Ulam Quaterly { Volume 2, Number 4, with Convergence Rate for Bilinear. Omar Zane. University of Kansas. Department of Mathematics

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1 Ulam Quaterly { Volume 2, Number 4, 1994 Identication of Parameters with Convergence Rate for Bilinear Stochastic Dierential Equations Omar Zane University of Kansas Department of Mathematics Lawrence, Kansas Abstract We consider the parameter identication problem for both drift and diffusion coecients of bilinear stochastic dierential equations. A strongly consistent estimator for the coecient of the drift is known for the case in which we have continuous observation of the state. The values of the parameters for the diusion coecients are known after an arbitrarily small positive interval of time: formulas for the actual computation of their values are given. For the case in which we observe the state only at discrete moments of time, we discretize the formulas. It is shown that the discretization of these estimators does not converge to the parameters but to quantities that depend both on the values of the parameters and on the disretization step. The expression of these quantities are given explicitly. Results on the rate of convergence of these estimators are given. Finally we give strongly consistent estimators for the discrete observations case. 1 Introduction We consider the bilinear stochastic dierential equations, for i = 1; ::; d, i (t) = b i X i (t)dt + P d j=1 ij X i (t) dw j (t); X = x ; (1.1) where b = (b 1 ; ::; b d ) is a constant vector, = ( ij ) d i;j=1 a constant matrix, and W a d-dimensional Brownian motion. This kind of equations has found applications in nancial mathematics as models for the evolution of prices in the stock market. In particular, an application to problems of optimal investment and consumption (see [K]) shows optimal policies that depend explicitly on the values of b and A :=. These parameters (average rate of return of the stocks and volatility of the market) are not known by the investor, who has the need for estimating their values in such a way that 65

2 66 Identication of Parameters with Convergence Rate he/she can dene the policy of investment and consumption. In this paper strongly consistent estimators for b and A are given both for the case in which we observe X(t) = (X 1 (t); ::; X d (t)) for every t and for the case in which the observations are made only at discrete moments t 2 ft k := kg1 k=, where is a positive constant. For the case in which we have continuous observations a strongly consistent estimator of b is known and so is its rate of convergence (see [DPD1], [DPD2]). We give a discretized version of this estimator and show that the discretization introduces a bias that depends on the discretization step and that can be found explicitly. This fact allows us to dene a strongly consistent estimator for the discrete observation case. The value of A is known after an arbitrarily small positive period of time for the case in which we have continuous observations; a formula for the computation is given. Just like in the previous case the discretization of this formula gives us a biased estimator for A; the value of the biased is found and a strongly consistent estimator for the discrete observation case is given. Finally results on the rate of convergence of the discretized estimators are given. The paper is organized as follows: in Section 2 estimators for both b and A are given for the case with continuous observations. The convergence of their discretization with results on the rate of convergence is given in Section 3. The results of the previous section are then used in Section 4 to prove the strong consistency of the estimators that are given for both b and A for the case in which we have discrete observations. The results are then tested through simulation in Section 5. 2 Continuous observations Let us consider the case in which we want to estimate the parameters b and A in (1.1) and we have continuous observations of the state X(t). The estimator of b, given for the case d = 1 in [BF], [DPD1], [K] and [S], can be used also for the case d > 1. In fact, we have the following result Proposition 2.1 For every i=1,..,d, let b i (t) be dened for t > by bi (t) = 1 t Z t i (s) X i (s) We have that these estimators are strongly consistent, i.e. (2.1) lim t!1 bi = b i a:s: (2.2) Proof. Using (1.1) we get = 1 t Z t b i ds + j=1 bi (t) = 1 t Z t i (s) X i (s) ij dw j (s) = b i + j=1 ij W j (t) t (2.3)

3 Omar Zane 67 and the result follows from the strong law of large numbers for Brownian motion. Remark 2.1 The rate of convergence of the estimator is discussed in [DPD2] using the law of iterated logarithm We can now move our attention to A: we observe that there is no need to estimate the entries of A. In fact these are known adter an arbitrarily small period of time of positive length provided that we have continuous observations of the state. It is enough to evaluate a stochastic integral to get the exact values Proposition 2.2 For every i,j = 1,..,d and t > Z A ij = 1 t d(x i (s)x j (s)) t X i (s)x j (s) b i (t) b j (t) (2.4) Proof. Applying Ito's dierential rule we get tr( 1 1 d(x i (s)x j (s)) = X i (s) j (s) + X j (s) i (s) (2.5) Xi (s) i1 :::X i (s) id Xi (s) i1 :::X i (s) id ) ds X j (s) j1 :::X j (s) jd X j (s) j1 :::X j (s) jd = X i (s) j (s) + X j (s) i (s) + k=1 ik jk X i (s)x j (s) ds dividing both sides by X i (s)x j (s) and integrating from to t we get the result. 3 Discrete observations I In this section we dene estimators for b and A for the case in which we have observations only at discrete moments ft k := kg1 k=, >. We exploit the fact that explicit solutions for equations (1.1) are known for i = 1; ::; d, and given by ([KS]) X i (t) = x expf(b i 1 2 ij 2 )t + j=1 j=1 ij W j (t)g (3.1) We proceed by giving a discretization of the estimators in (2.1)and (2.4); let ^b i (k) := 1 kx X i (t l ) X i(t ) l 1 k X i (t ) (3.2) l 1 ^A ij (k) := 1 k f kx l=1 l=1 X i (t l )X j(t l ) X i(t l 1 )X j(t l 1 ) X i (t l 1 )X j(t l 1 ) (3.3)

4 68 Identication of Parameters with Convergence Rate X i(t l ) X i(t l 1 ) X i (t l 1 ) Remark 3.1 If we dene i (k) by X j(tl ) X j(t ) l 1 X j (t ) g l 1 i (k) := X i(t k ) X i(t k 1 ) X i (t k 1 ) (3.4) then the denitions given in (3.2) and (3.3) are equivalent to the following ^b i (k) := 1 k ^A ij(k) := 1 k kx l=1 kx l=1 i (l) (3.5) i (l) j(l) (3.6) Remark 3.2 By dention of Ito stochastic integral for! we have that ^b i (k) and ^A ij (k) converge to b i (t k ) and A ij respectively. The estimators dened in (3.5) and (3.6) converge almost surely, as k approaches innity, to quantities that depend not only on b and A but also on. Theorem 3.1 For every > and i=1,..,d lim k!1 ^b i (k) = expf b ig 1 and for every > and i,j=1,..,d a:s: (3.7) lim k!1 ^A ij (k) = expf(b i + b j ) + A ij )g expf b i g expf b j g + 1 a:s: (3.8) Proof. Let Y i (t) := log(x i (t)); it follows from (3.1) that at time t k+1 the distribution of Y i is Y i (t k+1 ) Y i (t k ) + (b i 1 2 A ii) + p m=1 im w m (k + 1) (3.9) where fw(k + 1) := (w 1 (k + 1); ::; w d (k + 1))g 1 k= is a sequence of independent Gaussian random vectors and the symbol indicates that the random variables on the two sides have the same distribution. >From this it follows that i (l), dened in (3.4), is such that i (l) expf(b i 1 2 A ii) + p P d m=1 imw m (k + 1)g 1 (3.1) If we x i and, we have that the sequence f (l) i g 1 l=1 is a sequence of independent, identically distributed random variables. Applying the strong

5 law of large numbers (see [Sh]) and (3.5) we get 1 lim k!1 ^b i (k) = lim k!1 k that proves (3.7). For proving (3.8) observe that + p Omar Zane 69 kx l=1 i (l) = expf b ig 1 (3.11) i (l) j(l) = expf (b i + b j 1 2 (A ii + A jj )) (3.12) m=1 ( im + jm )w m (l)g i (l) j(l) + 1 and similarly to the previous case using (2.1), (2.16) and the strong law of large numbers we have the result. In order to get results on the rate of convergence of these estimators, we use a version of the law of iterated logarithm proved by Hartman and Wintner (see [Sh], pages ) Theorem 3.2 Let f i g 1 i=1 be a sequence of independent identically distributed P random variables with Ef i g = and Efi 2g = 2 > ; let n S n = i=1 i. Then lim sup n!1 js n j p = 1 a:s: (3.13) 2 2 n log log n Using the previous theorem, we can prove the following results Proposition 3.3 For every i = 1; ::; d, and > we have that where and j^b i lim sup (k) b1 i; j k!1 k p 1 2i; b a:s: (3.14) 2 b 1 i; = expfb ig 1 (3.15) b i; = Ef( i )2 g 2 (b 1 i;) 2 (3.16) Moreover, for every i; j = 1; ::; d, > and where and j ^A ij(k) A 1 i;j; lim sup j k!1 k p 1 2i;j; A a:s: (3.17) 2 A 1 i;j; = expf((b i + b j ) + A ij )g expf b i g expf b j g + 1 (3.18) A i;j; = Ef( i j) 2 g 2 (A 1 i;j;) 2 (3.19)

6 7 Identication of Parameters with Convergence Rate Proof. j^b i P flim sup (k) b1 i; j k!1 k p 1 2i;g b 2 j^b i P flim sup (k) b1 i; j p k p p 2 b k!1 log log k i;g P j 1 k P (l) i k l=1 1 k k l=1 = P flim sup b1 i; j p k p p 2 k!1 log log i; b g k = P flim sup k!1 P k j i b 1 l=1 i; q2 )j 1g = 1 a:s: bi; k log log k by theorem 3.2. The proof of the second part of the statement is analogous. Remark 3.3 The expected values in 3.16 and 3.19 can be easily evaluated using the fact that if X N(a; 2 ) then EfexpftXgg = expfat + 2 t 2 2 g A11, A12, A t(k) Fig. 1: delta=.1

7 Omar Zane b1, b t(k) Fig. 2: delta=.1 4 Discrete observations II Let us dene strongly consistent estimators for b and A in the case in which we have discrete obsrevations. For i; j = 1; ::; d, let b b i () and d A ij () be set equal to arbitrary constants. Let Q and R denote Q(; i; k) = ^b i (k) + 1 (4.1) R(; i; j; k) = A ^ ij (k) 1 + expf ^b i (k)g + expf ^b j (k)g (4.2) and dene recursively for every i = 1; ::; d and k > bb i (k) := 8 < : logfq(; i; k)g if Q(; i; k) > (4.3) bb i (k 1) otherwise and for i; j = 1; ::; d and k > d A ij (k) := 8 < : logfr(; i; j; k)g ^b i (k) ^b j (k) if R(; i; j; k) > (4.4) da ij (k 1) otherwise

8 72 Identication of Parameters with Convergence Rate A11, A12, A t(k) Fig. 3: delta=.1 We have that Theorem 4.1 For every > and i=1,..,d and for every > and i,j=1,..,d lim k!1 b b i (k) = b i a:s: (4.5) lim k!1 d A ij (k) = A ij a:s: (4.6) i.e., the estimators dened above are strongly consistent. Proof. The strong consistency of the estimators follows straightforwardly from Theorem Simulation In this section we analyze, through a simulation, the eect that dierent choices of (length of the interval between observations) have on the discretized estimators illustrating therefore the results of theorem 3.1. We generate a sample path of the solution of (1.1) using the property mentioned

9 Omar Zane b1, b t(k) Fig. 4: delta=.1 in 3.9. We simulate the system for the case in which d = 2; the discretization step is t = :1 and the time horizon is T = 25. Using the sample paths for X 1 (t) and X 2 (t) obtained from this simulation, we compute the values of the estimators for the cases in which we have 1) 1 observations per unit of time (i.e. = :1) 2) 1 observations per unit of time (i.e. = :1) The values of the parameters that have been used are :6 b := :5 :7 :7 = 1 :5 :98 1:5 A = 1:5 1:25 The theoretical values, obtained in theorem 3.1, for the three cases are given in Table 1.

10 74 Identication of Parameters with Convergence Rate b 1 1; b 1 2; A 1 1;1; A 1 1;2; A 1 2;2; Table 1 The gures 1-4 show that the results that are obtained are exactly what we are expecting from the theory. Acknowledgment I wish to thank Professor T.E. Duncan and Professor B. Pasik-Duncan for the stimulating conversations on this topic. [BF] [DPD1] [DPD2] [K] [KS] References Bielecki, T. and Frei, M. G., Identication and Control in the Partially Known Merton Portfolio Selection Model, Journal of Optimization Theory and Applications, 77, , (1993). Duncan, T. E., and Pasik- Duncan, B., Adaptive Control of a Continuous-Time Investment and Consumption Model, Journal of Optimization Theory and Applications, 61, 47-52, (1989). Duncan, T. E., and Pasik- Duncan, B., Rate of Convergence for an Estimator in a Portfolio and Consumption Model, Journal of Optimization Theory and Applications, 61, 53-59, (1989). Karatzas, I., Optimization Problems in the Theory of Continuous Trading, SIAM, Journal of Control and Optimization, 27, 1221{ 1259, Karatzas, I. and Shreve, S. E.,Brownian Motion and Stochastic Calculus, Springer-Verlag, (1988). [Sh] Shiryayev, A. N.,Probability, Springer-Verlag, (1984) [S] Stoyanov,J. M., Problems of Estimation in Continuous-Discrete Stochastic Models, Proceedings of the Seventh Conference on Probability Theory (ed. M. Iosifescu), E.A.R.S.R.,(1984). This electronic publication and its contents are ccopyright 1994 by Ulam Quarterly. Permission is hereby granted to give away the journal and it contents, but no one may \own" it. Any and all nancial interest is hereby assigned to the acknowledged authors of the individual texts. This notication must accompany all distribution of Ulam Quarterly.