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Hint By application of Ito's lemma, we have $$d(e^{kt}v_t)=\kappa e^{\kappa t}v_t\,dt+e^{\kappa t}dv_t+d(e^{\kappa t})dv_t$$ therefore $$v_t=v_0e^{-\kappa t}+\theta(1-e^{-\kappa t})+\sigma\int_{0}^{t}\sqrt{v_s}e^{-\kappa(t-s)}dB_{s}^{v}+\int_{0}^{t}e^{-\kappa(t-s)}J^v\,dN_{s}$$ $J_v$ is random jump size occurring at time $t_i$ and $N_t=N_t-N_0$ is the ...
Hint Let $\,H_0(x,t)=1$ , $H_1(x,t)=x$ and for every $n\ge 2$ set $${{H}_{n}}(x,t)=x {{H}_{n -1}}(x,t)-(n-1)\,t\,{{H}_{n-2}}(x,t)$$ then ${{H}_{n }}(W_t ,t)$ is a Martingale. For exapmple $$H_1(W_t,t)=W_t$$ $$\qquad H_2(W_t,t)=W_t^2-t$$ $$\qquad\qquad H_3(W_t,t)=W_t^3-3tW_t$$ $$\vdots$$