{"prompt": "Approximate the expectation of the solution of SDE $$dX_t = \\Bigl(-\\frac{\\sin(t^{-1/2})}{t^2}-\\frac{1+\\tfrac1t}{2t^{3/2}}\\cos(t^{-1/2})-\\frac{B_t}{2t^{3/2}}+\\frac1{t+1}+B_t^2\\bigl(6t^{-3}-\\frac1{(t+1)^2}\\bigr)-3\\frac{B_t^4}{t^4}\\Bigr)\\,dt + \\Bigl(\\frac1{\\sqrt t}+\\frac{2B_t}{t+1}+\\frac{4B_t^3}{t^3}\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{\\frac{1}{1 + \\sqrt{t}}}$", "parameters": "$t$", "type": "other", "index": 0}
{"prompt": "Approximate the expectation function of the solution of SDE $dX_t = \\Bigl[\\bigl(\\frac1t-\\frac1{t^2}+\\frac2{t^3}\\bigr)+\\ln(t)\\bigl(\\frac1{t^2}-\\frac4{t^3}\\bigr)-\\tfrac12\\ln(t)\\bigl(1-\\frac1t+\\frac2{t^2}\\bigr)\\Bigr]\\cos(B_t)\\,dt - \\ln(t)\\bigl(1-\\frac1t+\\frac2{t^2}\\bigr)\\sin(B_t)\\,dB_t$ in the limit $ t > 0$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{e^{-\\frac{t}{2}} \\ln(t) (1 - \\frac{1}{t} + \\frac{2}{t^2})}$", "parameters": "$t$", "type": "other", "index": 1}
{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t=\\exp\\bigl(\\sqrt{t}\\sin(2t)\\,B_t\\bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}+\\frac{B_t}{1+t^2}\\Bigl(\\frac{\\sin(2t)}{2\\sqrt{t}}+2\\sqrt{t}\\cos(2t)\\Bigr)+\\frac{(\\sqrt{t}\\sin(2t))^2}{2(1+t^2)}\\Bigr)\\,dt+\\frac{\\sqrt{t}\\sin(2t)}{1+t^2}\\exp\\bigl(\\sqrt{t}\\sin(2t)\\,B_t\\bigr)\\,dB_t$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.", "solution": "$\\boxed{u(x,t) = \\frac{1}{1 + t^2} e^{t^2 \\sin^2(2t)}}$", "parameters": "$t$", "type": "other", "index": 2}
{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t=\\exp\\bigl(\\cos(t)B_t\\bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}-\\frac{B_t\\sin(t)}{1+t^2}+\\frac{\\cos^2(t)}{2(1+t^2)}\\Bigr)\\,dt+\\frac{\\cos(t)}{1+t^2}\\exp\\bigl(\\cos(t)B_t\\bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^.", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\frac{t}{2}\\cos^2(t)}}$", "parameters": "$t$", "type": "other", "index": 3}
{"prompt": "Approximate the expectation function of the solution of SDE $$dX_t =\\exp\\!\\Bigl(\\tfrac{\\sqrt2\\,\\sin(t)}{t^{1/4}}\\,B_t\\Bigr)\\Bigl(-\\frac{2t}{(1+t^2)^2}+\\frac{B_t}{1+t^2}\\Bigl(\\frac{\\sqrt2\\,\\cos(t)}{t^{1/4}}-\\frac{\\sqrt2\\,\\sin(t)}{4\\,t^{5/4}}\\Bigr)+\\frac{\\sin^2(t)}{t^{1/2}(1+t^2)}\\Bigr)\\,dt+\\frac{\\sqrt2\\,\\sin(t)}{t^{1/4}(1+t^2)}\\exp\\!\\Bigl(\\tfrac{\\sqrt2\\,\\sin(t)}{t^{1/4}}\\,B_t\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^.", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\sqrt{t} \\sin^2(t)}}$", "parameters": "$t$", "type": "other", "index": 4}
{"prompt": "Approximate the expectation function of the solution of SDE $$ dX_t=\\exp\\Bigl(\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)B_t\\Bigr)\\Bigl(-\\tfrac{2B_t^2}{t^3(1+t^2)}-\\tfrac{2t(1+\\tfrac{B_t^2}{t^2})}{(1+t^2)^2}+\\tfrac{(1+\\tfrac{B_t^2}{t^2})B_t}{1+t^2}(\\tfrac{\\sin\\tfrac{t}{3}}{\\sqrt3}+\\tfrac{t\\cos\\tfrac{t}{3}}{3\\sqrt3})+\\tfrac12\\tfrac{1+\\tfrac{B_t^2}{t^2}}{1+t^2}\\tfrac{t^2}{3}\\sin^2\\bigl(\\tfrac{t}{3}\\bigr)\\Bigr)\\,dt+\\tfrac{1+\\tfrac{B_t^2}{t^2}}{1+t^2}\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)\\exp\\Bigl(\\tfrac{t}{\\sqrt3}\\sin\\bigl(\\tfrac{t}{3}\\bigr)B_t\\Bigr)\\,dB_t.$$ in the limit $t \\to \\infty$ Please provide your answer in LaTeX \\boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \\pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t. Do not use \\exp, but just explicitly write e^. Do not use X_0. Do not use enlarge any parenthesis. Keep paranthesis () without modifying anything", "solution": "$\\boxed{\\frac{1}{1 + t^2} e^{\\frac{t^3}{6} \\sin^2(t/3)}}$", "parameters": "$t$", "type": "other", "index": 5}