Efficient Simulations for the Exponential Integrals of Hölder Continuous Gaussian Random Fields

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1 Efficient Simulations for the Exponential Integrals of Hölder Continuous Gaussian Random Fields 0 Jingchen Liu and Gongjun Xu, Columbia University and University of Minnesota In this paper, we consider a Gaussian random field ft living on a compact set R d and the computation of the tail probabilities P eft dt > e b as b. We design asymptotically efficient importance sampling estimators for a general class of Hölder continuous Gaussian random fields. In addition to the variance control, we also analyze the bias relative to the interesting tail probabilities caused by the discretization.. INRODUCION In this paper, we focus on the design and the analysis of efficient Monte Carlo methods for computing the tail probabilities of integrals of exponential functions of Gaussian random fields living on a compact domain. Suppose that {ft : t } is a continuous Gaussian random field with zero mean and unit variance. hat is, for each finite subset {t,..., t n }, ft,..., ft n is a multivariate Gaussian random vector with Eft i = 0 and Varft i =, i =,, n. he domain is a d-dimensional compact subset of R d. Further conditions will be imposed on f in the later discussions. Define I = e σtft+µt dt, where µ and σ are two deterministic functions and σ is strictly positive. Our focus is on the tail probabilities.. Motivations wb = P I > e b = P e σtft+µt dt > e b as b. 2 he exponential integral of a Gaussian random field is a central object of several probability models. he integral is the limiting object of the weighted sum of correlated lognormal random variables. he tail event of the latter sum is an important topic. An incomplete list of works is given by [Ahsan 978; Duffie and Pan 997; Glasserman et al. 2000; Basak and Shapiro 200; Deutsch 2004; Foss and Richards 200]. In the portfolio risk analysis, consider a portfolio consisting of n assets S,..., S n each of which is associated with a weight e.g. his research is supported in part by NSF CMMI and NSF SES ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

2 0:2 Liu and Xu number of shares w,..., w n. One popular model assumes that log S,..., log S n is a multivariate Gaussian random vector. he value of the portfolio, S = n i= w is i, is then the sum of correlated log-normal random variables. Without loss of generality, we let w i = n. One interesting situation is that the portfolio size is large and the asset prices are usually highly correlated. One may employ a latent space approach used in the literature of social network. More specifically, we construct a Gaussian process {ft : t } and associate each asset i with a latent variable t i so that log S i = ft i. hen, the log asset prices fall into a subset of the continuous Gaussian process. Furthermore, there exists a deterministic process wt so that wt i = w i. hen, the unit share value of the portfolio is n wi S i = n wti e fti. For detailed discussion of latent state modeling, see [Hoff et al. 2002; Snijders 2002; Handcock et al. 2007; Xing et al. 200]. In the asymptotic regime that n and the correlations among the asset prices become close to one, the subset {t i : i =,..., n} becomes dense in. Ultimately, we obtain the limit n n w i S i i= wte ft htdt where ht indicates the limiting spatial distribution of {t i : i =,..., n} in. Let µt = log wt+log ht. hen the tail probability of the limiting unit share price is P e ft+µt dt > e b. Another application of the current study lies in option pricing. A stylized model for asset price St as a function of time t is geometric Brownian motion, that is, St = e W t, where W t is a Brownian motion; see [Black and Scholes 973; Merton 973]. hen the payoff of an Asian option is at expiration time is a function of the averaged price 0 ew t dt. For instance, the payoff of an Asian call option with strike price K is max 0 ew t dt K, 0; the expected payoff of a digital Asian call option is precisely P 0 ew t dt > K..2. Literature and contribution here is a rich rare-event simulation literature for the sum of independent random variables. he problems are mostly classified to light-tailed and heavy-tailed problems. An incomplete list of recent works includes [Asmussen and Kroese 2006; Dupuis et al. 2007; Blanchet and Glynn 2008; Blanchet and Liu 2008; Blanchet et al. 2008]. For the dependent case, [Snijders 20] proposes several efficient Monte Carlo estimators for the sum of finitely many correlated lognormal random variables. he current problem is a substantial generalization of this work. We employ a different change of measure that is a mixture of infinitely many exponential ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

3 Efficient Simulations for Gaussian Random Fields 0:3 change of measures suggesting that the rare-event is mostly caused by the abnormal behavior of the random field at one location. Another related literature is the extreme behaviors of Gaussian random fields. he results range from general bounds to sharp asymptotic approximations. An incomplete list of works includes [Landau and Shepp 970; Marcus and Shepp 970; Sudakov and sirelson 974; Borell 975; sirelson et al. 976; Berman 985; Hüsler 990; Ledoux and alagrand 99; alagrand 996]. A few lines of investigations on the supremum norm are given as follows. Assuming locally stationary structure, the double-sum method [Piterbarg 996] provides the exact asymptotic approximation of sup ft over a compact set, which is allowed to grow as the threshold tends to infinity. For almost surely at least twice differentiable fields, [Adler 98; aylor et al. 2005; Adler and aylor 2007] derive the analytic form of the expected Euler-Poicaré Characteristics of the excursion set which serves as a good approximation of the tail probability of the supremum. he tube method [Sun 993] takes advantage of the Karhune-Loève expansion and Weyl s formula. he Rice method [Azais and Wschebor 2005; 2008; 2009] provides an implicit description of sup ft. he discussions also go beyond the Gaussian fields such as Gaussian random fields with random variances [Hüsler et al. 20]. See also [Adler et al. 2009] for non-gaussian and heavy-tailed processes. he corresponding rare-event simulations have been studied in [Adler et al. 202]. he asymptotic tail behaviors of I have been studied recently in the literature with focus mostly on the analytic approximations of wb under smoothness conditions. [Liu 202] derives the asymptotic approximations of P eσft dt > e b as b when ft is a threetime differentiable and homogeneous Gaussian random field and σt takes constant value σ. [Liu and Xu 202c] further extends the results to the case when the process has a smooth varying mean function, i.e., P eσft+µt dt > e b. he tail asymptotics of I are difficult to develop when ft is non-differentiable. Under such a setting, accurate approximations of wb have not yet been developed except for some special cases such as ft is a Brownian motion c.f. [Yor 992; Dufresne 200]. herefore, rare-event simulation serves as an appealing alternative from the computational point of view in that the design and the analysis do not require very sharp approximations of wb. his paper, to the authors best knowledge, is the first analysis of I for a general class of non-differentiable and differentiable fields. he main contribution of this paper is to develop a provably efficient rare-event simulation algorithm to compute wb. he efficiency of the algorithm only requires that ft is uniformly Hölder continuous on the compact domain. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

4 0:4 Liu and Xu herefore, the results are applicable to essentially all the Gaussian processes practically in use. Central to the analysis is a change of measure Q in both a continuous form for the theoretical analysis and a discrete form for the simulation. he measure Q mimics the conditional distribution of f given the occurrence of the rare event {I > e b }. Importance sampling estimators are then constructed based on the measure Q. An appealing feature of the proposed method is that the specific simulation schemes do not vary substantially under different situations. hat is, the choice of the change of measure does not rely much on the specific mean, variance, or covariance structure of the process such as, constant mean/variance, varying mean/variance, multi- and uni-modal mean/variance, etc.. hus, this change of measure captures the common characteristics of the conditional distributions among all the processes in the class. With a fine tuning of the change of measure, it is conceivable that the efficiency can be further improved by taking into account more refined structures of the process such as homogeneity, smoothness, maxima of the mean and the variance function, etc. Nonetheless, a unified efficient simulation scheme is useful especially when the fine structures of the processes are not easy to obtain. We do not pursue this further improvement in this paper. he complexity analysis of the proposed estimators consists of two elements. Firstly, since f considered in this paper is continuous, exact simulation of the entire field is usually impossible. hus, we need to use a discrete object to approximate the continuous field and the bias caused by the discretization needs to be well controlled relative to wb. he second part of our analysis is the variance control, that is, to provide a bound of the second moments of the discrete importance sampling estimators. he rest of the paper is organized as follows. Section 2 provides the construction of our importance sampling algorithm and presents the main results of this paper. Section 3 includes simulation studies. Proofs of our main theorems are given in Section 4. he proofs of supporting lemmas are included in Section MAIN RESULS 2.. Preliminaries of rare-event simulations and importance sampling hroughout this paper, we are interested in computing wb 0 as b. In the context of rare-event simulations [Asmussen and Glynn 2007, Chapter VI], it is more meaningful to consider the computational error relative to wb. A well accepted efficiency concept is the so-called weak efficiency, also known as asymptotic efficiency and logarithmic efficiency [Asmussen and Glynn 2007]. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

5 Efficient Simulations for Gaussian Random Fields 0:5 Definition 2.. A Monte Carlo estimator L b is said to be weakly / asymptotically / logarithmically efficient in estimating wb if EL b = wb and lim b log EL 2 b =. 3 2 log wb Suppose that a weakly efficient estimator L b has been obtained. Let {L j n j= Lj b b : j =,..., n} be i.i.d. copies of L b and Z b = n be the averaged estimator that has a relative mean squared error Var /2 L b /n /2 wb. A simple consequence of Chebyshev s inequality yields P Z b /wb η VarL b nη 2 w 2 b. 4 he limit 3 suggests that for any λ > 0, VarL b EL 2 b = ow2 λ b. For any positive η and δ, in order to achieve the following relative accuracy P Z b /wb > η < δ, 5 it is sufficient to generate n = Oη 2 δ w λ b i.i.d. copies of L b for any λ > 0. o construct efficient estimators, in this paper, we use importance sampling for the variance reduction see [Asmussen and Glynn 2007, Chapter V]. It is based on the following identity wb = E I >eb = E Q dp I >eb, dq where Q is a probability measure on F such that dp/dq is well defined on the set {It > e b }. We use E and E Q to denote the expectations under the measures P and Q respectively. hen, the random variable dp L b = I >e b dq is an unbiased estimator of wb under the measure Q. If we choose Q = P I > e b to be the conditional distribution, then the corresponding likelihood ratio dp/dq wb on the set {I > e b } has zero variance under Q. hus, Q is also called the zero-variance change of measure. Note that this change of measure is of no practical value in that its implementation requires the computation of the probabilities wb. Nonetheless, the measure Q provides a guideline for the construction of an efficient change of measure to compute the probability wb. herefore, the task lies in constructing a change 6 ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

6 0:6 Liu and Xu of measure Q that is a good approximation of Q. In addition, from the computational point of view, we should also be able to numerically compute L b and to simulate f from Q. Besides the variance control, another important issue is the bias control. he random fields considered in this paper are continuous processes. Direct simulation is usually not feasible. herefore, we need to set up an appropriate discretization scheme to approximate the continuous objects. he bias caused by the discretization also needs to be controlled relative to wb. Suppose that a biased estimator L b has been constructed for wb such that E L b = wb. hus, the computation error can be decomposed as follows L b wb L b wb wb + wb wb. wb he first term on the right-hand-side is controlled by the relative variance of L b and the second term is the bias relative to wb. Both the bias and the variance control will be carefully analyzed in the following sections relative to wb. he overall computational complexity is then the necessary number of i.i.d. replicates for L b multiplied by the computational cost to generate one L b he change of measure and rare-event simulation We propose a change of measure Q on the continuous sample path space and further propose a discrete version, denoted by Q M, for the simulation purpose, where the subscript M indicates the size of the discretization. he description of Q requires the following quantities. For each b, we define u satisfying the following identity where 2π σ d 2 u d 2 e uσ = e b, 7 σ = sup σt. t he left-hand-side of 7 is in a similar form as the Lambert W -function. Note that when b is large, equation 7 generally has two solutions. One is on the order of b/σ ; the other one is close to zero. See Figure 2.2 for an example of the left-hand-side of 7 with d = and σ =. We choose u to be the larger solution. Remark 2.2. In the asymptotic regime that b tends to infinity, we need to have sup σtft + µt approximately exceeding level σ u so that the integral is above the ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

7 Efficient Simulations for Gaussian Random Fields 0:7 level b. We provide an intuitive calculation of 7. Conditional on that fτ = u for some τ, the conditional process is ft = Eft fτ = u + gt τ where g is a zeromean Gaussian process. It turns out that the variation of g is much smaller than that of the conditional mean. hen, we approximate the process by ft Eft fτ = u. For the special case that ft is stationary and differentiable admitting zero mean and a covariance function Covfs, ft + s = t 2 /2 + o t 2 then, σ =, one can compute eeft fτ=u dt = 2π d 2 u d 2 e u. For the precise calculations, see Section 3.4 in [Liu and Xu 202c]. In fact the pre-factor 2π d 2 u d 2 is not essential in the technical development. Weak efficiency holds if b σ u u 0.5 e u u Fig.. Plot of 2π 2 u 2 e u Furthermore, we define µ σ t = µt/σ and u t = u µ σ t. 8 We characterize the measure Q in two ways. First, we describe the simulation of the process f from Q following a three-step procedure. Simulate a random index τ uniformly over with respect to the Lebesgue measure. 2 Given the realized τ, simulate fτ Nu τ,, where u τ is defined as in 8. 3 Simulate the rest of the field {ft : t τ} from the original conditional distribution under P given τ, fτ. he above simulation description induces a measure Q. o derive the Radon-Nikodym derivative between Q and P, we need to realize that sampling f from the measure P may ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

8 0:8 Liu and Xu also follow a similar three-step procedure except that in Step 2 we sample fτ from the standard normal distribution. hus, the random variable τ and f are independent under P. We let F be the σ-field generated by f and F be that generated by f and τ. It is straightforward to verify that the Radon-Nikodym derivative dq/dp on F is Zf, τ = exp { 2 fτ u τ 2} exp { 2 fτ2}. hat is, for each A F, QA = E Zf, τ; A. hen, for each A C [ { exp 2 Qf A = E fτ u τ 2} ] { [ { exp exp { 2 fτ2} ; f A 2 = E E fτ u τ 2} ] } exp { f 2 fτ2} ; f A. 9 Note that τ and f are independent under P and τ is uniform on, thus the conditional expectation given f is written as [ { exp 2 Zf = E fτ u τ 2} ] exp { f 2 fτ2} = mes exp { 2 ft u t 2} exp { 2 ft2} dt, where mes denotes the Lebesgue measure. We insert the above form into 9 and obtain that Qf A = EZf; f A. herefore, Zf is the Radon-Nikodym derivative between Q and P restricted on F. Given that we are only interested in the tail event regarding f not τ, we work with Zf most of the time. By slightly abusing notation, we write dq dp = Zf = mes exp { 2 ft u t 2} exp { dt = 2ft2} { mes exp 2 } u2t + ftu t dt. 0 Remark 2.3. o better understand the connection between the above simulation procedure and the likelihood ratio 0, we present a discrete analogue for a finite dimensional multivariate Gaussian random vector X = X,..., X n. For the finite dimensional case, τ is uniformly distributed over {,..., n}. he density function of X under the change of measure is qx,..., x n = n n q τ x τ fx τ x τ τ= where x τ = x,..., x τ, x τ+,..., x n, q τ x is the sampling distribution of x τ given τ, and f is the density under the original measure. hus, the likelihood ratio is qx,..., x n fx,..., x n = n n τ= q τ x τ fx τ ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

9 Efficient Simulations for Gaussian Random Fields 0:9 that is the discrete analogue of 0. Based on the above discussion, we have the continuous version importance sampling estimator taking the form L b = I >e b and its second moment equals E Q [ L 2 ] b = E Q [ { mes exp 2 } u2t + ftu t dt { mes exp 2 } ] 2 u2t + ftu t dt ; I > e b. he measure Q is constructed such that the behavior of f under Q mimics the tail behavior of f given the rare event {I > e b } under P. According to the above simulation procedure, a random variable τ is first sampled uniformly over, then fτ is simulated with a large mean at level u τ. Under the zero-variance change of measure, the large value of the integral I is mostly caused by the high excursion of ft at one location. he random index τ searches the maximum of ft over the index set. It worth emphasizing that τ is not necessarily the exact maximum but should be very close to it. In the case when ft is differentiable and strictly stationary and µt 0 and σt, the zero-variance change of measure can be more precisely quantified. For the stationary case, u t is a constant and we write it as u. hen, the zero-variance change of measure is approximated as follows sup QA P f A sup γ u t > u 0 as b, A where Q = P f I > e b and γ u t = ft+ r 2 ft 2σu +κ 0. 2 ft is the Hessian matrix, r is the trace operator, and κ 0 is a constant only depending on the covariance function. See [Liu and Xu 202a] for the detailed description of the above results. According to the total variation approximation, the high excursion of I is almost the same as the high excursion of ft with a small correction depending on the Hessian matrix. he propose measure Q is different from Q mostly in two ways. First, under Q, the overshoot of γ u t ft over the level u is of order Ou ; under Q, the overshoot is of order O. Second, the measure Q does not tilt the distribution of 2 ft. hese are the main sources of inefficiency. On the other hand, the Radon-Nikodym derivative dq/dp takes a very friendly form that one can take advantage of the Jensen s inequality to prove weak efficiency for a ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

10 0:0 Liu and Xu general class of Gaussian processes, especially for non-differentiable processes, for which there is no quantitative result for the zero-variance change of measure he algorithm and efficiency results For the implementation, we introduce a suitable discretization scheme on. For any positive integer N, let G N,d be a subset of R d G N,d = { i N, i 2 N,..., i } d : i,..., i d Z, N where Z is the set of integers. hat is, G N,d is a regular lattice on R d. For each t = t,, t d G N,d, define N t = { s,, s d : s j t j /N, t j ] for j =,, d } that is the N -cube intersected with and upper-cornered at t. Furthermore, let N = {t G N,d : mes N t > 0}. Since is compact, N is a finite set. We enumerate the elements in N = {t,, t M }, where M mes N d. We use as an approximation of wb where I M = w M b = P I M > e b mes N t i e σtifti+µti. 2 i= We now state the main results that require the following technical conditions. A ft is almost surely continuous with respect to t and furthermore admits E[ft] = 0 and E[f 2 t] = for all t ; A2 here exist δ, κ H > 0, and β 0, ] such that, for all s t < δ, the mean and variance functions satisfy µt µs + σt σs κ H s t β ; ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

11 Efficient Simulations for Gaussian Random Fields 0: A3 For each s, t, define the covariance function Cs, t = Covfs, ft. For all s s < δ and t t < δ, the covariance function satisfies Ct, s Ct, s κ H t t 2β + s s 2β. Condition A assumes that f has zero mean and unit variance. For a general Gaussian random field with mean µt and variance σ 2 t, we treat the mean and the standard deviation as additional parameters. Conditions A2 and A3 essentially ensure that the process µt+σtft is uniformly Hölder continuous, that is, for s t < δ, Varfs ft 2κ H s t 2β. hese assumptions are weak enough such that they accommodate essentially all Gaussian processes practically in use, such as fractional Brownian motion, smooth Gaussian processes, etc. herefore, the algorithm developed in this paper is suitable for a wide range of applications. he first result controls the relative bias of w M b. HEOREM 2.4. Consider a Gaussian random field f satisfying conditions A-A3. For any 0 < ɛ < /2, there exists a constant κ 0 such that for any η > 0, b >, and the lattice size N = κ /β 0 log η /β η /β b 2+ɛ/β, where β is given as in conditions A2 and A3. w M b wb wb < η, hanks to the continuity of ft, as N tends to infinity, the lattice N becomes dense in. he finite sum I M converges in probability to I and therefore w M b/wb converges to. On the other hand, the convergence of w M b/wb is not uniform in b. he above theorem provides a lower bound of N such that w M b/wb is close enough to unity and the relative bias can be controlled. With N chosen as in heorem 2.4, we proceed to estimating w M b by importance sampling, which is based on the change of measure proposed in 0. We make the corresponding adaptation under the above discretization N. In particular we define a measure Q M as the discrete version of Q such that dq M /dp takes the form: dq M dp = M i= e 2 fti ut i 2 = M e 2 fti2 i= M eut i fti 2 u2 t i. 3 ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

12 0:2 Liu and Xu he computation of the above likelihood ratio and the event I M > b only consists of {ft i : i =,..., M}. For the simulation under the measure Q M, we propose the following algorithm. he algorithm has two steps: Step : Simulate ι uniformly from {,..., M} and generate ft ι from Nu tι,. Given ι, ft ι, simulate ft,, ft ι, ft ι+,, ft M from the original conditional distribution under the measure P. Step 2: Compute and output L b = M i= {IM >e b }. 4 M eu t i ft i 2 u2 t i It is not hard to verify that the above simulation procedure is consistent with the likelihood ratio 3 and thus L b = {IM >e b } dp dq M is an unbiased estimator of w M b. he next theorem controls the variance of the estimator L b. HEOREM 2.5. Suppose that f is a Gaussian random field satisfying conditions A-A3. If N is chosen as in heorem 2.4, then log E Q M L2 lim b b 2 log w M b =. he above results show that the estimator L b in Algorithm is asymptotically efficient in estimating w M b. o estimate wb, we simulate n i.i.d. copies of L b, { L j b : j =,..., n} and the final estimator is Z b = n j n j= L b. he estimation error is Z b wb w M b wb + Z b w M b. 5 he first term is controlled by heorem 2.4, i.e., w M b wb ηwb if we choose the discretization size N = O log η /β η /β b 2+ɛ/β. he second term of 5 is controlled by the discussion as in 4 if we choose the number of replicates n = Oη 2 δ w λ b. he simulation of L b consists of generating a random vector of dimension M = ON d. Note that the complexity of computing the eigenvalues and eigenvectors or the Cholesky decomposi- ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

13 Efficient Simulations for Gaussian Random Fields 0:3 tion of an M-dimension matrix is OM 3 = ON 3d. hus, the total computational complexity to achieve the prescribed accuracy in 5 is ON 3d η 2 δ w η b. he current choice of measure Q as in 0 does not depend on the particular form of the covariance structure. When there is more knowledge available, we can further tune and adapt the measure Q to more refined structures and to improve the efficiency. Notice that the measure Q twists the process f at the random location τ. he main tuning parameters are the distribution of τ which is currently chosen to be uniform and the distribution of fτ which is currently chosen to be Nu τ,. herefore, the general form of the change of measure Q is dq dp = ht g tft ϕ t ft dt where ht is the density of τ, g t is the sampling density of fτ given that τ = t, and ϕ t is the density of ft under P. We may refine the choice of g t and h to take into account the specific structures of f. For instance, if µt has one unique maximum attained at t, then it would be more efficient to choose ht concentrating around t. he theoretical analysis also needs to be adapted to the specific choices of h and g t. he most difficult part of the analysis lies in obtaining more accurate asymptotic approximations of wb so as to justify stronger type of efficiency. his is particularly challenging when ft is not differentiable and it is beyond the topic of the current paper. In this paper, we stick to the unified simulation scheme that is applicable to a large class of processes and admits an acceptable efficiency property. Remark 2.6. One limitation of the current analysis is that the total computational complexity grows exponentially fast with dimension d. his is mainly due to the regular discretization method. One may alternatively use other numerical methods such as quasi-monte Carlo, randomized quasi-monte Carlo, or Monte Carlo, whose complexities do not depend on dimension, to approximate the integral e ft dt. In the literature of quasi-monte Carlo and randomized quasi-monte Carlo low-discrepancy sequences have been developed. hese more refined choices may further reduce bias compared to the regular lattice G N,d. For more detailed analysis, see [L Ecuyer and Munger 200; L Ecuyer et al. 200; L Ecuyer 2009]. We do not perform rigorous analysis along this line that is beyond the scope of this paper. 3. SIMULAION o illustrate the proposed algorithm, we first apply it to homogeneous Gaussian random fields {ft, t = [0, ] d } with dimension d = and 2. For the not-so-small tail probabilities, ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

14 0:4 Liu and Xu we also use crude Monte Carlo to compute them as a validation of the importance sampling algorithm. For each case, we assume that f has zero mean and covariance function Cs, t = e t s α, 6 where is the L 2 -norm and α is taken to be and 2 corresponding to different covariance structures of f. When α = 2, f is infinitely differentiable; when α =, f is non-differentiable. We discretize following the procedure in Section 2.2. For d =, we take N = {i/00 : i =,, 00} with discretization size N = 00. he detailed simulation is described in the next steps.. Generate a random variable ι Uniform {, 2,, 00}. 2. Simulate f ι 00 Nu,, where u is calculated from equation Given ι and f ι 00, simulate {f i 00, i =,, ι, ι +,, 00} under the covariance structure specified in 6. First, we consider the one-dimensional case with constant σt and µt. Let µt = 0 and σt =. hen the tail probability of interest takes the form wb = P 0 e ft dt > e b. he estimated tail probabilities wb along with the estimated standard deviations Std Q L b = Var Q L b are shown in able I. All the results are based on 0 4 independent simulations. he standard deviation of the final estimate in the column Est. is the reported standard deviation in the column of Std. divided by 00. Comparing the simulation results of α = and 2, we can see that the algorithm has a smaller relative error when α = 2. he CPU time to generate 0 4 samples is less than one second. o validate the simulation results, we use crude Monte Carlo for b = 3 and 5. Based on 0 6 independent simulations, for b = 3, the estimated tail probabilities are 4.7e-4 Std. 2e-5 and 8.2e-4 Std. 3e-5 when α = and 2, respectively; based on 0 9 independent simulations, for b = 5, the estimated tail probabilities are.2e-8 Std. 3e-9 and 7.7e-8 Std. 9e-9 when α = and 2, respectively. hese results are consistent with those computed by the importance sampling estimators. We also consider the non-constant mean and variances. In particular, we choose σt = t and µt = 2t. he corresponding simulation results for wb = ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

15 Efficient Simulations for Gaussian Random Fields 0:5 able I. Estimates of wb on = [0, ], σt =, and µt = 0, based on 0 4 independent simulations. b Est. Std. Std./Est. b = e-04.4e α = b = 5.3e e b = 7.80e-5.66e b = e e α = 2 b = e e b = e e he standard deviation of the estimate is Std./00. P 0 eσtft+µt dt > e b are shown in able II. he CPU time to generate 0 4 samples is less than one second. For b = 3, crude Monte Carlo estimator based on 0 6 independent simulations gives the estimated tail probabilities.4e-3 Std. 4e-5 and 2.3e-3 Std. 5e-5 when α = and α = 2, respectively; for b = 5, crude Monte Carlo estimator based on 0 9 independent simulations gives the estimated tail probabilities.8e-8 Std. 4e-9 and.4e-7 Std. e-8 when α = and α = 2, respectively. For the case that d = 2, we start with constant mean and variance, µt = 0 and σt =. hen the tail probability of interest takes the form wb = P [0,] 2 e ft dt > e b. Similarly, we discretize and take N = {i, j/00 : i, j =,, 00} with the discretization size N = able III shows the estimated tail probabilities wb along with Std Q L b. In addition, we perform crude Monte Carlo for b = 3. he estimated tail probabilities, based on 0 5 independent simulations, are.8e-4 Std. 4e-5 and 5.3e-4 Std. 7e-5 when α = and α = 2, respectively. Furthermore, able IV gives estimated tail probabilities for non-constant functions σt = t 0.5, and µt = 2t,, where and 2 are the L - and L 2 -norms respectively. he CPU time to generate 0 4 samples varies from from to 5 minutes. For b = 3, the crude Monte Carlo based on 0 5 independent simulations gives 3.3e-3 Std. 2e-4 and 5.6e-3 Std. 2e-4 for α = and α = 2, respectively. he simulation results show that the coefficients of variation Std Q L b /wb increase as the tail probabilities become smaller. Nonetheless, the coefficients of variation stay reasonably small when the probability is as small as 0 7. he continuity of the process and the dimension do affect the empirical performance of the algorithm. More precisely, the algorithm admits smaller coefficients of variation when the process is more continuous corresponding to a larger value of α and the domain is of a lower dimension. In addition, for stationary processes, the algorithm has a slightly better performance than the nonstationary cases. his is because, for the stationary cases, the uniform distribution of τ is closer to the distribution of the maximum of f under Q. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

16 0:6 Liu and Xu able II. Estimates of wb on = [0, ], σt = t 0.5 2, and µt = 2t, based on 0 4 independent simulations. b Est. Std. Std./Est. b = 3.39e e α = b = e-08.3e b = e e b = e e α = 2 b = 5.3e e b = 7.9e e he standard deviation of the estimate is Std./00. able III. Estimates of wb on = [0, ] 2, σt =, and µt = 0, based on 0 4 independent simulations. b Est. Std. Std./Est. b = 3 2.0e-04.77e α = b = 5.6e-09.58e b = 7.5e e b = e e α = 2 b = 5.46e e b = e-5 3.e he standard deviation of the estimate is Std./00. able IV. Estimates of wb on = [0, ] 2, σt = t 0.5, 0.5 2, and µt = 2t,, based on 0 4 independent simulations. b Est. Std. Std./Est. b = e-03.8e α = b = e e-08.6 b = 7.30e e b = 3 5.5e-03.58e α = 2 b = e e b = e-5 3.6e he standard deviation of the estimate is Std./ PROOF OF HE HEOREMS he proofs of the theorems need several supporting lemmas. o smooth the discussion, we provide their statements at places where they are used and delay their proofs to Section 5. PROOF OF HEOREM 2.4. later, let L = { f C : For any η, ɛ > 0 and a large constant κ 0 > 0 to be determined } sup σsfs + µs σtft + µt ηκ /3 0 b +ɛ, 7 s,t: t s dn where C is the set of all continuous functions on and N is chosen as in the statement of the theorem. he following lemma suggests that we only need to focus on the set L. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

17 Efficient Simulations for Gaussian Random Fields 0:7 LEMMA 4.. Under the conditions in heorem 2.4, for any ɛ > 0, there exists a constant κ 0 such that, for any η > 0, b >, and N = κ /β 0 log η /β η /β b 2+2ɛ/β, we have P I > e b, L c ηwb, P I M > e b, L c ηwb. 8 With N chosen as in Lemma 4., we have that w M b wb 9 P I M > e b, I < e b, L + P I M < e b, I > e b, L + 2ηwb. On the set L, we have that I M I [ = e σtifti+µti e σtft+µt] dt i= N t i 2 mes N t i e σtifti+µti sup σt i ft i + µt i σtft + µt t t i dn i= 2ηκ /3 0 b +ɛ I M. Recall that t i is the corner point of N t i. he first inequality in the above display is an application of aylor s expansion. A similar argument yields that hus, the first term in 9 is bounded by P I M > e b, I < e b, L P and the second term is bounded by P I M < e b, I > e b, L P I M I 2ηκ /3 0 b +ɛ I. e b 2ηκ /3 0 b +ɛ < I < e b, L e b < I < e b + 2ηκ /3 0 b +ɛ, L. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

18 0:8 Liu and Xu We insert the above bounds back to 9. here exists some constant c > 0 such that w M b wb P e b 2ηκ /3 0 b +ɛ < I < e b + 2ηκ /3 0 b +ɛ, L + 2ηwb P b + log 2ηκ /3 0 b +ɛ < log I < b + log + 2ηκ /3 0 b +ɛ, L + 2ηwb cηκ /3 0 b +ɛ F b cηκ /3 0 b +ɛ + 2ηwb, 20 where F x is the probability density function of log I. he following lemma provides an upper bound of the density function F x. LEMMA 4.2. Under the conditions of heorem 2.4, let F x be the probability density function of log I. hen F x exists almost everywhere. Moreover, for all ɛ > 0 and λ > 0, F x = ox +ɛ/2 w x + x λ as x. 2 We apply Lemma 4.2 to 20 by setting x = b cηκ /3 0 b +ɛ. Furthermore, we choose λ small such that x + x λ = b and thus F b cηκ /3 0 b +ɛ ob +ɛ/2 wb. We insert the above bound to the right-hand-side of 20 and obtain that for κ 0 large enough, w M b wb cκ /3 0 ηb +ɛ ob +ɛ/2 wb + 2ηwb 3ηwb. hen, we can redefine η and κ 0 and conclude the conclusion. PROOF OF HEOREM 2.5. he main idea. he key element of this proof is an application of the Jensen s inequality that for all u and ft M [ e ufti M i= e fti] u. i= hus, if M M i= efti > e b, then M M i= eufti > e ub. he following technical proof is to write the likelihood ratio and I M in a form that the above inequality is applicable. hus, we are able to develop an upper bound of the second moment of the likelihood ratio on the set {I M > e b }. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

19 Efficient Simulations for Gaussian Random Fields 0:9 he technical proof. We first present a lemma that provides not-so-accurate but useful bounds of wb. LEMMA 4.3. Under Conditions A-A3, there exist constants c 0, c and c 2 such that exp b2 + c b log b + c 0 2σ 2 wb exp b2 2σ 2 + c 2 b. Consequently, log wb lim b b 2 2σ 2. Under the change of measure, the discrete likelihood ratio is dq M dp = M i= e 2 fti ut i 2 = M e 2 fti2 i= M e 2 u2 t i +ft iu ti. Note that most N t i are rectangles and there are just a few on the boundary of that are not rectangles. If we let κ = 2mes, then I M is bounded from the above by κ N d e σtifti+µti e σtifti+µti mes N t i = I M. i= i= hat is, those N t i s on the boundary of are replaced by rectangles. hus, the second moment of the estimator is bounded by E Q M [ dp 2 ; I M > e b ] dq M M 2 E Q M M e 2 u2 t +ft i iu ti ; κ N d i= e σtifti+µti > e b For the next step, we wish to take the term 2 u2 t i in the exponential out of the expectation. Note that 2 u2 t i 2 u min t µ σ t 2 and we continue the above calculation M 2 e u min t µ σt 2 E Q M M eftiut i ; κ N d i= i= 22 e σtifti+µti > e b. 23 i= ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

20 0:20 Liu and Xu We now split the set {t,..., t M } into {t i : ft i > 0} and {t i : ft i 0}. Furthermore, there exists κ such that N d {t i:ft i 0} e σtifti+µti κ. hus, there exists δ 0 > 0 such that, for b sufficiently large and on the set { κ N d M i= eσtifti+µti > e b }, we have κ < δ 0 e b and {i:ft i>0} herefore, κ N d M i= eσtifti+µti > e b implies that M e σ ft i+max t µt M i= M e σtifti+µti δ 0 e σtifti+µti. 24 {i:ft i 0} i= e σtifti+µti δ 0N d Mκ e b. 25 We now consider the behavior of the likelihood ratio on the set { κ N d M i= eσtifti+µti > e b }. For ft > 0, we have that u t ft u max µ σ tft = u max µ σt [σ ft + max µt] u max µ σ t max µ σ t. σ For u large enough, we have that M e ut i fti M i= Notice that M that = M i:ft i>0 i:ft i>0 e u max µσtfti 26 { u maxt µ σ t exp σ [σ ft i + max t } µt] u max µ σ t max µ σt t t i:ft i 0 eu max µσtfti. We continue the above calculations and obtain e u maxt µσt max t µ σt M i= { } u maxt µ σ t exp [σ ft i + max σ µt]. t ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

21 Efficient Simulations for Gaussian Random Fields 0:2 We apply Jensen s inequality to the above summation and obtain that M i= e ut i fti e u max t µσ t σ max t µt [ M u maxt µσt/σ e σ ft i+max t µt] i= We apply the result in 25 and continue the above calculations. here exist κ 2, κ 3 > 0 such that on the set { κ N d M i= eσtifti+µti > e b } M i= e ut i fti e κ2u δ0 N d u maxt µ σt/σ e b Mκ e κ3u log b+u2. 27 For the last step, we use the definition of u as in 7 and obtain that σ u κ 4 log b b σ u where κ 4 > 0 and further lim b b σ u =. Combining 22, 23, and 27, there exists a constant κ 5 > 0 such that M 2 e u min t µ σt 2 E Q M M eftiut i ; κ N d i= e σtifti+µti > e b Combining the above results with heorem 2.4 and Lemma 4.3, we have lim inf b i= log E Q M L2 b b 2 /σ 2 lim inf + κ 5b log b 2 log w M b b b 2 /σ 2 =. e σ 2 b2 +κ 5b log b. On the other hand, Jensen s inequality suggests that log E Q M L2 b 2 log w M b and lim sup b log E Q M L2 b 2 log w M b. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

22 0:22 Liu and Xu 5. PROOFS OF LEMMAS In this section, we present the proofs of the supporting lemmas in the main proof. he first lemma is known as the Borel-IS lemma, which is proved independently by [Borell 975; sirelson et al. 976]. LEMMA 5. BOREL-IS. Let ft, t U U is a parameter set be a mean zero Gaussian random field. f is almost surely bounded on U. hen, E[sup U ft] <, and P where σ 2 U = sup t U Var[ft]. sup t U f t E[sup f t] b exp b2 t U 2σU 2, he Borel-IS lemma provides a very general bound of the tail probability P sup ft > b exp t U b E[sup t U f t] 2 In most cases, E[sup t U f t] is much smaller than b. hus, for b sufficiently large, the tail probability can be further bounded by 2σ 2 U P sup ft > b exp b2 t U 4σU 2. he following result by [Dudley 973] c.f. heorem 6.7 in [Adler et al. 202] is often used to control E[sup t U f t]. LEMMA 5.2. Let U be a compact subset of R n, and let {ft : t U} be a mean zero, continuous Gaussian random field. Define the canonical metric d on U as. d s, t = E[f t f s] 2 and put diam U = sup s,t U d s, t, which is assumed to be finite. hen there exists a finite universal constant κ > 0 such that E[max t U f t] κ diamu/2 0 [log N ε] /2 dε, where the entropy N ε is the smallest number of d balls of radius ε whose union covers U. By means of the above lemma, one can often establish that E[max t U f t] = Oδ log δ where δ 2 = sup t U Varft. We now proceed to the proof of the supporting lemmas. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

23 Efficient Simulations for Gaussian Random Fields 0:23 PROOF OF LEMMA 4.. We present the proof of the first bound in the lemma. he proof of the second bound is completely analogous. Consider the change of measure dq dp = With a similar argument as in 23, we have P I > e b, L c [ ] dp = E Q dq ; I > eb, L c mes exp u t ft u2 t dt e u mint µσt2 /2 E Q [ mes eutft dt ; I > eb, L c ], where E Q is the expectation under measure Q. Note that I > e b implies that for all large b, e σ ft+max t µt dt mes e σtft+µt dt mes {ft 0} mes eb Furthermore, on the set {I > e b }, we have that e utft dt mes mes {ft<0} e σtft+µt dt mes eb e max t µt. 29 { exp mes {ft>0} u max t } µ σ tft dt. u maxt µσt We add and subtract the term σ max t µt in the exponent and continue the above calculation = { u maxt µ σ t exp σ ft + max mes {ft>0} σ µt u max } t µ σ t max t σ µt dt t Since µσtft mes dt, the above display is bounded from below by {ft 0} eu maxt { u maxt µ σ t exp σ ft + max mes σ µt u max } t µ σ t max t σ µt dt t ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

24 0:24 Liu and Xu By means of Jensen s inequality and the lower bound in 29, the above display is further lower bounded by { exp u max t µ σ t σ } [ max µt t { exp u max t µ σ t max σ µt t herefore, we have the following bound mes } [ e b mes emax t µt P I > e b, L c [ ] e u mint µσt2 /2 E Q mes eutft dt ; I > eb, L c ] u maxt µ σt/σ exp{σ ft + max µt}dt t ] u maxt µ σt/σ. 30 [ e u mint µσt2 /2 e u max t µσ t σ max t µt e b u maxt µσ t ] mes emax t µt σ Q I > e b, L c Notice the facts that u min t µ σ t 2 /2 = u 2 /2 + Ou and [ e b ] u max t µσ t σ mes emax t µt = e b/σ +Ou+O. By the facts that u = b/σ + Olog b and Q I > e b, L c QL c, there exists a constant c such that for b sufficiently large, P I > e b, L c { exp b2 2σ 2 } + c b log b QL c. hen Lemma 5.3 presented momentarily together with Lemma 4.3 implies that there exist constants κ and λ such that P I > e b, L c η exp b2 2σ 2 + c b log b λb +ɛ κ ηwb. Note that κ can be chosen such that the above inequality holds for all b and η. hen, we can redefine η and the constant κ 0 and obtain the conclusion. he proof of the second inequality in 8 is similar. he only difference is that the integral in the above derivation is replaced by a summation over discretization N. Hence, we omit the details. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

25 Efficient Simulations for Gaussian Random Fields 0:25 LEMMA 5.3. Under the conditions in heorem 2.4, there exists a constant λ such that Q L c η exp λb +ɛ, where L is the set defined as in 7. PROOF OF LEMMA 5.3. We focus on a given τ and define another process f t = ft u τ Ct, τ. hen under the measure of Q, f t has the same distribution as that of ft under the original measure P. Under the above notation and N = κ /β 0 log η /β η /β b 2+2ɛ/β, we have that L c = { sup σsf s + σsu τ Cs, τ + µs s,t: t s dn σtf t + σtu τ Ct, τ + µt > ηκ /3 0 b +ɛ }. According to the Borel-IS lemma, Q sup f t > /2 log η κ /2 0 b /2+ɛ log η κ 0 b /2+ɛ E Q [sup t f t] 2 exp t 2σ 2 2. Since E Q [sup t f t] = O, there exists λ > 0 such that for large κ 0 Q sup f t > log η κ /2 0 b /2+ɛ exp λ log η κ 0 b +2ɛ = η λκ0b+2ɛ η exp λ κ 0 b +2ɛ. t 3 In what follows, we bound the tail of L c {sup t f t log η κ /2 0 b /2+ɛ }. For all s, t satisfying t s β d β N β = d β κ 0 log η ηb 2+2ɛ and b >, there exist positive constants c, c 2, and c such that σsf s + σsu τ Cs, τ + µs σtf t + σtu τ Ct, τ + µt σs f s f t + f t σs σt +u τ σs Cs, τ Ct, τ + u τ Ct, τ σs σt + µs µt 32 σ f s f t + cκ /2 0 log η /2 ηb 2ɛ. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

26 0:26 Liu and Xu o obtain the last inequality in the above display, we need to notice that for 0 < ɛ < /2, f t = O log η κ /2 0 b /2+ɛ }, and σs σt = O log η ηb 2+2ɛ f t σs σt = O log η κ /2 0 b /2+ɛ log η ηb 2+2ɛ = O log η /2 ηb 3/2 ɛ = O log η /2 ηb 2ɛ. he last three terms of 32 are bounded by u τ σs Cs, τ Ct, τ + u τ Ct, τ σs σt + µs µt = O log η ηb +2ɛ. Applying the above bound, we obtain that Q Q L c, sup t f t log η κ /2 0 b /2+ɛ sup σ f s f t + cκ /2 0 log η /2 ηb 2ɛ > ηκ /3 0 b +ɛ s,t: t s dn Q sup σ f s f t > s,t: t s dn 2 ηκ /3 0 b. +ɛ For the component f s f t and t s dn, the variance function has an upper bound: Varf s f t = 2 Cs, t 2κ H s t 2β 2κ H d 2β N 2β = 2κ H d 2β κ 2 0 log η 2 η 2 b 4+ɛ. We apply Lemma 5.2 to the double-indexed process ξs, t = f s f t on the set U = {s, t : s, t, s t dn }. hen, logn ε = O log ε and diamu = O log η ηb 2+ε. he result of Lemma 5.2 yields that E Q[ ] sup f s f t = Oηb 2+ɛ log b. t s dn ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

27 Efficient Simulations for Gaussian Random Fields 0:27 We apply the Borel-IS inequality Q sup σ f s f t > s,t: t s dn 2 ηκ /3 0 b +ɛ 2 exp 2 ηκ /3 0 b +ɛ E Q [sup t s dn f s f t] 4σ 2 κ Hd 2β κ 2 0 log η 2 η 2 b 4+ɛ exp λ 2 log η 2 κ 4/3 0 b 2+ɛ 33 for some λ 2 > 0. Combining the results in 3 and 33, there exists λ such that for large κ 0, QL c exp λ log η κ 0 b +2ɛ + exp λ 2 log η 2 κ 4/3 0 b 2+ɛ η exp λb +ɛ. 2 PROOF OF LEMMA 4.2. Let Φ be the cumulative distribution function of the standard Gaussian distribution. Define cumulative distribution function of I and the associated quantiles with respect to Φ as F x = P log I x and t x = Φ F x. We cite one result Proposition in [Liu and Xu 202b] that gives an upper bound of F x: LEMMA 5.4. Under the conditions of Lemma 4.2, F x exists almost everywhere. Choose y < x depending on x in a way that x y 0 and xx y when we send x to infinity. hen, where σ = sup t σt. lim sup x σ 2 2πσ exp t 2 y + 2x yy F x, 34 2σ 2 Lemma 5.4 is equivalent to F x + o 2πσ exp σ2 t2 y + 2x yy 2σ ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

28 0:28 Liu and Xu Y Y = σ x t x x Fig. 2. Plot of t x o prove Lemma 4.2, it is sufficient to show that the right-hand-side of 35 is ox +ɛ/2 wx + x λ. Lemma 4.3 implies that t x lim x x = σ. 36 hen, for λ > 0 we let x = x + x λ and y = x log log x/x. With such choices of x and y, the following holds t 2 x t 2 y + 2σ 2 x yy = 2 + oσ x t x t y 2σ 2 x yy = 2 + oσ xx y t x t y x y x y x y σ + o. 37 According to heorem 4.4. in [Bogachev 998] see also [Ehrhard 983], t x is a concave function of x, and thus t z t x /z x is a monotone non-increasing function of z when z > x and lim z x+ t z t x /z x exists. Figure 5 illustrates the function t x. hen for y = x log log x/x and x = x + x λ, we have that he concavity of t x t x t y / x y lim z y+ t z t y /z y. 38 and 36 imply that lim z y+ t z t y /z y is a non-increasing function of y and converges to σ. Sending x to infinity on both sides of 38 gives lim sup x t x t y / x y σ. hus, the first term in the parenthesis of 37 is bounded by σ + o. herefore, there exists a constant c 0 > 0 such that t 2 x t 2 y + 2σ 2 x yy c 0 xx y = c 0 log log x. ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

29 Efficient Simulations for Gaussian Random Fields 0:29 hen, we obtain an upper bound for 35: F x exp σ2 t2 y + 2x yy 2πσ 2σ 2 exp t2 x c 0 log log x. 2πσ 2 Note that w x = F x = + o 2πt x exp t2 x = + o σ exp t2 x. 2 2πx 2 herefore, for any ɛ > 0, we have that F x + o σ 2 xlog xc0/2 w x = ox +ɛ/2 wx + x λ. PROOF OF LEMMA 4.3. We start to derive a lower bound for wb. Let σt = sup t σt = σ. On the set L as defined in 7 with η = η 0 fixed, there exists a δ 0 > 0 such that e σtft+µt dt δ 0 N d e σ ft. t t <N hen, on the set L if σ ft > b + d log N log δ 0 then e σtft+µt dt e b. t t <N herefore, we have the lower bound wb P I > e b, L P e σtft+µt dt > e b, L t t <N P σ ft > b + d log N log δ 0, L P σ ft > b + d log N log δ 0 P L σ ft > b + d log N log δ 0 39 here exist δ 0, δ > 0 such that the conditional probability P L σ ft > b + d log N log δ 0 > δ. herefore, wb δ P σ ft > b + d log N log δ ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.

30 0:30 Liu and Xu Note that N = Ob +ɛ/β and ft follows a standard normal distribution. hus, there exist positive constants c 0 and c such that wb exp b2 + c b log b+ c 0. 2σ 2 We continue to construct an upper bound. Since I > e b implies that sup t {σtft} > b max t µt log mes. herefore, there exists a constant c 2 such that wb P sup{σtft} > b max µt log mes exp b2 t 2σ 2 Combining 40 and 4, we conclude the proof. + c 2 b. 4 REFERENCES ADLER, R. 98. he Geometry of Random Fields. Wiley, Chichester, U.K.; New York, U.S.A. ADLER, R., BLANCHE, J., AND LIU, J Efficient Monte Carlo for large excursions of Gaussian random fields. Annals of Applied Probability 22, 3, ADLER, R., SAMORODNISKY, G., AND AYLOR, J High level excursion set geometry for non-gaussian infinitely divisible random fields. preprint. ADLER, R. AND AYLOR, J Random fields and geometry. Springer. AHSAN, S Portfolio selection in a lognormal securities market. Zeitschrift Fur Nationalokonomie-Journal of Economics 38, -2, ASMUSSEN, S. AND GLYNN, P Stochastic Simulation: Algorithms and Analysis. Springer, New York, NY, USA. ASMUSSEN, S. AND KROESE, D Improved algorithms for rare event simulation with heavy tails. Advances in Applied Probability 38, AZAIS, J. M. AND WSCHEBOR, M On the distribution of the maximum of a Gaussian field with d parameters. Annals of Applied Probability 5, A, AZAIS, J. M. AND WSCHEBOR, M A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stochastic Processes and heir Applications 8, 7, AZAIS, J. M. AND WSCHEBOR, M Level sets and extrema of random processes and fields. Wiley, Hoboken, N.J. BASAK, S. AND SHAPIRO, A Value-at-risk-based risk management: Optimal policies and asset prices. Review of Financial Studies 4, 2, BERMAN, S. M An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. Journal of Applied Probability 22, 2, BLACK, F. AND SCHOLES, M Pricing of options and corporate liabilities. Journal of Political Economy 8, 3, BLANCHE, J. AND GLYNN, P Effcient rare event simulation for the maximum of heavy-tailed random walks. to appear in Annals of Applied Probability 8, 4, BLANCHE, J., GLYNN, P., AND LEDER, K Strongly efficient algorithms for light-tailed random walks: An old folk song sung to a faster new tune. In MCQMC 2008, P. LEcuyer and A. Owen, Eds. Springer, ACM ransactions on Modeling and Computer Simulation, Vol. 0, No. 0, Article 0, Publication date: 202.