(y2,y3))×(∏i=4kpsi−si−1(yi−1,yi))pT−sk(yk,z),⋮J2k−1=(∏i=1k−1pηn(si)−ηn(si−1)(yi−1,yi))×(psk−ηn(sk−1)(yk−1,yk)−pηn(sk)−ηn(sk−1)(yk−1,yk))pT−sk(yk,z),J2k=(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))(pT−sk(yk,z)−pT−ηn(sk)(yk,z)).
Our way to estimate each of 2k terms in (4) is mostly the same, but its realization is different for the first, the last, and the intermediate terms. Let us estimate the first term in (4):
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=1kdsi|=|∫0T∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|≤|∫0T/n∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|+|∫T/nT∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|
Let us consider each term in detail:
|∫0T/n∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|≤∫0T/n∫Sk−1,0,T∫(Rd)k+1|(∏i=1kh(yi))f(z)J1|dz∏j=1kdyj∏i=2kdsids1≤‖h‖k‖f‖∫0T/n∫Sk−1,0,T∫(Rd)k+1|J1|dz∏j=1kdyj∏i=2kdsids1≤1(k−1)!‖h‖k‖f‖Tk−1∫0T/n∫Rd|ps1(x,y1)−pηn(s1)(x,y1)|dy1ds1≤2(k−1)!‖h‖k‖f‖Tk1n.
Next, we have
|∫T/nT∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|=|∫T/nT∫ηn(s1)s1∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)∂upu(x,y1)×(∏i=2kpsi−si−1(yi−1,yi))pT−sk(yk,z)dz∏j=1kdyj∏i=2kdsiduds1|≤∫T/nT∫ηn(s1)s1∫Sk−1,0,T∫(Rd)k+1|(∏i=1kh(yi))f(z)∂upu(x,y1)×(∏i=2kpsi−si−1(yi−1,yi))pT−sk(yk,z)|dz∏j=1kdyj∏i=2kdsi×duds1≤‖h‖k‖f‖∫T/nT∫ηn(s1)s1∫Sk−1,0,T∫(Rd)k+1|∂upu(x,y1)|(∏i=2kpsi−si−1(yi−1,yi))×pT−sk(yk,z)dz∏j=1kdyj∏i=2kdsiduds1.
Integrating over z,yk,yk−1,…,y2 and then over sk,sk−1,…,s2, we derive
|∫T/nT∫Sk−1,s1,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=2kdsids1|≤1(k−1)!‖h‖k‖f‖Tk−1∫T/nT∫ηn(s1)s1∫Rd|∂upu(x,y1)|dy1duds1≤CT1(k−1)!‖h‖k‖f‖Tk−1∫T/nT∫ηn(s1)s1∫Rdu−1qu,x(y1)dy1duds1=CT1(k−1)!‖h‖k‖f‖Tk−1∫T/nT∫ηn(s1)s1u−1duds1=CT1(k−1)!‖h‖k‖f‖Tk−1∑i=1n−1∫iT/n(i+1)T/n∫iT/ns1u−1duds1=CT1(k−1)!‖h‖k‖f‖Tk−1∑i=1n−1∫iT/n(i+1)T/n∫u(i+1)T/nu−1ds1du≤CT1(k−1)!‖h‖k‖f‖Tk1n∑i=1n−1∫iT/n(i+1)T/nu−1du=CT1(k−1)!‖h‖k‖f‖Tk1n∫T/nTu−1du=CT1(k−1)!‖h‖k‖f‖Tklognn.
Therefore,
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)J1dz∏j=1kdyj∏i=1kdsi|≤3CT1(k−1)!‖h‖k‖f‖Tklognn.
Now we are ready to estimate the last summand in (4):
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1kdsi|=|∫0T∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|≤|∫0T−T/n∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|+|∫T−T/nT∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|
Let us estimate each term separately. We get
|∫T−T/nT∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|≤∫T−T/nT∫Sk−1,0,T∫(Rd)k+1|(∏i=1kh(yi))f(z)J2k|dz∏j=1kdyj∏i=1k−1dsidsk≤‖h‖k‖f‖∫T−T/nT∫Sk−1,0,T∫(Rd)k+1|J2k|dz∏j=1kdyj∏i=1k−1dsidsk≤2(k−1)!‖h‖k‖f‖Tk1n.
For the other term, we obtain:
|∫0T−T/n∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|=|∫0T−T/n∫ηn(sk)sk∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))∂T−upT−u(yk,z)dz∏j=1kdyj∏i=1k−1dsidudsk|≤∫0T−T/n∫ηn(sk)sk∫Sk−1,0,T∫(Rd)k+1|(∏i=1kh(yi))f(z)×(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))∂T−upT−u(yk,z)|dz∏j=1kdyj∏i=1k−1dsidudsk≤‖h‖k‖f‖∫0T−T/n∫ηn(sk)sk∫Sk−1,0,T∫(Rd)k+1(∏i=1kpηn(si)−ηn(si−1)(yi−1,yi))×|∂T−upT−u(yk,z)|dz∏j=1kdyj∏i=1k−1dsidudsk.
Let us rewrite this expression in the form
‖h‖k‖f‖∫Sk−1,0,T∫(Rd)k−1(∏i=1k−1pηn(si)−ηn(si−1)(yi−1,yi))×∫0T−T/n∫ηn(sk)sk∫(Rd)2pηn(sk)−ηn(sk−1)(yk−1,yk)×|∂T−upT−u(yk,z)|dzdykdudsk∏j=1k−1dyj∏i=1k−1dsi
and consider the inner integral
∫0T−T/n∫ηn(sk)sk∫(Rd)2pηn(sk)−ηn(sk−1)(yk−1,yk)|∂T−upT−u(yk,z)|dzdykdudsk=∑i=0n−2∫iT/n(i+1)T/n∫iT/nsk∫(Rd)2piT/n−ηn(sk−1)(yk−1,yk)×|∂T−upT−u(yk,z)|dzdykdudsk=∑i=0n−2∫iT/n(i+1)T/n∫u(i+1)T/n∫(Rd)2piT/n−ηn(sk−1)(yk−1,yk)×|∂T−upT−u(yk,z)|dzdykdskdu≤Tn∑i=0n−2∫iT/n(i+1)T/n∫(Rd)2piT/n−ηn(sk−1)(yk−1,yk)|∂T−upT−u(yk,z)|dzdykdu≤CTTn∑i=0n−2∫iT/n(i+1)T/n∫RdpiT/n−ηn(sk−1)(yk−1,yk)(T−u)−1dykdu=CTTn∑i=0n−2∫iT/n(i+1)T/n(T−u)−1du=CTTn∫0T−T/n(T−u)−1du=TCTlognn.
Therefore, we have:
|∫0T−T/n∫Sk−1,0,sk∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1k−1dsidsk|≤CT1(k−1)!‖h‖k‖f‖Tklognn
and
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)J2kdz∏j=1kdyj∏i=1kdsi|≤3CT1(k−1)!‖h‖k‖f‖Tklognn.
To complete the proof, we should additionally consider the following terms in (4):
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))×(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)dz∏q=1kdyq∏r=1kdsr|
and
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))×(psj−ηn(sj−1)(yj−1,yj)−pηn(sj)−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)dz∏q=1kdyq∏r=1kdsr|,
where j=2,k‾.
Consider (5) in more detail. We rewrite it in the form
|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−2sj∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2|≤|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−2sj−T/n∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2|+|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−T/nsj∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2|.
We estimate each term separately:
|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−T/nsj∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2|≤‖h‖k‖f‖×∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−T/nsj∫Sk−j,sj,T∫(Rd)j(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))×|psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj)|×∏q=1jdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2≤2(k−1)!‖h‖k‖f‖Tk1n.
For the other term, we have
|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−2sj−T/n∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdsj−1dsj∏v=1j−3dsvdsj−2|=|∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−2sj−T/n∫ηn(sj−1)sj−1∫Sk−j,sj,T∫(Rd)k+1(∏i=1kh(yi))f(z)×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))∂sj−upsj−u(yj−1,yj)×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)×dz∏q=1kdyq∏r=j+1kdsrdudsj−1dsj∏v=1j−3dsvdsj−2|≤‖h‖k‖f‖×∫0T∫Sj−3,0,sj−2∫sj−2T∫sj−2sj−T/n∫ηn(sj−1)sj−1∫Sk−j,sj,T∫(Rd)j|∂sj−upsj−u(yj−1,yj)|×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))∏q=1jdyq∏r=j+1kdsrdudsj−1dsj∏v=1j−3dsvdsj−2≤‖h‖k‖f‖×∫0T∫Sj−3,0,sj−2∫sj−2T∫Sk−j,sj,T∫0sj−T/n∫ηn(sj−1)sj−1∫(Rd)j|∂sj−upsj−u(yj−1,yj)|×(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))∏q=1jdyqdudsj−1∏r=j+1kdsrdsj∏v=1j−3dsrdsj−2.
Again, we consider the inner integral:
∫0ηn(sj)−T/n∫ηn(sj−1)sj−1∫(Rd)2pηn(sj−1)−ηn(sj−2)(yj−2,yj−1)×|∂sj−upsj−u(yj−1,yj)|dyjdyj−1dudsj−1=∑i=0ηn(sj)n/T−2∫iT/n(i+1)T/n∫iT/nsj−1∫(Rd)2piT/n−ηn(sj−2)(yj−2,yj−1)×|∂sj−upsj−u(yj−1,yj)|dyjdyj−1dudsj−1=∑i=0ηn(sj)n/T−2∫iT/n(i+1)T/n∫u(i+1)T/n∫(Rd)2piT/n−ηn(sj−2)(yj−2,yj−1)×|∂sj−upsj−u(yj−1,yj)|dyjdyj−1dsj−1du≤Tn∑i=0ηn(sj)n/T−2∫iT/n(i+1)T/n∫(Rd)2piT/n−ηn(sj−2)(yj−2,yj−1)×|∂sj−upsj−u(yj−1,yj)|dyjdyj−1du≤CTTn∑i=0ηn(sj)n/T−2∫iT/n(i+1)T/n∫RdpiT/n−ηn(sj−2)(yj−2,yj−1)(sj−u)−1dyj−1du=CTTn∑i=0ηn(sj)n/T−2∫iT/n(i+1)T/n(sj−u)−1du=CTTn∫0ηn(sj)−T/n(sj−u)−1du.
We have
∫0sj−T/n∫ηn(sj−1)sj−1∫(Rd)2pηn(sj−1)−ηn(sj−2)(yj−2,yj−1)×|∂sj−upsj−u(yj−1,yj)|dyjdyj−1dudsj−1≤CTTn∫0sj−T/n(sj−u)−1du≤TCTlognn.
Therefore, we obtain
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))×(psj−sj−1(yj−1,yj)−psj−ηn(sj−1)(yj−1,yj))(∏m=j+1kpsm−sm−1(ym−1,ym))×pT−sk(yk,z)dz∏q=1kdyq∏r=1kdsr|≤3CT1(k−1)!‖h‖k‖f‖Tklognn.
Analogously, we also have:
|∫Sk,0,T∫(Rd)k+1(∏i=1kh(yi))f(z)(∏l=1j−1pηn(sl)−ηn(sl−1)(yl−1,yl))×(psj−ηn(sj−1)(yj−1,yj)−pηn(sj)−ηn(sj−1)(yj−1,yj))×(∏m=j+1kpsm−sm−1(ym−1,ym))pT−sk(yk,z)dz∏q=1kdyq∏r=1kdsr|≤3CT1(k−1)!‖h‖k‖f‖Tklognn.
Therefore, we finally obtain
|Ex[(IT(h))k−(IT,n(h))k]f(XT)|≤6k2CTTk‖h‖k(lognn)‖f‖,
which completes the proof. □
ApplicationsDiscrete approximation of the Feynman–Kac semigroup
Let X be a Brownian motion with values in Rd. Then condition X holds with
qt,x(y)=c1t−d/2exp(−c2t−1|x−y|2),
where c1,c2 are some positive constants.
Let h be a bounded measurable function. Then, it is known (see, e.g., [6], Chapter 1) that the family of operators
Rthf(x)=Ex[f(Xt)exp{λIt(h)}]
forms a semigroup on Lp(Rd),p≥1, and its generator equals
Ahf=12Δf+λhf.
This semigroup is called the Feynman–Kac semigroup.
Denote
Rt,nhf(x)=Ex[f(Xt)exp{λIt,n(h)}].
Then, using the Taylor expansion of the exponential function and Theorem 1, we have the following statement.
For any bounded functionsf,hand real positive number λ, we have:|Rthf(x)−Rt,nhf(x)|≤CT,λ,h(lognn)‖f‖,whereCT,λ,h=6CTλ‖h‖T(1+λ‖h‖T)exp{λ‖h‖T}.
Therefore, the main result of this paper provides an approximation of the Feynman–Kac semigroup with accuracy (logn)/n.
Approximation of the price of an occupation-time option
Let the price of an asset S={St,t≥0} be of the form
St=S0exp(Xt),
where X is a one-dimensional Markov process satisfying condition X. The time spent by S in a defined set J⊂R (or the time spent by X in a set J′={x:ex∈J}) from time 0 to time T is given by
∫0TI{St∈J}dt=∫0TI{Xt∈J′}dt.
We consider an occupation-time option (see [5]) whose price depends on the time spent by the process S in a set J. In contrast to the traditional barrier options, which are activated or canceled when the process S hits a defined level (barrier), the payoff of an occupation-time option depends on the time spent by the price of the asset above/below this level.
For the strike price K, the barrier L, and the knock-out rate ρ, the payoff of a down-and-out call occupation-time option is given by
exp(−ρ∫0TI{St≤L}dt)(ST−K)+.
Then, for the risk-free interest rate r, its price is given by
C(T)=exp(−rT)E[exp(−ρ∫0TI{St≤L}dt)(ST−K)+].
Denote
Cn(T)=exp(−rT)E[exp(−ρT/n∑k=0n−1I{SkT/n≤L}dt)(ST−K)+].
We provide the following corollary of Theorem 1.
Suppose thatXholds and there existsu>11$]]>such thatG:=Eexp(uXT)=ESTu<+∞. Then|Cn(T)−C(T)|≤3max{CT,ρ,G}exp(−rT)(lognn1−1/u),whereCT,ρ=6CTρT(1+ρT)exp(ρT).
For some N>00$]]>, we denote
CN(T)=exp(−rT)E[exp(−ρ∫0TI{St≤L}dt)((ST−K)+∧N)],CnN(T)=exp(−rT)E[exp(−ρT/n∑k=0n−1I{SkT/n≤L}dt)((ST−K)+∧N)].
Then
|Cn(T)−C(T)|≤|CnN(T)−CN(T)|+|C(T)−CN(T)|+|Cn(T)−CnN(T)|.
We estimate each term separately. According to Corollary 2,
|CnN(T)−CN(T)|≤NCT,ρexp(−rT)(lognn).
For other terms, we have:
|C(T)−CN(T)|+|Cn(T)−CnN(T)|≤2exp(−rT)E[(ST−K)+−(ST−K)+∧N]≤2exp(−rT)E[STI{ST>N}]=2exp(−rT)E[STNu−1I{ST>N}Nu−1]≤2GNu−1exp(−rT).N\}}]\\{} & \displaystyle \hspace{1em}=2\exp (-rT)E\bigg[\frac{S_{T}{N}^{u-1}\mathbb{I}_{\{S_{T}>N\}}}{{N}^{u-1}}\bigg]\le \frac{2G}{{N}^{u-1}}\exp (-rT).\end{array}\]]]>
Now, putting N=n1/u completes the proof. □
Therefore, the main result of this paper provides the approximate value Cn(T) of the price of an occupation-time option C(T) with accuracy of order (logn)/n1−1/u for the class of processes X satisfying X and the condition Eexp(uXT)<+∞ for some u>11$]]>.
Acknowledgments
The authors are deeply grateful to Arturo Kohatsu-Higa for discussion and valuable suggestions about the possible area of applications of the main result of the paper.
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