## DJ / PRODUCER

Exercises on Brownian Motion⁄ Dr. Iqbal Owadallyy November 18, 2002 Elementary Problems Q1. By direct integration X(t) = x0 +„t+¾W(t) and hence X(t) is normally distributed, with mean x0 +„t and variance ¾2t. Brownian motion is a Gaussian process and is therefore uniquely characterised by its Exercise 2.6 For (B t) t 0 standard Brownian Motion, de ne the stochastic process X t:= (tB 1=t for t>0 0 for t= 0: Show that (X t) t 0 is also a Brownian Motion. This fact leads to another construction of the Brownian motion.1,2 One may start with a chain of simple random walks S(m) = (S(m) n)n (on a single probability space) such that each S(m) is embedded into S(m+1). Almostsurely,limsup t!1 B t= +1andliminf t!1 B t= 1 . Let (B t) t 0 be a standard Brownian motion. 2 Brownian Motion (with drift) Deﬂnition. Introduction and Techniques Exercises in Financial Mathematics List 2 UiO-STK4510 Solutions and Hints Autumn 2015 Teacher: S. Ortiz-Latorre Brownian Motion and Stochastic Calculus Recall –rst some de–nitions given in class. W 0 = 0;P-a.s., 3. Exercises on Brownian Motion⁄ Dr. Iqbal Owadallyy November 18, 2002 Elementary Problems Q1. The goal … Brownian motion. Show that P 0(˝ 2 … W has continuous paths P-a.s., 2. is��Tb�|Wh|���J��|��Dhô��~J��/̔���Q4���� �����֪n��]�^����E&��o�����l���M|�o�.>&/I��x�Hf,{��ku�18�~���#@��P�Hd�8c,}P! (ii) A process fYt;t ‚ 0g satisﬁes Yt = 1+0:1t+0:3Bt. Its density function is De–nition 1 (Def. Forany x2R, wedeﬁnethe stoppingtime˝ x= infft 0: X t= xg. ��ݮM Almostsurely,onanyinterval,t7!B t isnotmonotonic. >> ]��?��OTC�j��i��i��6���ʚv'��:����2 �_^��c�!-���ô�m�jSe8��m1�uHT�"���cm��*8�b� Gaussian) A standard Brownian motion is a process satisfying a) W has continuous paths P-a.s., We call µ the drift. 6D@��+ߟ�Cqk?���g���QO�����-�pP���>� hŭ���BS�o�΍_���N a�����E�Pu�/xԢC�7�b���a�*l��][}��Z�}�p{�Rb�@7�\$ub�HL�j Q"�� Calculate P[Y10 > 1 j Y0 = 1]. Show that P 0(˝ 1 = 0) = 1. X = (B t+ t: t 0) isthenaBrownianmotionwithdrift . ����`�j Let gbe the continuous function de ned on @B(0;1) by g(x) = ( 0; if jxj= 1 1; if x= 0: Prove that the Dirichlet problem in B(0;1) with boundary condition ghas no solution. Lett 0 2R+. (a) Prove that any perfect set in R (a closed subset of R without isolated points) has the cardinality of the continuum. Exercise 3 (Another Brownian motion) [5 Pts.Let (B t) t 0 be a standard Brownian motion and let Zbe an independent random variable such that P(Z= 1) = P(Z= 1) = 1 2; and let t 2[0;1):De ne (W t) t 0 by W t:= B t1 ft 1 j Y0 = 1]. Standard Brownian motion (deﬁned above) is a martingale. /Length 2831 Thus, almost surely, the zero set of one-dimensional Brownian motion has the cardinality of the continuum. Proposition8. Let B be a Brownian motion on a probability space (W,F,P)such that B 0 = 0. Exercise 4 (Brownian bridge) [5 Pts.A stochastic process (X (ii) Let ˝ 2:= infft>0 : B t = 0g. Let B be a standard Brownian motion and ﬁx a constant >0. 2. ��a�~ �w�t�i�Ӈ����wA�>��S>�]��V�OoٖI?�ټ;�rӖ�"�n���I��T=��>;;sO�G+b�=���S�2�c���8�̑�+�q?s�� ���(+��f��w��{��] %ƒ=\$�&������{W��wy`�Vu�f�[ ��\ٓN�8��z�(2:Ļ+>�MF��^� ���}���] �;1��k,8гQ@��fO�B 2. ���ľ���m���.���!I�0"��7mu�P�U޴�C��c��ؔ���Pĵ��m�y.1秚n�� @m(&�9�s.��|,�E�CQ?��?�����ɏ'\$�3ҫ�j`Zd����w8��w KDAo��9�a ������N^X[`*# Brownian Motion and Partial Di erential Equations. (b) Solve exercise 2.6 from the Brownian motion book. X is a martingale if µ = 0. � (A)’ W 0 = 0. Exercise: Verify this. �B#�h@�x!ԏ���xmK5�ll��� (�P��4�'4*e���kŧ�3^˖�V��Z�^����Q3Q6�����i�b0�h�r�&�X���l�F�\���T�?a�]���Ff�~�.���c�3p#:Hm�A4�҅�(]R(!��X�G��U�� �����I�@o��F�z}(֩ R���ؑ!E��U��-����� _����G���