7th International Workshop on Rare Event Simulation On the mathematical theory of splitting and Russian roulette techniques Viatcheslav B. Melas St.Petersburg State University, Russia Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation 1. Introduction • Splitting is an universal and potentially very powerful technique for increasing efficiency of simulation studies. The idea of this technique goes back to von Neumann (see [Kan and Harris,1951]). A theory of multilevel splitting is rather well developed (see e.g.[L’Ecuryer et al.2006] ). However, in the most of papers strongly restrictive assumptions on transition probabilities are imposed. • In [Melas, 1993, 1997, 2002] and [Ermakov and Melas, 1995] a more general theory is developed. It is based on the introduction of a probability measure governing the procedures of splitting and Russian roulette. In the present talk we will describe basic ideas and results of this theory. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation 2. Formulation of the problem Let random variables X 1 , X 2 , . . . are defined on probability space ( X , A , P ) and form a homogenous Markov chaim with transition function P ( x , dy ) = P { X 1 ∈ dy | X 0 = x } measurable in the forst � argument and such that X P ( x , dy ) = 1. We will consider Harris positively recurrent Markov chains (see, f.e., (Nummelin, 1984) for definitions). In fact, we need only a mild restrictions that secure existence of the stationary distribution (denote it by π ( dx ) ) and asymptotic unbiasedness of usual Monte Carlo estimators. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation It is required to estimate one or several functionals of the form � J i = h i ( x ) π ( dx ) , i = 1 , 2 , . . . , l X with respect to simulation results. In case of chains with finite state space these functionals have the form m � J i = h i ( x ) π x , i = 1 , 2 , . . . , l . x =0 Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation 3. Branching technique Let { X n } be a Markov chain of the general form described above. Let η n chains { X n } be simulated at the n –th step. A procedure for regulating η n can be introduced in the following way. Definition. A measurable function β ( x ) on ( X , A ) with the following properties: 1 β ( x ) ≥ 0 , x ∈ Ω, 2 � B β ( x ) π ( dx ) > 0, if π ( B ) > 0 is called a branching density. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Introduce the random variable r ( x , y ) = ⌊ β ( y ) /β ( x ) ⌋ , with probability 1 − β ( y ) /β ( x ) and r ( x , y ) = ⌊ β ( y ) /β ( x ) ⌋ + 1 , with probability β ( y ) /β ( x ) , where ⌊ a ⌋ , with a = β ( y ) /β ( x ), means the integer part of a and a means the whole part of a . At step zero, set η 0 = 1. At each following step ( n + 1 = 1 , 2 , . . . ) if X γ n = x , X γ n +1 = y , then when β ( y ) /β ( x ) < 1 the simulation of path γ discontinues with probability 1 − β ( y ) /β ( x ). When β ( y ) /β ( x ) ≥ 1 we simulate r ( x , y ) − 1 additional paths, beginning from point x , where k -th steps of these chains will be denoted by index n + k . Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Let us simulate k loops: { X γ n ( i ); γ = 1 , 2 , . . . , η n ( i ) } , i = 1 , 2 , . . . , k . Let us determine estimators of the functionals J j ( j = 1 , 2 , . . . , l ) by the formula � k k � � J k ,β ( j ) = 1 � � Y β ( i , j ) Y β ( i , j ) , k i =1 i =1 where N ( i ) η n � � h j ( X γ n ( i )) / β ( X γ Y β ( i , j ) = n ( i )) , n =0 γ =1 � Y β ( i , j ) = Y β ( i , j ) with h i ( x ) ≡ 1 . Note that when β ( x ) ≡ 1 the process reduces to direct simulation of the chain { X n } , and the estimators � J i ,β become the known ratio estimators of the regeneration method (see Crane, Iglehart, 1974). Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Denote � �� l � � J k ,β ( i ) , � D ( β ) = k →∞ kCov lim J k ,β ( j ) , i , j =1 k � T ( β ) = lim T i ,β / k . k →∞ i =1 By virtue of Law of Large Numbers, T ( β ) is the expectation of the number of steps for all chains in one loop. Set ˆ h j ( x ) = � J k ,β ( j ) , j = 1 , 2 , . . . , l , under the condition that all paths begin at x ( X 1 0 ( i ) = x , i = 1 , . . . , k ) . Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation 4. Expression for covariances Theorem 1. When regularity conditions (see Melas(1993)) are valid for general Markov chains, estimators � J k ,β ( i ) are asymptotically unbiased and �� � l β − 1 ( x ) π ( dx ) ( d ij ( x ) + r β ) D ( β ) = , i , j =1 � T ( β ) = β ( x ) π ( dx ) , where d ij ( x ) = E ˆ h i ( x ) ˆ h j ( x ) − E ˆ h i ( x ) E ˆ h j ( x ) , E is the expectation operator, and r β is a remainder term. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Theorem 1 permits us to introduce optimality criteria for the choice of the branching measure. To simplify the notation let us consider that π ( dx ) = π ( x ) dx . Set �� τ ( x ) = π ( x ) β ( x ) β ( x ) π ( x ) dx (1) �� τ x = π x β x β x π x (finite chains) . (2) x Then T ( τ ) = 1 for any β . Dropping the remainder term, we obtain the matrix �� � τ − 1 ( x ) d ij ( x ) π 2 ( x ) dx D ( τ ) = , � m � � τ − 1 x π 2 D ( τ ) = x d ij ( x ) (finite chains) . x =0 Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation As an estimation accuracy criterion let us take quantities trA D ( τ ) , (3) det D ( τ ) (4) where A is an arbitrary nonnegative-definite matrix. Note that when A = I (where I is identity matrix) and l = 1, trA D ( τ ) is just the variance of the estimator � J 1 . The probability measure τ ( x ) is called a design of the simulation experiment. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation 5. Optimal experimental designs Consider the minimization problem for criteria (3) and (4) in the class of probability measures τ induced by branching densities. The optimal design for criterion (3) can be found with the help of the Schwarz inequality. The direct application of this inequality brings the following result. Theorem 2. Let the hypothesis of Theorem 1 be satisfied. Then the experimental design minimizing the value of tr A D ( τ ) is given by formula (1), where � β ( x ) = trA D ( τ ) . Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Theorem 3. For finite regular Markov chains and functions h i ( x ) of the form h i ( x ) = δ ix , i = 1 , . . . , m, there exists an experiment design minimizing det D ( τ ) . This design is unique and satisfies the relation � √ m , � � 1 / 2 π x tr B x B − 1 ( τ ∗ ) τ ( x ) = where ( p xy δ yz − p xy p xz ) m B x = y , z =1 , x = 0 , 1 , . . . , m , � m � � π 2 B ( τ ) = x /τ x B x . x =0 The design described in Theorem 3 will be called D -optimal. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation In order to construct a D-optimal design the following iterative method can be used. Set τ 0 ( y ) = π y , y = 0 , . . . , m , τ k +1 ( y , α ) = (1 − α ) τ k ( y ) + ατ ( x ) ( y ) , k = 0 , 1 , 2 , . . . where τ ( x ) ( y ) = δ xy π 2 k ( x ) tr B x B − 1 ( τ k ) . x x = arg max τ 2 x Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Set α k = arg min α ∈ [0 , 1] det B ( { τ k +1 ( y , α ) } ) , τ k +1 ( y ) = τ k +1 ( y , α k ) , y = 0 , 1 , . . . , m . Theorem 4. Under the hypothesis of Theorem 3 for k → ∞ det D ( τ k ) → det D ( τ ∗ ) , where det D ( τ ∗ ) = min det D ( τ ) . Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
7th International Workshop on Rare Event Simulation Consider a random walk process on a line S 1 = 0 , S n +1 = S n + X n , k = 1 , 2 , . . . , where X n , k = 1 , 2 , . . . are independent random variables with common density function f ( x ), and connect it with the waiting process W 1 = 0 , W n +1 = max(0 , W n + X n ) , n = 1 , 2 , . . . It is known (see Feller, 1970) that the quantities W n and M n = max { S 1 , . . . , S n } have the same distribution. Under an additional condition on EX 1 , M n → M and W n → W in distribution, where M and W are random variables. Viatcheslav B. Melas On the mathematical theory of splitting and Russian roulette
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